\label{t.main}
	Let $n,s,\ell,c$ be positive integers such that $n=2s+c=3s-\ell$, and $c,\ell\in [s-1]$. Then $$e(n, s) = \max\big\{|\mathcal{P}(s, \ell)|, |\mathcal{P}'(\ell)|, |\mathcal{Q}(s, \ell)|, |\mathcal{W}(s, \ell)|\big\}.$$ Moreover, if $s \geq 3$, ${\mathcal F}\subset 2^{[n]}$ is shifted, has no $s$-matching and $|{\mathcal F}|=e(n,s)$, then ${\mathcal F}$ must coincide with one of the families above.

\begin{thm}[\cite{FK9,FK8}]\label{thmfk} \ $e(sm+s-\ell,s) = |\pp(s,m,\ell)|$ holds for
\begin{align*} &\mathrm{(i)}\ \ \ \ \ell = 2, \\
    &\mathrm{(ii)}\ \ \  m=1,\\
     &\mathrm{(iii)}  \ \  s\ge \ell m+3\ell+3.
\end{align*}
\end{thm}

\begin{gypo}[\cite{FK9}]\label{conj1}
Suppose that $s\ge 2, m\ge 1$, and $n = sm+s-\ell$ for some integer $0<\ell\le \lceil \frac s2\rceil$. Then
\begin{equation}\label{eq007} e(sm+s-\ell,s) = |\pp(s,m,\ell)|.
\end{equation}
\end{gypo}

Let $[n] := \{1,2,\ldots, n\}$ and, more generally, $[a,b]=\{a,a+1,\ldots, b\}$. For a set $X$ and an integer $k$, let  $2^{X}$, ${X\choose k}$ and ${X\choose \geq k}$ stand for the power set of $X$, the set of its $k$-element subsets and the set of its subsets with size at least $k$, respectively. Any collection of sets is called a {\it family.} A {\it matching} is a collection of pairwise disjoint sets. An {\it $s$-matching} is a matching of size $s$. Given a family ${\mathcal F},$ its {\it matching number}
$\nu(\mathcal F)$ is the size of the largest matching in ${\mathcal F}$.

One of the classical topics in extremal set theory is the study of {\it intersecting} families, that is, families with matching number $1$. Erd\H os, Ko and Rado~\cite{EKR} showed that the largest intersecting family ${\mathcal F}\subset 2^{[n]}$ has size at most $2^{n-1}$, and that for $n\ge 2k$ the largest intersecting family ${\mathcal F}\subset {[n]\choose k}$ has size ${n-1\choose k-1}.$ In the several years that followed, Erd\H os asked for the size of the largest family avoiding an $s$-matching. Let us introduce the following two quantities.
\begin{align*}
    e(n,s)&=\max\big\{|{\mathcal F}|: {\mathcal F}\subset 2^{[n]}, \nu({\mathcal F})<s\big\},\\
    e_k(n,s)&=\max\Big\{|{\mathcal F}|: {\mathcal F}\subset {[n]\choose k}, \nu({\mathcal F})<s\Big\}.
\end{align*}
\subsection{The uniform case}
The value of $e_k(n,s)$ is the subject of the famous Erd\H os Matching Conjecture. Let us define the families $\aaa_i^{(k)}(n,s):$
\begin{equation}\label{eq0011} \aaa_i^{(k)}(n,s-1) := \Bigl\{F\in {[n]\choose k}:|F\cap [si-1]|\ge i\Bigr\}, \ \ \ 1\le i\le k.\end{equation}
It is not difficult to see that $\nu(\aaa_i^{(k)}(n,s-1)\le s-1$ for any $n,s$.
\begin{gypo}[Erd\H os Matching Conjecture \cite{E}]\label{conj2} For $n\ge sk$
\begin{equation}\label{eq008} e_k(n,s) = \max\bigl\{|\aaa_1^{(k)}(n,s-1)|,|\aaa_k^{(k)}(n,s-1)|\bigr\}.
\end{equation}
\end{gypo}
Let us comment on the statement of the conjecture. First, the condition $n\ge sk$ is required in order for ${[n]\choose k}$ to contain an $s$-matching. Second, note that the conjecture suggests that only two out of $k+1$ families $\aaa_i$ could be extremal. This is different from some other similar-looking questions, notably, the Complete $t$-Intersection Theorem for ${[n]\choose k}$ \cite{AK}. There, all families could be extremal depending on the parameters $n,k,t.$

The conjecture (\ref{eq008}) is known to be true for $k= 2$ \cite{EG}, and $k=3$ \cite{LM,F11}. In fact, an elegant proof of Frankl of the $k=2$ case of the conjecture, initially proven by Erd\H os and Gallai, will play an important role in this paper. We will recite it later.

The conjecture was proved for $n\ge n_0(s,k)$ by Erd\H os \cite{E}. Many researchers worked on extending the range in which the conjecture is valid:  \cite{E}, \cite{BDE}, \cite{HLS}, \cite{FLM},  \cite{F4}, \cite{FK21}. Notably, in the last two papers, it is proved that
\begin{equation}\label{eq009} e_k(n,s) = |\aaa_1|={n\choose k}-{n-s+1\choose k} \end{equation}
for $n\ge (2s+1)k-s$ and for ($n\ge \frac 53sk$, $s\ge s_0$), respectively. In \cite{F7} and \cite{KoKu}, the authors studied the problem from the other end and showed that $\aaa_k$ is extremal for ($n\le s(k+1/100k)$, $s\ge ck^3$ with some absolute $c>0$). We refer the reader to \cite{aletal, FK21} for the connections of the Erd\H os Matching Conjecture and other questions, such as Dirac thresholds and small deviations in probability theory. In \cite{HLS}, \cite{K49}, the multi-family variant of the EMC was addressed.  In \cite{FK6}, a Hilton--Milner type stability result for the EMC is obtained.

\subsection{The non-uniform case}
The study of $e(n,s)$ was also initiated by Erd\H os at around the same time. The behavior of $e(n,s)$ heavily depends on $n\ ({\rm mod\ } s)$.  Answering a question of Erd\H os, Kleitman proved the following theorem.
\begin{thm}[Kleitman \cite{Kl}]\label{thmkl}
\begin{align}\label{eq001} e(sm-1,s) &= \sum_{t=m}^{sm-1}{sm-1\choose t},\\
\label{eq002} e(sm,s) &= {sm-1\choose m}+\sum_{t=m+1}^{sm}{sm\choose t}.
\end{align}
\end{thm}
The matching example for the first case is the family ${[n]\choose \ge m}$ of all subsets of $[n]$ of size at least $m$. It is also  not difficult to see that $e(sm,s) = 2e(sm-1,s)$. In general, $e(n+1,s)\ge 2e(n,s)$ because of the {\it doubling} construction. Given a family ${\mathcal F}\subset 2^{[n]}$ with $\nu({\mathcal F})<s$, we may construct the {\it doubling} ${\mathcal F}'$ of ${\mathcal F}$ as follows: ${\mathcal F}' = \{F\subset [n+1]: F\cap [n]\in {\mathcal F}\}$. It is easy to see that $\nu({\mathcal F}')<s$ as well.

In \cite{FK9}, Frankl and the first author put forth the following conjecture.
\begin{opr}\label{def3} Let $n = sm+s-\ell$, $0<\ell\le s$. Set
\[{\mathcal P}(s,m,\ell) := \bigl\{P\subset 2^{[n]}: |P|+|P\cap [\ell-1]|\ge m+1\bigr\}.\]
\end{opr}
It is not difficult to check (cf. \cite{FK9}) that $\nu( \mathcal P(s,m,\ell))<s.$
\begin{gypo}[\cite{FK9}]\label{conj1}
Suppose that $s\ge 2, m\ge 1$, and $n = sm+s-\ell$ for some integer $0<\ell\le \lceil \frac s2\rceil$. Then
\begin{equation}\label{eq007} e(sm+s-\ell,s) = |{\mathcal P}(s,m,\ell)|.
\end{equation}
\end{gypo}
They confirmed the conjecture for a variety of cases:
\begin{thm}[\cite{FK9,FK8}]\label{thmfk} \ $e(sm+s-\ell,s) = |{\mathcal P}(s,m,\ell)|$ holds for
\begin{align*} &\mathrm{(i)}\ \ \ \ \ell = 2, \\
    &\mathrm{(ii)}\ \ \  m=1,\\
     &\mathrm{(iii)}  \ \  s\ge \ell m+3\ell+3.
\end{align*}
\end{thm}
The case of $\ell=2,s=3$ was solved much earlier by Quinn \cite{Q} in his PhD dissertation. We also note that a sum-type inequality for $s$ families avoiding a matching is much easier to obtain, see \cite{FK6}.

Let us return to the definition of $\mathcal P(s,m,\ell)$. The two main uniformities to understand the construction are $m$ and $m+1$. One sees that the family contains all sets of size $m+1$, as well as all sets of size $m$ that intersect $[\ell-1]$, i.e., the family $\aaa_1^{(m)}(n,\ell-1)$. Thus, in fact, the sets of uniformity $m$ form a family with no matching of size $\ell$. We should note here that the sets in $\mathcal P(s,m,\ell)\cap {[n]\choose \le m}$, a bulk of which is  $\aaa_1^{(m)}(n,\ell-1)$, could be replaced by any other family in ${[n]\choose m}$ with no matching of size $\ell$. As a result, we would obtain a construction of a family in $2^{[n]}$ with no $s$-matching.

Here, one sees that the problems of determining $e(n,s)$ and $e_k(n,s)$ are, in fact, closely connected. In particular, let us give  the reason for the constraint $\ell\le s/2$ in Conjecture~\ref{conj1}: in this regime, $n\ge 2\ell m,$ and the result of Frankl \cite{F4} guarantees that the largest family in ${[n]\choose m}$ with no matching of size $\ell$ is indeed $\aaa_1^{(m)}(n,\ell-1)$.

Incidentally, one could also interpret the extremal family for $n=sm = sm+s-s$ in the same way: the extremal example consists of all sets of size at least $m+1$, as well as of the family of sets of size $m$ with no $s$-matching. The construction for the sets of uniformity $m$ is, however, different: it is $\aaa_k^{(m)}(n,s-1)$. Actually, the situation is even more complicated, as is shown in \cite{FK9}: already for $n = sm+1$ there is a construction in which some  $(m+1)$-element sets are missing and that is bigger than constructions with all $(m+1)$-element sets present. Namely, it is the doubling construction.

Concluding the discussion, the authors of \cite{FK9} state that it is hard to formulate a general conjecture, and that they had already struggled with the case $n=2s+t$ for some small values of $t$.
\subsection{Main result}
In this paper, we find the value of $e(n,s)$ for all $n,s$, provided $n\le 3s.$ The easy case $n<2s$ is solved in Theorem~\ref{thmfk}. The cases $n=2s$ and $n=3s$ are answered in \eqref{eq002}. Thus, we are left to deal with the cases $2s<n<3s$. For the rest of the paper, unless otherwise stated, we use notation $n=2s+c=3s-\ell$, where $c, \ell\in[s-1]$, $c+\ell=s$.

In order to state the result, we describe the prospective extremal families. We have already seen the first example.
$$\mathcal{P}(s, \ell) := \mathcal P(s,2,\ell)=\big\{P \in 2^{[n]} : |P| + |P \cap [\ell - 1]| \geq 3\big\}.$$
The second family also contains all sets of size at least $3$, but differs on the $2$-uniform layer. It has the other extremal example for $2$-uniform families avoiding an $\ell$-matching: a clique on $2\ell-1$ vertices.
$$\mathcal{P}'(s, \ell) := {[n] \choose \geq 3} \cup {[2\ell-1] \choose 2}.$$
The third family misses some $3$-element sets, and instead has a larger $2$-uniform layer. It is also a clique. $$\mathcal{Q}(s, \ell) := {[n] \choose \geq 3} \cup {[s+\ell-1] \choose 2} \setminus {[s+\ell, n] \choose 3}.$$
Finally, the fourth family is an iterated doubling of the extremal family for $n=2s-1$.
$$\mathcal{W}(s, \ell) := \big\{P \in 2^{[n]} : |P \cap [2s - 1]| \geq 2\big\}.$$ We note that none of these families contain $0$- or $1$-element sets. In the next section, we show that all these families indeed avoid $s$-matchings.

The main result of the paper is the following theorem.

One of the classical topics in extremal set theory is the study of {\it intersecting} families, that is, families with matching number $1$. Erd\H os, Ko and Rado~\cite{EKR} showed that the largest intersecting family ${\mathcal F}\subset 2^{[n]}$ has size at most $2^{n-1}$, and that for $n\ge 2k$ the largest intersecting family ${\mathcal F}\subset {[n]\choose k}$ has size ${n-1\choose k-1}.$ In the several years that followed, Erd\H os asked for the size of the largest family avoiding an $s$-matching. Let us introduce the following two quantities.
\begin{align*}
 e(n,s)&=\max\big\{|{\mathcal F}|: {\mathcal F}\subset 2^{[n]}, \nu({\mathcal F})<s\big\},\\
 e_k(n,s)&=\max\Big\{|{\mathcal F}|: {\mathcal F}\subset {[n]\choose k}, \nu({\mathcal F})<s\Big\}.
\end{align*}
\subsection{The uniform case}
The value of $e_k(n,s)$ is the subject of the famous Erd\H os Matching Conjecture. Let us define the families $\aaa_i^{(k)}(n,s):$
\begin{equation}\label{eq0011} \aaa_i^{(k)}(n,s-1) := \Bigl\{F\in {[n]\choose k}:|F\cap [si-1]|\ge i\Bigr\}, \ \ \ 1\le i\le k.\end{equation}
It is not difficult to see that $\nu(\aaa_i^{(k)}(n,s-1)\le s-1$ for any $n,s$.
\begin{gypo}[Erd\H os Matching Conjecture \cite{E}]\label{conj2} For $n\ge sk$
\begin{equation}\label{eq008} e_k(n,s) = \max\bigl\{|\aaa_1^{(k)}(n,s-1)|,|\aaa_k^{(k)}(n,s-1)|\bigr\}.
\end{equation}
\end{gypo}
Let us comment on the statement of the conjecture. First, the condition $n\ge sk$ is required in order for ${[n]\choose k}$ to contain an $s$-matching. Second, note that the conjecture suggests that only two out of $k+1$ families $\aaa_i$ could be extremal. This is different from some other similar-looking questions, notably, the Complete $t$-Intersection Theorem for ${[n]\choose k}$ \cite{AK}. There, all families could be extremal depending on the parameters $n,k,t.$

\subsection{The non-uniform case}
The study of $e(n,s)$ was also initiated by Erd\H os at around the same time. The behavior of $e(n,s)$ heavily depends on $n\ ({\rm mod\ } s)$. Answering a question of Erd\H os, Kleitman proved the following theorem.
\begin{thm}[Kleitman \cite{Kl}]\label{thmkl}
\begin{align}\label{eq001} e(sm-1,s) &= \sum_{t=m}^{sm-1}{sm-1\choose t},\\
\label{eq002} e(sm,s) &= {sm-1\choose m}+\sum_{t=m+1}^{sm}{sm\choose t}.
\end{align}
\end{thm}
The matching example for the first case is the family ${[n]\choose \ge m}$ of all subsets of $[n]$ of size at least $m$. It is also not difficult to see that $e(sm,s) = 2e(sm-1,s)$. In general, $e(n+1,s)\ge 2e(n,s)$ because of the {\it doubling} construction. Given a family ${\mathcal F}\subset 2^{[n]}$ with $\nu({\mathcal F})<s$, we may construct the {\it doubling} ${\mathcal F}'$ of ${\mathcal F}$ as follows: ${\mathcal F}' = \{F\subset [n+1]: F\cap [n]\in {\mathcal F}\}$. It is easy to see that $\nu({\mathcal F}')<s$ as well.

In \cite{FK9}, Frankl and the first author put forth the following conjecture.
\begin{opr}\label{def3} Let $n = sm+s-\ell$, $0<\ell\le s$. Set
\[{\mathcal P}(s,m,\ell) := \bigl\{P\subset 2^{[n]}: |P|+|P\cap [\ell-1]|\ge m+1\bigr\}.\]
\end{opr}
It is not difficult to check (cf. \cite{FK9}) that $\nu( \mathcal P(s,m,\ell))<s.$
\begin{gypo}[\cite{FK9}]\label{conj1}
Suppose that $s\ge 2, m\ge 1$, and $n = sm+s-\ell$ for some integer $0<\ell\le \lceil \frac s2\rceil$. Then
\begin{equation}\label{eq007} e(sm+s-\ell,s) = |{\mathcal P}(s,m,\ell)|.
\end{equation}
\end{gypo}
They confirmed the conjecture for a variety of cases:
\begin{thm}[\cite{FK9,FK8}]\label{thmfk} \ $e(sm+s-\ell,s) = |{\mathcal P}(s,m,\ell)|$ holds for
\begin{align*} &\mathrm{(i)}\ \ \ \ \ell = 2, \\
 &\mathrm{(ii)}\ \ \ m=1,\\
 &\mathrm{(iii)} \ \ s\ge \ell m+3\ell+3.
\end{align*}
\end{thm}
The case of $\ell=2,s=3$ was solved much earlier by Quinn \cite{Q} in his PhD dissertation. We also note that a sum-type inequality for $s$ families avoiding a matching is much easier to obtain, see \cite{FK6}.

Let us return to the definition of $\mathcal P(s,m,\ell)$. The two main uniformities to understand the construction are $m$ and $m+1$. One sees that the family contains all sets of size $m+1$, as well as all sets of size $m$ that intersect $[\ell-1]$, i.e., the family $\aaa_1^{(m)}(n,\ell-1)$. Thus, in fact, the sets of uniformity $m$ form a family with no matching of size $\ell$. We should note here that the sets in $\mathcal P(s,m,\ell)\cap {[n]\choose \le m}$, a bulk of which is $\aaa_1^{(m)}(n,\ell-1)$, could be replaced by any other family in ${[n]\choose m}$ with no matching of size $\ell$. As a result, we would obtain a construction of a family in $2^{[n]}$ with no $s$-matching.

In order to state the result, we describe the prospective extremal families. We have already seen the first example.
$$\mathcal{P}(s, \ell) := \mathcal P(s,2,\ell)=\big\{P \in 2^{[n]} : |P| + |P \cap [\ell - 1]| \geq 3\big\}.$$
The second family also contains all sets of size at least $3$, but differs on the $2$-uniform layer. It has the other extremal example for $2$-uniform families avoiding an $\ell$-matching: a clique on $2\ell-1$ vertices.
$$\mathcal{P}'(s, \ell) := {[n] \choose \geq 3} \cup {[2\ell-1] \choose 2}.$$
The third family misses some $3$-element sets, and instead has a larger $2$-uniform layer. It is also a clique. $$\mathcal{Q}(s, \ell) := {[n] \choose \geq 3} \cup {[s+\ell-1] \choose 2} \setminus {[s+\ell, n] \choose 3}.$$
Finally, the fourth family is an iterated doubling of the extremal family for $n=2s-1$.
$$\mathcal{W}(s, \ell) := \big\{P \in 2^{[n]} : |P \cap [2s - 1]| \geq 2\big\}.$$ We note that none of these families contain $0$- or $1$-element sets. In the next section, we show that all these families indeed avoid $s$-matchings.

The main result of the paper is the following theorem.

The main result of the paper is the following theorem.

\begin{figure}

\begin{theorem} \label{t.truncated_lattice}
 Let $n,s,\ell,c$ be positive integers such that $n=2s+c=3s-\ell$, and $c,\ell\in [s-1]$. If $\mathcal{F} \subset {[n] \choose \leq 3}$ has no $s$-matching, then
 $$|\mathcal{F}| \leq \max\big\{|\mathcal{P}(s, \ell)^{(\leq 3)}|, |\mathcal{P}'(s, \ell)^{(\leq 3)}|, |\mathcal{Q}(s, \ell)^{(\leq 3)}|, |\mathcal{W}(s, \ell)^{(\leq 3)}|\big\}.$$
\end{theorem}

\begin{theorem} \label{t.Erdos_Gallai}
 Let $n,s$ be positive integers, $n\geq 2s$. Suppose that $\mathcal{F} \subset 2^{[n]}$ satisfies $\nu(\mathcal{F}^{(2)}) < s$. Then $y(2) \geq \min({n - s + 1 \choose 2}, {n \choose 2} - {2s-1 \choose 2})$.
\end{theorem}
\begin{proof}
 As we have mentioned, we may assume that $\ff$ is shifted, cf.~\cite{F3}.
 Consider the sets $\{i, 2s+1-i\}$, $i\in [s]$. These sets form an $s$-matching, and thus at least one of them is not in $\mathcal{F}$. By Lemma \ref{l.EG}, we get $$y(2) \geq \frac{(n+2s+1-3i)(n-2s+i)}{2}$$ for some $i \in [1, s]$. The right-hand side is upward convex as a function of $i$. Thus, its minimum in the interval $[1, s]$ is attained at one of its ends. We get the bound
 $$y(2) \geq \min\Big\{{n - s + 1 \choose 2}, \frac{(n+2s-2)(n-2s+1)}{2}\Big\}.$$ This is the claimed bound since ${n \choose 2} - {2s-1 \choose 2} = \frac{(n+2s-2)(n-2s+1)}{2}$.
\end{proof}

\begin{thm}\label{t.relaxed} Let $n,s,\ell$ be non-negative integers such that $n = 3s-\ell$ and $0\le \ell\le s$. Suppose that $\mathcal{F} \subset 2^{[n]}$ is a shifted up-set with $\nu(\mathcal{F}) < s$ and $\mathcal{F} \cap {[n] \choose 1} = \emptyset$. Then $|\mathcal{F}| \leq \max\big\{|\mathcal{P}(s, \ell)|, |\mathcal{P}'(s, \ell)|, |\mathcal{Q}(s, \ell)|, |\mathcal{W}(s, \ell)|\big\}$. Moreover, equality is achieved only if $\mathcal{F}$ coincides with one of these families.
\end{thm}
\begin{proof}[Proof of Theorem~\ref{t.main} using Theorem~\ref{t.relaxed}]
We prove Theorem~\ref{t.main} by induction on $\ell$. For $\ell \in \{0, 1\}$ the statement is given by Theorem~\ref{thmkl}, together with uniqueness that is proven in \cite{Kl}. In what follows, we assume that $\ell\ge 2.$

If $\{1\} \notin \mathcal{F}$ then by shiftedness $\mathcal{F} \cap {[n] \choose 1} = \emptyset$ and we are done.
If $\{1\} \in \mathcal{F}$ then we apply induction. Let $\mathcal{F'} = \mathcal{F} \cap 2^{[2, n]}$. The family $\m F'$ is a family of subsets of an $(n-1)$-element set, $\nu(\mathcal{F'}) < s-1$, and we have $n-1 =3(s-1)-(\ell-2).$ Note that $\ell-2\ge 0$ and $\ell-2<s-1$, and thus we may apply induction to $\m F'$, getting that
$$|\overline{\mathcal{F'}}| \geq \min\big\{|\overline{\mathcal{P'}(s-1, \ell-2)}|, |\overline{\mathcal{P}(s-1, \ell-2)}|, |\overline{\mathcal{Q}(s-1, \ell-2)}|, |\overline{\mathcal{W}(s-1, \ell-2)}|\big\},$$
where the complement is taken with respect to $2^{[2,n]}.$ Clearly, $|\overline{\mathcal{F}}|\ge |\overline{\mathcal{F'}}|$, and so Theorem~\ref{t.main} is implied by the following claim.
\begin{restatable}{cla}{claimeleven} We have
 \begin{multline*}
 \min\big\{|\overline{\mathcal{P'}(s-1, \ell-2)}|, |\overline{\mathcal{P}(s-1, \ell-2)}|, |\overline{\mathcal{Q}(s-1, \ell-2)}|, \\|\overline{\mathcal{W}(s-1, \ell-2)}|\big\} >\min\big\{|\overline{\mathcal{P'}(s, \ell)}|, |\overline{\mathcal{P}(s, \ell)}|, |\overline{\mathcal{Q}(s, \ell)}|, |\overline{\mathcal{W}(s, \ell)}|\big\}.
 \end{multline*}
 Moreover,
 \begin{multline*}
 \min\big\{|\overline{\mathcal{P'}(s-1, \ell-2)}^{(\leq 3)}|, |\overline{\mathcal{P}(s-1, \ell-2)}^{(\leq 3)}|, |\overline{\mathcal{Q}(s-1, \ell-2)}^{(\leq 3)}|, \\|\overline{\mathcal{W}(s-1, \ell-2)}^{(\leq 3)}|\big\} >\min\big\{|\overline{\mathcal{P'}(s, \ell)}^{(\leq 3)}|, |\overline{\mathcal{P}(s, \ell)}^{(\leq 3)}|, |\overline{\mathcal{Q}(s, \ell)}^{(\leq 3)}|, |\overline{\mathcal{W}(s, \ell)}^{(\leq 3)}|\big\}.
 \end{multline*}
\label{c.no_siggletons}
\end{restatable}

The case of odd $d$ requires a more careful analysis. In this case, we use the inequality \eqref{eqd2c} which states that $d(\mathcal{F}) \leq 2c$.
\begin{lemma} \label{l.odd_d_to_y2}
 Let $d$ be a positive odd integer, $d \leq 2c$. If $\mathcal{F} \subset 2^{[n]}$ is a shifted family with $d(\mathcal{F}) = d$, then {\small \begin{equation}\label{eqy22}y(2) \geq \min\Big\{\frac{(4\ell+3c+d-2)(3c-d+1)}{2}, \frac{(\ell+3c-\frac{d-1}{2})(\ell+3c-\frac{d+1}2)}{2}\Big\}.\end{equation}} Moreover, equality is achieved only if $\mathcal{F}^{(2)} = {[2\ell+d-1] \choose 2}$ or $\mathcal{F}^{(2)} = \{F \in {n \choose 2 }: F\cap[\ell+\frac{d-1}{2}] \neq \emptyset\}$.
\end{lemma}

\subsection{$c \in \{3, 4\}$}
\begin{lemma} \label{l.c_eq_3}
 Let $n = 2s+3$ and $\mathcal{F} \subset 2^{[n]}$ is a shifted up-set with $\nu(\mathcal{F}) < s$ and $\mathcal{F} \cap {[n] \choose 1} = \emptyset$. Then $|\mathcal{F}| \leq \max(|\mathcal{P}(s, \ell)|, |\mathcal{P}'(s, \ell)|, |\mathcal{Q}(s, \ell)|)$. Moreover, equality is achieved only if $\mathcal{F}$ is one of the families $\mathcal{P}(s, \ell), \mathcal{P}'(s, \ell), \mathcal{Q}(s, \ell)$.
\end{lemma}

\begin{figure}

\centering

\includegraphics[width=0.8\linewidth]{extremal_families.png}

\caption{Extremal families. $\mathcal{W}$ is one of the extremal families for $c = 1, s \geq 5$.}

\label{fig1}

\end{figure}

\begin{figure}

\centering

\includegraphics[width=0.6\linewidth]{extremal_families_2.png}

\caption{Extremal families for small $c,s$. Note that for $c=1, s=5$ there are $3$ different extremal families. For $\ell = 1$, that is, $s = c + 1$, the family $\mathcal{P}'$ coincides with $\mathcal{P}$ (both are equal to ${n \choose \geq (m + 1)}$) and thus is formally extremal. When $\mathcal{P}' \neq \mathcal{P}$, for $c \leq 9$ $\mathcal{P}'$ cannot be extremal.}

\label{fig2}

\end{figure}
On Figures~\ref{fig1} and~\ref{fig2} we show, which families are extremal for different regimes of the parameters $s,c$. For some values we get that three different families are extremal at the same time.

We define shifted families in the next section. We should note that actually there are rather natural examples of families with no $s$-matching interpolating between $\mathcal{P}'(s, l)$ and $\mathcal{Q}(s, l)$ in a somewhat similar way as $\aaa_i$ interpolate between $\aaa_0$ and $\aaa_k$, but, as in the case of the EMC, there is a certain convexity that leads to the fact that it is the endpoints that must be extremal.

In the proof we will work only with sets of size $3$ or less. Therefore, any family, avoiding $s$-matching, must miss at least as many sets of size $3$ or less, as the extremal family. We thus get the following theorem about the truncated boolean lattice, confirming a conjecture of Frankl and the first author \cite{FK9} in our regime of the parameters.

\begin{theorem} \label{t.truncated_lattice}
    Let $n,s,\ell,c$ be positive integers such that $n=2s+c=3s-\ell$, and $c,\ell\in [s-1]$. If $\mathcal{F} \subset {[n] \choose \leq 3}$ has no $s$-matching, then
    $$|\mathcal{F}| \leq \max\big\{|\mathcal{P}(s, \ell)^{(\leq 3)}|, |\mathcal{P}'(s, \ell)^{(\leq 3)}|, |\mathcal{Q}(s, \ell)^{(\leq 3)}|, |\mathcal{W}(s, \ell)^{(\leq 3)}|\big\}.$$
\end{theorem}

Note that a similar statement about $2$ first layers of boolean lattice is obviously false. Indeed, one of the families $\aaa_1^{(2)}(n,s-1), \aaa_2^{(2)}(n,s-1)$ has a larger cardinality than families $\mathcal{P}(s, \ell)^{(\leq 2)}, \mathcal{P}'(s, \ell)^{(\leq 2)}, \mathcal{Q}(s, \ell)^{(\leq 2)}, \mathcal{W}(s, \ell)^{(\leq 2)}$.

In Section~\ref{sec2}, we prove several easy facts and make the necessary preparations for the proof of the main theorem. In Section~\ref{sec3} we describe the strategy of the proof of the main theorem.

To prove Theorem~\ref{t.main}, the authors note that they will "work only with sets of size $3$ or less." Hence, "any family, avoiding $s$-matching, must miss at least as many sets of size $3$ or less, as the extremal family." This reduction leads to an auxiliary result on the truncated Boolean lattice: Theorem~\ref{t.truncated_lattice}, which bounds $|\mathcal F\subset {[n]\choose \le 3}|$ with no $s$-matching by the maximum of the truncated versions of the candidate extremal families $\mathcal P,\mathcal P',\mathcal Q,\mathcal W$. They also mention that while there are "natural examples" interpolating between $\mathcal P'$ and $\mathcal Q$, "there is a certain convexity" implying "it is the endpoints that must be extremal." Finally, they indicate the paper structure: in Section~\ref{sec2} they "prove several easy facts and make the necessary preparations" and in Section~\ref{sec3} they "describe the strategy of the proof of the main theorem."

To prove the main theorem, the authors note that they will "work only with sets of size $3$ or less." Hence, "any family, avoiding $s$-matching, must miss at least as many sets of size $3$ or less, as the extremal family." This reduction leads to an auxiliary result on the truncated Boolean lattice: Theorem~\ref{t.truncated_lattice}, which bounds $|\mathcal F\subset {[n]\choose \le 3}|$ with no $s$-matching by the maximum of the truncated versions of the candidate extremal families $\mathcal P,\mathcal P',\mathcal Q,\mathcal W$. They also mention that while there are "natural examples" interpolating between $\mathcal P'$ and $\mathcal Q$, "there is a certain convexity" implying "it is the endpoints that must be extremal." Finally, they indicate the paper structure: next they "prove several easy facts and make the necessary preparations" and later they "describe the strategy of the proof of the main theorem."

Classification or Bijection
Equality or Bound

Question:
Let \([n]=\{1,\dots,n\}\), and define
\[
e(n,s)=\max\{\,|\mathcal F|: \mathcal F\subset 2^{[n]},\ \nu(\mathcal F)<s\,\},
\]
where \(\nu(\mathcal F)\) is the largest size of a pairwise disjoint subfamily of \(\mathcal F\). A family \(\mathcal F\subset 2^{[n]}\) is called shifted if whenever \(A\in\mathcal F\), \(j\in A\), \(i\notin A\), and \(1\le i<j\le n\), then \((A\setminus\{j\})\cup\{i\}\in\mathcal F\). For positive integers \(n,s,\ell,c\) satisfying \(n=2s+c=3s-\ell\) and \(c,\ell\in\{1,\dots,s-1\}\), let
\[
\mathcal P(s,\ell)=\{A\subset [n]: |A|+|A\cap[\ell-1]|\ge 3\},
\]
and let \(\mathcal P'(s,\ell),\mathcal Q(s,\ell),\mathcal W(s,\ell)\) be the other three specified candidate families. Which statement gives the correct extremal description?

One has
\[
e(n,s)=\max\big\{|\mathcal P(s,\ell)|,\ |\mathcal P'(s,\ell)|,\ |\mathcal Q(s,\ell)|,\ |\mathcal W(s,\ell)|\big\}.
\]
Moreover, if \(s\ge 3\), \(\mathcal F\subset 2^{[n]}\) is shifted, \(\nu(\mathcal F)<s\), and \(|\mathcal F|=e(n,s)\), then \(\mathcal F\) is exactly one of \(\mathcal P(s,\ell),\mathcal P'(s,\ell),\mathcal Q(s,\ell),\mathcal W(s,\ell)\).

Choices:
B.
One has
\[
e(n,s)=\max\big\{|\mathcal P(s,\ell)|,\ |\mathcal P'(s,\ell)|,\ |\mathcal Q(s,\ell)|,\ |\mathcal W(s,\ell)|\big\}.
\]
Moreover, if \(s\ge 3\), \(\mathcal F\subset 2^{[n]}\) is shifted, \(\nu(\mathcal F)<s\), and \(|\mathcal F|=e(n,s)\), then \(\mathcal F\) is exactly one of \(\mathcal P(s,\ell),\mathcal P'(s,\ell),\mathcal Q(s,\ell),\mathcal W(s,\ell)\), and this extremal family is unique for the given parameters \((n,s,\ell,c)\).
C.
One has
\[
e(n,s)=\max\big\{|\mathcal P(s,\ell)|,\ |\mathcal P'(s,\ell)|,\ |\mathcal Q(s,\ell)|,\ |\mathcal W(s,\ell)|\big\}.
\]
D.
One has
\[
e(n,s)=\max\big\{|\mathcal P(s,\ell)|,\ |\mathcal P'(s,\ell)|,\ |\mathcal Q(s,\ell)|,\ |\mathcal W(s,\ell)|\big\}.
\]
Moreover, if \(s\ge 3\), \(\mathcal F\subset 2^{[n]}\) satisfies \(\nu(\mathcal F)<s\) and \(|\mathcal F|=e(n,s)\), then \(\mathcal F\) is exactly one of \(\mathcal P(s,\ell),\mathcal P'(s,\ell),\mathcal Q(s,\ell),\mathcal W(s,\ell)\).
E.
One has
\[
e(n,s)=\max\big\{|\mathcal P(s,\ell)|,\ |\mathcal P'(s,\ell)|,\ |\mathcal Q(s,\ell)|,\ |\mathcal W(s,\ell)|\big\}.
\]
Moreover, if \(s\ge 3\), \(\mathcal F\subset 2^{[n]}\) is shifted, \(\nu(\mathcal F)<s\), and \(|\mathcal F|=e(n,s)\), then \(\mathcal F\) is exactly one of the truncated families \(\mathcal P(s,\ell)\cap {[n]\choose \le 3},\ \mathcal P'(s,\ell)\cap {[n]\choose \le 3},\ \mathcal Q(s,\ell)\cap {[n]\choose \le 3},\ \mathcal W(s,\ell)\cap {[n]\choose \le 3}\).

multiple-endpoint extremals vs uniqueness

dropped shifted-extremal classification clause

shifted hypothesis in uniqueness statement

truncated-lattice reduction promoted to full classification

The stem does not state the correct conclusion outright and does not single out choice A. It only sets up the objects and asks for the correct extremal description.

This is close to theorem-statement recognition: the correct option is essentially the exact extremal theorem, with distractors formed by slight logical modifications. It is not fully tautological, but it is only a mild reformulation of a known result.

Some reasoning is needed to distinguish the exact classification from stronger, weaker, or hypothesis-altered variants, especially around uniqueness and the shifted assumption. However, it mainly tests precise recall/statement matching rather than deeper derivation.

The distractors are plausible and mathematically meaningful: one is a weaker true statement, one overstrengthens to uniqueness, one drops a crucial hypothesis, and one introduces an unjustified truncation. These reflect realistic failure modes.

A solid theorem-recognition MCQ with strong distractors and little answer leakage, but it leans more toward precise statement identification than genuinely generative mathematical reasoning.


\label{MainTheorem}
    Let $g \in L^2(\mathbb{R})$ with Fourier transform satisfies the super-exponential decay condition:
    \begin{equation*}
        |\hat{g}(\xi)| \leq C e^{-a|\xi|^m}
    \end{equation*}
    for some constants $m>1$, $C>0$, $a \in (1,\infty)$. Let $\tau_1, \tau_2 \in \mathbb{R}^+$ satisfy
    \begin{equation*}
        \tau_1 < \big(\frac{2}{(2\pi)^{m/(m-1)}(ma)^{-1/(m-1)}e}\big)^{\frac{m-1}{m}}, \quad \quad \quad \tau_2< (\frac{2}{ame})^{\frac{1}{m}}.
    \end{equation*}
    Define the set $\Lambda$ such that:
    \begin{equation*}
        \Lambda =  \{(\pm\tau_1n^{\frac{m-1}{m}},\pm\tau_2n^{\frac{1}{m}}),n \in \mathbb{N}\}
    \end{equation*}
    Then the following statements are equivalent for every $f, h \in L^2(\mathbb{R}):$
    \begin{enumerate}
        \item $|V_g(f)(\lambda)| = |V_gh(\lambda)|$ for every $\lambda \in \Lambda$.
        \item $f = e^{i \alpha}h$ for some $\alpha \in [0,2\pi).$
    \end{enumerate}

\begin{theorem}\label{MainTheorem}
    Let $g \in L^2(\R)$ with Fourier transform satisfies the super-exponential decay condition:
    \begin{equation*}
        |\hat{g}(\xi)| \leq C e^{-a|\xi|^m}
    \end{equation*}
    for some constants $m>1$, $C>0$, $a \in (1,\infty)$. Let $\tau_1, \tau_2 \in \R^+$ satisfy
    \begin{equation*}
        \tau_1 < \big(\frac{2}{(2\pi)^{m/(m-1)}(ma)^{-1/(m-1)}e}\big)^{\frac{m-1}{m}}, \quad \quad \quad \tau_2< (\frac{2}{ame})^{\frac{1}{m}}.
    \end{equation*}
    Define the set $\Lambda$ such that:
    \begin{equation*}
        \Lambda =  \{(\pm\tau_1n^{\frac{m-1}{m}},\pm\tau_2n^{\frac{1}{m}}),n \in \N\}
    \end{equation*}
    Then the following statements are equivalent for every $f, h \in L^2(\R):$
    \begin{enumerate}
        \item $|V_g(f)(\lambda)| = |V_gh(\lambda)|$ for every $\lambda \in \Lambda$.
        \item $f = e^{i \alpha}h$ for some $\alpha \in [0,2\pi).$
    \end{enumerate}
\end{theorem}

Context:
\label{sec:1}
\input{introduction}

\label{sec:1}
\input{introduction}

\section{Preliminary}
\label{sec:2}
A central principle in Fourier analysis is the connection between the smoothness of a function and the decay rate of its Fourier transform at infinity. The Paley-Wiener theorem and its variants are the primary tools that precisely characterize this relationship, providing the foundation for extending functions into the complex domain based on the behavior of their Fourier transforms.
\subsection{Paley-Wiener Type Theorem}
The Fourier transform of a Lebesgue integrable function $f$ on $\R$ is given by
\begin{equation*}
 \mathcal{F}(f):=\hat{f} (\xi) = \int_{\R} f(t) e^{-2 \pi i t \cdot\xi} dt, \quad \xi \in \R.
\end{equation*}

A standard density argument can be used to extend this definition so that $\mathcal{F}$ is an isometry on $L^2(\R)$. The Fourier transform of a function $f \in L^2(\R)$ is said to be of \emph{super-exponential decay} if
\begin{equation}\label{FourierDecay}
 |\hat{f}(\xi)| \lesssim Ce^{-a|\xi|^m}
\end{equation}
for some $m>1$, $C>0$ and $a \in (1,\infty)$. In this subsection we introduce a slightly modified version of Paley-Wiener theorem. In particular, we show that if the Fourier transform of a function $f$ decays super-exponential, then $f$ can
be extended to an entire function whose order and type depend on the parameters $a$ and $m$. To establish this result, we make use of Morera’s Theorem, which we now recall.
\begin{mainthm}\cite[Theorem 5.2]{stein2010complex}\label{Morera}
 If $\{ f_n \}_{n \in \N}$ is a sequence of holomorphic functions that converges uniformly to a function $f$ in every compact subset of $\Omega$, then $f$ is holomorphic in $\Omega$.
\end{mainthm}

Similarly, fix $z=x+iy$ and consider $z' = x'+iy'\subset K'$, where $K'$ is compact. Then we have
\begin{equation}\label{fixz}
 \int_{-\infty}^{\infty} |\overline{g(t-z)}|^2 |e^{2\pi i z' \xi}|^2 dt \leq C\int_{-\infty}^{\infty} e^{-2a|t-x|^{\rho}+2b|y|^{\rho}-4\pi y't}dt.
\end{equation}
Since $z=x+iy$ is fixed, for $z' \in K'$ it follows that \eqref{fixz} converges for $z' \in K$. Following a statement from \cite[Page 68]{boas2011entire}, the order and type of an entire function is retained under integral, therefore the orders and types for $|V_gf (z,z')|$ can be computed directly using Lemma \ref{OrderType}. It is noteworthy that our computation for orders is compatible with the result from \cite[Lemma 3.4]{grohs2024phaseless} when $m = 2$.
\end{proof}
\section{Main Result}
Denote the ring of holomorphic functions by $\mathcal{O}(\C)$. Consider the collection of entire functions with $\rho>1$ and $a,b>0$:
$$\fG_{a,b,\rho} = \{ f \in \mathcal{O}(\C)\big| |f(x+iy)| \lesssim e^{-a|x|^{\rho}+b|y|^{\rho}} \}.$$
It is straightforward to see that $\fG_{a,b,\rho}$ is a linear space. The objective is to establish a uniqueness set so that whenever two functions $f, g \in \fG_{a,b,\rho}$ agree on the set, they agree \textit{everywhere}. Recent literature shows that such functions cannot be uniquely determined from their values on integer lattices \cite{alaifari2021phase}. For the purpose of notation, let $U \subset \mathcal{O(\C)}$ denote a linear function space. A set $\Lambda \subset \C$ is called a uniqueness set of $U$ if for any $f \in U$, it holds that
$$(f( \lambda ) = 0 \; \text{for all} \; \lambda \in \Lambda) \Longrightarrow f = 0 \; \text{everywhere on } \: \C.$$
Since zero of any entire function can't be an accumulation point, any open set in $\C$ is trivially a uniqueness set. Without this property, determining discrete uniqueness sets of $\fG_{a,b,\rho}$ contained in $\R$ is a nontrivial task. We first propose a sufficient condition for such a set.\\

\input{references}
\end{document}
OrderType.tex0000664000000000000000000001265715107436001012225 0ustar rootroot
\begin{lemma}\label{OrderType}
Let $f \in L^2(\mathbb{R})$ be a function whose Fourier transform satisfies $|\hat{f}(\xi)| \le C e^{-a|\xi|^m}$ for some $m>1$ and $a>0$. Then its analytic continuation $f(z)$ is an entire function of type at most $\tau = \frac{m-1}{m}(2\pi)^{m/(m-1)}(am)^{-1/(m-1)}$ corresponding to maximum possible order $\rho = \frac{m}{m-1}$.
\end{lemma}
\begin{proof}
The analytic continuation of the function, $f(z)$, is given by the inverse Fourier transform:
\begin{equation*}
 f(z) = \int_{-\infty}^\infty \hat{f}(\xi) e^{2\pi i z \xi} d\xi
\end{equation*}
The Taylor coefficients of $f(z)$ are given by
\begin{equation*}
c_n = \frac{f^{(n)}(0)}{n!} = \frac{(2\pi i)^n}{n!} \int_{-\infty}^\infty \xi^n \hat{f}(\xi) d\xi
\end{equation*}
We can now bound the magnitude of the coefficients using the given decay of $\hat{f}(\xi)$:
\begin{equation*}
|c_n| \le \frac{(2\pi)^n}{n!} \int_{-\infty}^\infty |\xi|^n |\hat{f}(\xi)| d\xi \le \frac{C(2\pi)^n}{n!} \int_{-\infty}^\infty |\xi|^n e^{-a|\xi|^m} d\xi
\end{equation*}
The integral can be evaluated using the Gamma function:
\begin{equation*}
 \int_{-\infty}^\infty |\xi|^n e^{-a|\xi|^m} d\xi = \frac{2}{m a^{(n+1)/m}} \Gamma\left(\frac{n+1}{m}\right).
\end{equation*}
Using Stirling's formula for Gamma function, we obtain
\begin{equation*}
 \Gamma(\frac{n+1}{m}) = \sqrt{\frac{2 \pi m}{n+1}}(\frac{n+1}{me})^{(n+1)/m}\big(1+ O(\frac{m}{n+1})\big).
\end{equation*}
and
\begin{equation*}
 n! \approx \sqrt{2 \pi n}(\frac{n}{e})^n
\end{equation*}
Then, for some $C' >0$, we derive the following expression:
 \begin{align}\label{GeneralTaylorCoefficient}
 |c_n|
 &\lesssim \Big((2\pi)^n \cdot n^{-\frac{1}{2}-n}\cdot e^{n -\frac{n+1}{m}}\cdot a^{-\frac{n+1}{m}}\cdot m ^{-\frac{1}{2}-\frac{n+1}{m}}\cdot (n+1)^{\frac{n+1}{m}-\frac{1}{2}}\Big)\cdot\Big(1 + \frac{C'm}{n+1}\Big) \\
 &= R(a,n,m)\cdot n^{-n}\cdot(n+1)^{\frac{n+1}{m}}\cdot\Big(1 + \frac{C'm}{n+1}\Big) \nonumber
 \end{align}
The term $R(a,n,m)$ collects all remaining exponential and polynomial factors. It follows that:
\begin{align*}
 \rho &= \limsup_{n\to\infty} \frac{n \log n}{-\log|c_n|} \lesssim \limsup_{n\to\infty} \frac{n\log n}{-\log R(a,m,n) + n\log n -\frac{n+1}{m}\log(n+1)- \log(1+\frac{C'm}{n+1})}\\
 &\lesssim \limsup_{n\to\infty} \frac{1}{\frac{-\log R(a,m,n)}{n\log n} + 1 -\frac{n+1}{nm}\frac{\log(n+1)}{\log n}- \frac{\log(1+\frac{C'm}{n+1})}{n\log n}}.
\end{align*}
A series of computations shows that $$\begin{cases} \lim_{n\to \infty}\frac{\log R(a,m,n)}{n\log n} =0\\ \lim_{n\to \infty}\frac{\log(1+ \frac{C'm}{n+1})}{n\log n}=0\end{cases}.$$ Therefore, the remaining terms are:
\begin{equation*}
 \rho \leq \limsup_{n\to \infty}\frac{1}{1-\frac{n+1}{mn}\frac{\log(n+1)}{\log n}}= \frac{1}{1- \frac{1}{m}} = \frac{m}{m-1}.
\end{equation*}
Let $\tilde{\rho} = \frac{m}{m-1}$, and choose $A,R >0$ such that
\begin{equation}\label{TypeWRTorder}
 \max_{|z|= r}|f(z)| < Re^{Ar^{\tilde{\rho}}}.
\end{equation}
Now we compute the type $\tau$ of $f(z)$ with respect to order $\tilde{\rho}$, i.e., the greatest lower bound of $A$ in Equation~\eqref{TypeWRTorder}. We note that

\begin{figure}[h!]
    \centering
    \includegraphics[width=0.7\textwidth]{rhosampling.png}
    \caption{An illustration of sampling points $\Lambda$ in time-frequency plane for $g$ with Fourier decay parameter $m=\frac{3}{2}$ and $a=1$}
\end{figure}
{The rest of the paper is organized as follows: In Section 2 we introduce some necessary preliminaries. Subsequently we prove Theorem~\ref{MainTheorem} in Section 3. To this end, we first establish a sufficient condition for a discrete set to be a uniqueness set for the relevant class of entire functions (Proposition 1) and use Proposition 2 to show that our result is in a sense "sharp". We then use these technical results to prove Theorem~\ref{MainTheorem}, which provides a concrete discrete sampling set $\Lambda$ that guarantees the unique recovery of $f$ up to a global phase.

To prove Theorem~\ref{MainTheorem}, the paper (in Section 3) proceeds by first establishing “a sufficient condition for a discrete set to be a uniqueness set for the relevant class of entire functions (Proposition 1),” then using “Proposition 2 to show that our result is in a sense sharp,” and finally applying “these technical results to prove Theorem~\ref{MainTheorem},” yielding “a concrete discrete sampling set $\Lambda$ that guarantees the unique recovery of $f$ up to a global phase.”

To prove the main theorem, the paper proceeds by first establishing a sufficient condition for a discrete set to be a uniqueness set for the relevant class of entire functions (Proposition 1), then using Proposition 2 to show that the result is in a sense sharp, and finally applying these technical results to prove the main theorem, yielding a concrete discrete sampling set $\Lambda$ that guarantees the unique recovery of $f$ up to a global phase.

Biconditional or Equivalence
Classification or Bijection

Question:
Let the short-time Fourier transform with window \(g\) be \(V_g f(x,\xi)=\int_{\mathbb R} f(t)\overline{g(t-x)}e^{-2\pi i\xi t}\,dt\). Suppose \(g\in L^2(\mathbb R)\) has Fourier transform satisfying
\[
|\widehat g(\xi)|\le C e^{-a|\xi|^m}
\]
for some \(m>1\), \(C>0\), and \(a>1\). Let \(\tau_1,\tau_2>0\) satisfy
\[
\tau_1<\Big(\frac{2}{(2\pi)^{m/(m-1)}(ma)^{-1/(m-1)}e}\Big)^{\frac{m-1}{m}},
\qquad
\tau_2<\Big(\frac{2}{ame}\Big)^{1/m},
\]
and define
\[
\Lambda=\{(\pm \tau_1 n^{(m-1)/m},\, \pm \tau_2 n^{1/m}) : n\in\mathbb N\}\subset \mathbb R^2,
\]
with all sign choices allowed. Which statement exactly characterizes the pairs \(f,h\in L^2(\mathbb R)\) for which
\(|V_g f(\lambda)|=|V_g h(\lambda)|\) for every \(\lambda\in\Lambda\)?

They are exactly the pairs differing by a global phase: \(|V_g f(\lambda)|=|V_g h(\lambda)|\) for all \(\lambda\in\Lambda\) if and only if \(f=e^{i\alpha}h\) for some \(\alpha\in[0,2\pi)\).

Choices:
B. They are exactly the pairs related by a time-frequency shift and a global phase: \(|V_g f(\lambda)|=|V_g h(\lambda)|\) for all \(\lambda\in\Lambda\) if and only if there exist \(x_0,\xi_0\in\mathbb R\) and \(\alpha\in[0,2\pi)\) such that \(f(t)=e^{i\alpha}e^{2\pi i \xi_0 t}h(t-x_0)\) a.e.
C. If \(f=e^{i\alpha}h\) for some \(\alpha\in[0,2\pi)\), then \(|V_g f(\lambda)|=|V_g h(\lambda)|\) for every \(\lambda\in\Lambda\).
D. They are exactly the pairs differing by a global phase provided the equalities hold on one fixed-sign branch of the sampling set, i.e. if and only if \(|V_g f(\tau_1 n^{(m-1)/m},\tau_2 n^{1/m})|=|V_g h(\tau_1 n^{(m-1)/m},\tau_2 n^{1/m})|\) for every \(n\in\mathbb N\), then \(f=e^{i\alpha}h\) for some \(\alpha\in[0,2\pi)\).
E. They are exactly the pairs differing by a global phase for any choice of \(\tau_1,\tau_2>0\): \(|V_g f(\lambda)|=|V_g h(\lambda)|\) for all \(\lambda\in\Lambda\) if and only if \(f=e^{i\alpha}h\) for some \(\alpha\in[0,2\pi)\), without requiring the upper bounds on \(\tau_1\) and \(\tau_2\).

The stem does not state or strongly hint at the correct conclusion; it only gives the hypotheses and asks for the exact characterization.

This is close to a theorem-recall item: the stem essentially presents the assumptions and asks for the precise conclusion. It is not a pure tautology because the options vary in quantifiers, hypotheses, and equivalence strength.

Some reasoning is needed to distinguish the exact iff statement from weaker, stronger, or hypothesis-altered variants, but the task is mainly precise theorem recognition rather than substantial derivation.

The distractors are plausible and target real failure modes: confusing global phase with time-frequency symmetry, accepting only one implication, dropping the four-sign sampling requirement, or removing the tau bounds.

A solid theorem-identification MCQ with strong distractors and no answer leakage, though it leans more toward exact statement recall than deeper generative reasoning.


[Chitrao-Ghate \cite{CG23}]\label{Main theorem in the second part of my thesis}
      For $k \in [3, p + 1]$ and for primes $p \geq 5$, the semi-simplification of the reduction mod $p$ of the
      semi-stable representation $V_{k, \mathcal{L}}$ on $\mathrm{Gal}(\brqp/\mathbb{Q}_p)$ is given by an alternating
      sequence of irreducible and reducible representations:
    \[
    \overline{V}_{k, \mathcal{L}} \sim
    \begin{cases}
        \mathrm{ind}\> (\omega_2^{r+1 + (i-1)(p-1)}), & \text{ if $(i-1) - r/2 < \nu < i - r/2$} \\
        \mu_{\lambda_i}\omega^{r+1-i} \oplus \mu_{\lambda_i^{-1}}\omega^{i}, & \text{ if $\nu = i - r/2$},
    \end{cases}
\]
where
  \begin{eqnarray*}
    \begin{cases}
      1 \leq i \leq (r+1)/2 & \text{if $r$ is odd} \\
                          1 \leq i \leq (r+2)/2 & \text{if $r$ is even,}
    \end{cases}
 \end{eqnarray*}
  and the mod $p$ constants $\lambda_i$ are determined by the formulas:
    \begin{eqnarray*}
        \lambda_i & = & \br{(-1)^i \> i {r+1-i \choose i}\frac{(\mathcal{L} - H_{-} - H_{+})}{p^{i-r/2}}}, \quad \text{ if } 1 \leq i < \dfrac{r + 1}{2} \\
        \lambda_{i} + \lambda_i^{-1} & = & \br{(-1)^i \> i{r + 1 - i \choose i}\frac{(\mathcal{L} - H_{-} - H_{+})}{p^{i-r/2}}}, \quad \text{ if } i = \dfrac{r + 1}{2} \text{ and } r \text{ is odd}.
    \end{eqnarray*}

\begin{theorem}[Chitrao-Ghate \cite{CG23}]\label{Main theorem in the second part of my thesis}
      For $k \in [3, p + 1]$ and for primes $p \geq 5$, the semi-simplification of the reduction mod $p$ of the
      semi-stable representation $V_{k, \cL}$ on $\mathrm{Gal}(\brqp/\qp)$ is given by an alternating
      sequence of irreducible and reducible representations:
    \[
    \br{V}_{k, \cL} \sim
    \begin{cases}
        \ind (\omega_2^{r+1 + (i-1)(p-1)}), & \text{ if $(i-1) - r/2 < \nu < i - r/2$} \\
        \mu_{\lambda_i}\omega^{r+1-i} \oplus \mu_{\lambda_i^{-1}}\omega^{i}, & \text{ if $\nu = i - r/2$},
    \end{cases}
\]
where
  \begin{eqnarray*}
    \begin{cases}
      1 \leq i \leq (r+1)/2 & \text{if $r$ is odd} \\
                          1 \leq i \leq (r+2)/2 & \text{if $r$ is even,}
    \end{cases}
 \end{eqnarray*}
  and the mod $p$ constants $\lambda_i$ are determined by the formulas:
    \begin{eqnarray*}
        \lambda_i & = & \br{(-1)^i \> i {r+1-i \choose i}\frac{(\cL - H_{-} - H_{+})}{p^{i-r/2}}}, \quad \text{ if } 1 \leq i < \dfrac{r + 1}{2} \\
        \lambda_{i} + \lambda_i^{-1} & = & \br{(-1)^i \> i{r + 1 - i \choose i}\frac{(\cL - H_{-} - H_{+})}{p^{i-r/2}}}, \quad \text{ if } i = \dfrac{r + 1}{2} \text{ and } r \text{ is odd}.
    \end{eqnarray*}
\end{theorem}

Let $p$ be a prime. It has been over 10 years now since I started working on mathematics connected with the $p$-adic Local
Langlands correspondence for $p$ a prime. The theory was initiated by Breuil about 20 years ago, and caused a mini-revolution
in the field of number theory. I remember him speaking about his vision at the ICM in Hyderabad in 2010 \cite{Breuil ICM}.
A few years later Colmez made the
next quantum leap forward by making Breuil's correspondences functorial \cite{Col10b}. I was fortunate to have a ringside view of these
developments especially since some of them were made during the years 2007-2010 when we ran a joint Indo-French CEFIPRA project.
Meanwhile many other prominent mathematicians such as Berger, Dospinescu, Pa\u{s}k\={u}n{a}s, to name just a few,
contributed deep results along the way.

I entered the subject for the following reason. About 13 years ago, I was trying to show that certain mod $p$ Galois representations
attached to modular forms had large image. Indeed, I was trying to generalize Serre's famous conjecture
that the global mod $p$ Galois representation attached to a non-CM rational elliptic curve has full image for all primes $p$ larger than
an absolute constant to the setting of modular forms. It turns out that the na\"ive generalization of this statement is necessarily false
but Pierre Parent and I could state a variant of this conjecture and make some mild progress on it for weight $2$ forms
\cite{GP12}.

One way to tackle this problem is to show that the corresponding restricted local mod $p$ Galois representation has large image.
Nothing like showing a group is large by showing that it has a large subgroup. This approach works to some extent but not entirely, not least because it turns out that the mod $p$ reductions of local Galois
representations attached to modular forms have not yet been written down in all cases. It was very disconcerting to me that there was such a glaring
gap in the subject.
I decided to devote a good chunk of my future research time in making some
headway with this question.

It turns out  the $p$-adic Local Langlands program is ideally suited to studying the mod $p$ images of local
modular Galois representations.
Indeed, Breuil invented his theory to tackle exactly this problem. In the beginning he restricted to the so called crystalline case
where $p$ does not divide the level of the modular form. Let us from now on always restrict to odd primes $p$ (though in
some results in this Introduction we assume without warning that $p \geq 5$). Breuil showed that one could compute the local mod $p$ reductions for all
forms of weights $k \leq 2p+1$ and positive slope \cite{Bre03}, \cite{Bre03b}. Here the slope of a form is the $p$-adic valuation of its $p$-th Fourier
coefficient, where the valuation is normalized so that the valuation of $p$ is 1.
(The case of slope $0$ and all weights is classical and is due to Deligne.)
This generalized the work of his advisor Fontaine, although the details were worked out in \cite{Edi92},
who had earlier computed the reductions for weights $k \leq p+1$ and positive slope.

One obvious restriction
in these theorems is that the weight is bounded above, but there is no restriction on the slope. In an orthogonal direction, in a
short but influential paper, Buzzard and Gee \cite{BG09} used the $p$-adic Local Langlands correspondence to compute the mod $p$
reductions of crystalline Galois representations of slopes in the small range
$(0,1)$, but for all weights. Some
difficulties encountered at slope $1/2$ were only cleared up in a second paper \cite{BG13}.
(The case of $p=2$ for slopes in this range is being treated in a forthcoming thesis of Arathy Venugopal.)
For historical completeness, let us
mention that earlier and using a different method, Berger, Li and Zhu \cite{BLZ04}
had treated the case of slopes which are large compared to the weight, namely slopes which are larger
than $\lfloor \frac{k-2}{p-1} \rfloor$ (an interesting variant of this results was recently proved by Bergdall-Levin \cite{BL22}
who treated the
case of slope larger than $\lfloor \frac{k-1}{p} \rfloor$).

This is where I entered the problem. In a series of papers with my coauthors (students, postdocs, colleagues), I first extended the result of
Buzzard-Gee to all slopes in $(0,2)$. This does not seem like much, but this marginal gain of a unit interval's worth of slopes
was to consume me for the better part of the first half of the
following decade. At first, the answers we got for the reduction seemed unpredictable, almost as if one was entering a fractal. There
were no general guidelines (other than a folklore conjecture of Breuil, Buzzard and Emerton which said that for fractional
slopes and even weights the reduction should be irreducible). Murphy's law (if things can go wrong, they will)
seemed to rule the roost - if a particular exception to a general rule for the shape of the reduction was in principle possible,
then it always wound up occurring.

Some initial headway for the case of slopes in $(1,2)$ was made with Abhik Ganguli for bounded weights \cite{GG15}, and then in a very nice paper with
Shalini Bhattacharya for all weights \cite{BG15} but again we were only able to partially treat the case of slope $3/2$.
The missing case of slope $1$ was then
treated with Shalini Bhattacharya and Sandra Rozensztajn \cite{BGR18}. The complete picture for
slope $3/2$ was finally provided only a few years ago with Vivek Rai, though the paper \cite{GR19} appeared just this year.
(Beyond this range, I also wish to mention forthcoming work of Sudipta Majumder for slope $2$, some partial
results of Nagell-Pande and Arsovski for slopes in $(2,3)$, and a
forthcoming project \cite{BGR25} with Shalini Bhattacharya and Ravitheja Vangala which aims to treat all fractional slopes
in the range $(0,p)$ building on the foundations laid in \cite{GV22}.)
All these papers use the functoriality of
the $p$-adic Local Langlands Correspondence with respect to reduction (established by Berger if one is willing to work up to semi-simplification - as we mostly
were - and in general by Colmez), to reduce the question of studying the reductions of local crystalline Galois representations
to studying the reduction of the standard lattice in a certain $p$-adic Banach space. The slope 1 paper \cite{BGR18} also computes the reductions
of several other lattices, and in particular establishes criteria to distinguish between {\it peu} and {\it tr\`es ramifi\'ee}
cases.

Part of the problem encountered at the half-integral slopes $1/2$, $1$, $3/2, \ldots$ was that the reduction seemed to behave even more
erratically than usual at the so called exceptional weights $k$ (these are weights which are congruent to two more
than twice the slope modulo $(p-1)$).
Based on the results in slope $1/2$ and $1$, and some cautionary computations of Rozensztajn,
I eventually wound up making a conjecture
which I called the {\it zig-zag conjecture} which described the behaviour of the reduction for all positive half-integral
slopes less than or equal to $\frac{p-1}{2}$ and all sufficiently large exceptional weights.
Roughly, the conjecture predicted that the reductions varied through an
alternating sequence of irreducible and reducible representations
depending on the (relative) sizes of two parameters. The statement appeared in a proceedings of an annual number theory conference
at RIMS in Kyoto, Japan \cite{Gha21}, where I was thrown to the wolves by my kind host Shinichi Kobayashi as the opening speaker (I thank him for this honor).
At the time it had become my  mission to settle the case of slope $3/2$ if only to prove that the conjecture
had some merit. However, over the years there was an uncomfortable truth that
began to emerge. The paper \cite{BG13} for slope $1/2$ was about 10 pages long, the one \cite{BGR18} for slope $1$ was about 50 pages long (though not all of it dealt with zig-zag), and the (unabridged arXiv version of the)
one \cite{GR19} for slope $3/2$ was just under 80 pages long (its entire focus was zig-zag at $3/2$). So clearly another approach would be required to prove the conjecture in general.
To my complete surprise this was to surface a few years later.

To explain this, let us return to Breuil's early foundational work. Apart from treating the crystalline case, Breuil also wrote some important papers in
the so called semi-stable case. This case occurs for modular forms for which $p$ exactly divides the level of the form (and does not divide the conductor
of the nebentypus character). In an important work with M\'ezard dating to more than 20 years ago, Breuil computed the mod $p$ reductions
of all semi-stable Galois representations of {\it even} weight $k \leq p-1$ using techniques from integral $p$-adic Hodge theory \cite{BM}. The case of {\it odd} weights
in this range was completed much later (about 6 years ago) in another {\it tour de force} by Guerberoff and Park \cite{GP} (and Lee and Park \cite{LP22}), though several constants
remained to be determined completely. For completeness, we also mention that an alternate algebro-geometric approach to computing the reduction involving the global sections of certain
bundles on the $p$-adic upper half-plane was developed by Breuil-M\'ezard in some cases \cite{BM10}, though this approach does indeed seem to require that the weight $k$ be even
since it involved $k/2$-powers of certain line bundles.

In any case, some time in 2022, I realized that all of these works in the parallel universe of semi-stable representations could be used to give a proof
of the zig-zag conjecture in the crystalline world, at least for most slopes and on the inertia group. This was perhaps one of the more important
observations that I have made in the past 10 years. Let us explain how it came about. A bit earlier than this, Anand Chitrao, Seidai Yasuda and I had been trying to use the above
mentioned works in the crystalline world to try and deduce results about $V_{k,\mathcal{L}}$ in the semi-stable world, using a limiting argument in Colmez's blow-up space of
non-split rank $2$ trianguline $(\varphi,\Gamma)$-modules. We wrote a nice paper about this which appeared this year \cite{CGY21} and
which allowed one, for instance, to predict the exact shape of
some of the above mentioned missing constants for small odd weights (e.g., for $k=5$ from the slope ${3}/{2}$ paper),
and, in general, to recover the work of
Breuil-M\'ezard and Guerberoff-Park on inertia {\it assuming} my zig-zag conjecture. The breakthrough came when I realized that one could reverse the
entire argument and instead deduce information about crystalline representations - in particular, a large portion of the zig-zag phenomenon -
from the literature in the semi-stable case. In fact, after this realization I could
immediately prove zig-zag up to slope $\frac{p-3}{2}$
(though in the first instance I could only prove it on the inertia group since, as already mentioned, the constants in the semi-stable world had not yet been
completely determined for odd weights). The missing cases of slope $\frac{p-2}{2}$ and $\frac{p-1}{2}$
would require extending the work of Guerberoff-Park \cite{GP} to the odd weight $k=p$ and the classical work of Breuil-M\'ezard \cite{BM} to the case of the even weight $k = p+1$.

The possible extension to these two weights was more than just a technicality. There was a theoretical obstruction. It turned out that the strongly
divisible modules occurring in integral $p$-adic Hodge theory were either not as well behaved ($k = p$,  see
\cite{Gao17}) or not even available (for $k \geq p+1$)
(although since then a theory of Breuil-Kisin modules has become available which works for all weights $k$). So an entirely new perspective was required.
Based on my experience with computing the reduction using the functoriality of the $p$-adic Local Langlands Correspondences in the crystalline world (a method initiated by Breuil and
Buzzard-Gee), I wondered whether Anand Chitrao and I might
be able to tackle the reduction problem in the semi-stable case in a similar manner. We were given to understand
that one
might have to wait for a very long time for this hope to be realized.

However, not ones to shy away from a challenge, Chitrao and I started work on this ambitious project. Initial gains were few and
far between. But, to make a long story short, we were eventually able to
compute the mod $p$ reductions of all semi-stable representations for weights $k \leq p+1$, including the cases of weight $k = p$
and $k = p+1$. We were also able to provide a complete and uniform treatment of all the constants involved.
The goal of this expository paper is to
explain this result and to expand on some of the mathematical background that goes into its proof.
But before I go further, let me record that this work
allowed one to complete the proof of my zig-zag conjecture (i.e., to extend the initial proof up to slope $\frac{p-1}{2}$,
and to determine all the constants that occur in the unramified characters on the decomposition group in the reduction, see \cite{Gha22}).

Already, in the early days, Breuil had written two important papers describing the Banach space attached to
a semi-stable representation $V_{k,\mathcal{L}}$ of weight $k$ and $\mathcal{L}$-invariant $\mathcal{L}$ under the $p$-adic Local Langlands Correspondence. The first description, denoted by
$B(k,\mathcal{L})$ in \cite{Bre04},
involved some work of Schneider and Teitelbaum and used Morita duality
and seemed a bit abstract to us. But the second description, denoted by $\tilde{B}(k, \mathcal{L})$ in \cite{Bre10}, `only' used $p$-adic
functional analysis (a beautiful summary of which can be found in \cite{Col10a}, see $\S 3$) and this definition
seemed much more amenable to computation.
In this model, the Banach space $\tilde{B}(k, \mathcal{L})$ was nothing but the space of differentiable functions
on $\mathbb{Q}_p$ of order $r/2$ where $r = k-2$ (more precisely of type $\mathscr{C}^{r/2}$, these notions are slightly different),
with a similar differentiability condition at $\infty$, modulo polynomial functions of degree at most $r$ and certain finite sums of  polynomial
times logarithmic ({poly$\cdot$log!}) functions (with the polynomial part having degree less than $r/2$). This description was something that you
could explain to a clever high school student learning calculus. Using it, we began to search for an integral structure (lattice)
on this Banach space.

It was not immediately clear how to proceed, but we soon realized that one could uniformize this
Banach space (much as in the crystalline world) by the compactly induced space $\mathrm{ind}_{IZ}^G \mathrm{Sym}^{k-2} E^2$ of
certain rational polynomial valued functions allowing us to define the (standard) lattice
in the Banach space as the (closure of the) image of the space $\mathrm{ind}_{IZ}^G \mathrm{Sym}^{k-2}{\mathcal O}_E^2$ of
integral polynomial valued functions under this uniformization. Here
$I$ is the Iwahori subgroup of $G = \GL_2(\mathbb{Q}_p)$ and $Z$ is the center of $G$. It then `remained' to compute the reduction of this
lattice. In hindsight, this required several new creative computations. After much effort we were able to compute all
the Jordan-H\"older (JH) factors in the reduction
in terms of certain mod $p$ compactly induced spaces modulo the images of certain Iwahori-Hecke operators. Applying a
reformulation of Breuil's mod $p$ Local Langlands correspondence worked out by Chitrao, which is
called the Iwahori mod $p$ Local Langlands correspondence in \cite{Chi23},
we could then return to the Galois side and compute the reductions of the semi-stable Galois representations $V_{k,\mathcal{L}}$.
Our target to treat all weights in the range $3 \leq k \leq p+1$ was achieved in the paper \cite{CG23}, though we remark that
in principle our method can be used to treat  all weights $k \geq 3$ at least in principle (unlike the initial approach with strongly divisible modules).
Indeed, Anand and I have just begun to see whether our approach to computing the reduction of semi-stable representations
using $p$-adic Langlands can be used to recover very recent work of Bergdall, Levin, and Liu \cite{BLL22}
which uses Breuil-Kisin modules
to study the reduction of $V_{k,\mathcal{L}}$ with $\mathcal{L}$-invariant having very negative $p$-adic valuation.

Let us state the main result in the paper \cite{CG23} with Anand Chitrao.
Let $p \geq 5$ be a prime and $E$ be a finite extension of $\mathbb{Q}_p$ containing $\sqrt{p}$.
We describe completely the semi-simplification of the reduction mod $p$ of
the irreducible two-dimensional semi-stable representation $V_{k, \mathcal{L}}$ of $\mathrm{Gal}(\brqp/\mathbb{Q}_p)$
over $E$ with Hodge-Tate weights $(0,k-1)$ for $k \in [3, p + 1]$ and
$\mathcal{L}$-invariant $\mathcal{L} \in E$. To do this
we need a little bit more notation. Let as usual $r = k-2$ so that $1 \leq r \leq p-1$.
    Let $v_-$ and $v_+$ be
    the largest and smallest integers, respectively, such that $v_- < r/2 < v_+$ for $r \geq 1$.
    For $n \geq 1$, let $H_n = \sum_{i = 1}^{n}\frac{1}{i}$ be the $n$-th partial harmonic sum.
    Set $H_0 = 0$ and $H_{\pm} = H_{v_{\pm}}$. Let $v_p$ be the $p$-adic valuation on $\brqp$
    normalized such that $v_p(p) = 1$.
    Let $$\nu = v_p(\mathcal{L} - H_{-} - H_{+})$$
    be the $p$-adic valuation of $\mathcal{L}$ shifted by the
    partial harmonic sums $H_{-}$ and $H_{+}$.
    Everything hinges on the size of the parameter $\nu$. Let $\omega$ be the fundamental character of
    $\mathrm{Gal}(\br{{\mathbb Q}}_p/\mathbb{Q}_p)$ of
    level $1$.     Similarly, for an integer $c$ (with
    $p + 1 \nmid c$), let $\omega_2^c$ be an extension from the inertia subgroup $I_{\mathbb{Q}_p}$ to $\mathrm{Gal}(\brqp/\bq_{p^2})$
    of the $c$-th power of the
    fundamental character $\omega_2$     of level $2$ chosen
    so that the (irreducible) representation $\mathrm{ind}\> (\omega_2^c)$ obtained
    by inducing this character from $\mathrm{Gal}(\brqp/\bq_{p^2})$ to $\mathrm{Gal}(\brqp/\mathbb{Q}_p)$ has
    determinant $\omega^c$. Let $\mu_{\lambda}$ be the unramified character of $\mathrm{Gal}(\brqp/\mathbb{Q}_p)$     sending geometric Frobenius to $\lambda \in \br{\mathbb{F}}_p^{*}$.     Our main theorem is the following result.

To explain this, let us return to Breuil's early foundational work. Apart from treating the crystalline case, Breuil also wrote some important papers in
the so called semi-stable case. This case occurs for modular forms for which $p$ exactly divides the level of the form (and does not divide the conductor
of the nebentypus character). In an important work with M\'ezard dating to more than 20 years ago, Breuil computed the mod $p$ reductions
of all semi-stable Galois representations of {\it even} weight $k \leq p-1$ using techniques from integral $p$-adic Hodge theory \cite{BM}. The case of {\it odd} weights
in this range was completed much later (about 6 years ago) in another {\it tour de force} by Guerberoff and Park \cite{GP} (and Lee and Park \cite{LP22}), though several constants
remained to be determined completely. For completeness, we also mention that an alternate algebro-geometric approach to computing the reduction involving the global sections of certain
bundles on the $p$-adic upper half-plane was developed by Breuil-M\'ezard in some cases \cite{BM10}, though this approach does indeed seem to require that the weight $k$ be even
since it involved $k/2$-powers of certain line bundles.

However, not ones to shy away from a challenge, Chitrao and I started work on this ambitious project. Initial gains were few and
far between. But, to make a long story short, we were eventually able to
compute the mod $p$ reductions of all semi-stable representations for weights $k \leq p+1$, including the cases of weight $k = p$
and $k = p+1$. We were also able to provide a complete and uniform treatment of all the constants involved.
The goal of this expository paper is to
explain this result and to expand on some of the mathematical background that goes into its proof.
But before I go further, let me record that this work
allowed one to complete the proof of my zig-zag conjecture (i.e., to extend the initial proof up to slope $\frac{p-1}{2}$,
and to determine all the constants that occur in the unramified characters on the decomposition group in the reduction, see \cite{Gha22}).

It was not immediately clear how to proceed, but we soon realized that one could uniformize this
Banach space (much as in the crystalline world) by the compactly induced space $\mathrm{ind}_{IZ}^G \mathrm{Sym}^{k-2} E^2$ of
certain rational polynomial valued functions allowing us to define the (standard) lattice
in the Banach space as the (closure of the) image of the space $\mathrm{ind}_{IZ}^G \mathrm{Sym}^{k-2}{\mathcal O}_E^2$ of
integral polynomial valued functions under this uniformization. Here
$I$ is the Iwahori subgroup of $G = \GL_2(\mathbb{Q}_p)$ and $Z$ is the center of $G$. It then `remained' to compute the reduction of this
lattice. In hindsight, this required several new creative computations. After much effort we were able to compute all
the Jordan-H\"older (JH) factors in the reduction
in terms of certain mod $p$ compactly induced spaces modulo the images of certain Iwahori-Hecke operators. Applying a
reformulation of Breuil's mod $p$ Local Langlands correspondence worked out by Chitrao, which is
called the Iwahori mod $p$ Local Langlands correspondence in \cite{Chi23},
we could then return to the Galois side and compute the reductions of the semi-stable Galois representations $V_{k,\mathcal{L}}$.
Our target to treat all weights in the range $3 \leq k \leq p+1$ was achieved in the paper \cite{CG23}, though we remark that
in principle our method can be used to treat all weights $k \geq 3$ at least in principle (unlike the initial approach with strongly divisible modules).
Indeed, Anand and I have just begun to see whether our approach to computing the reduction of semi-stable representations
using $p$-adic Langlands can be used to recover very recent work of Bergdall, Levin, and Liu \cite{BLL22}
which uses Breuil-Kisin modules
to study the reduction of $V_{k,\mathcal{L}}$ with $\mathcal{L}$-invariant having very negative $p$-adic valuation.

Let us state the main result in the paper \cite{CG23} with Anand Chitrao.
Let $p \geq 5$ be a prime and $E$ be a finite extension of $\mathbb{Q}_p$ containing $\sqrt{p}$.
We describe completely the semi-simplification of the reduction mod $p$ of
the irreducible two-dimensional semi-stable representation $V_{k, \mathcal{L}}$ of $\mathrm{Gal}(\brqp/\mathbb{Q}_p)$
over $E$ with Hodge-Tate weights $(0,k-1)$ for $k \in [3, p + 1]$ and
$\mathcal{L}$-invariant $\mathcal{L} \in E$. To do this
we need a little bit more notation. Let as usual $r = k-2$ so that $1 \leq r \leq p-1$.
 Let $v_-$ and $v_+$ be
 the largest and smallest integers, respectively, such that $v_- < r/2 < v_+$ for $r \geq 1$.
 For $n \geq 1$, let $H_n = \sum_{i = 1}^{n}\frac{1}{i}$ be the $n$-th partial harmonic sum.
 Set $H_0 = 0$ and $H_{\pm} = H_{v_{\pm}}$. Let $v_p$ be the $p$-adic valuation on $\brqp$
 normalized such that $v_p(p) = 1$.
 Let $$\nu = v_p(\mathcal{L} - H_{-} - H_{+})$$
 be the $p$-adic valuation of $\mathcal{L}$ shifted by the
 partial harmonic sums $H_{-}$ and $H_{+}$.
 Everything hinges on the size of the parameter $\nu$. Let $\omega$ be the fundamental character of
 $\mathrm{Gal}(\br{{\mathbb Q}}_p/\mathbb{Q}_p)$ of
 level $1$. Similarly, for an integer $c$ (with
 $p + 1 \nmid c$), let $\omega_2^c$ be an extension from the inertia subgroup $I_{\mathbb{Q}_p}$ to $\mathrm{Gal}(\brqp/\bq_{p^2})$
 of the $c$-th power of the
 fundamental character $\omega_2$ of level $2$ chosen
 so that the (irreducible) representation $\mathrm{ind}\> (\omega_2^c)$ obtained
 by inducing this character from $\mathrm{Gal}(\brqp/\bq_{p^2})$ to $\mathrm{Gal}(\brqp/\mathbb{Q}_p)$ has
 determinant $\omega^c$. Let $\mu_{\lambda}$ be the unramified character of $\mathrm{Gal}(\brqp/\mathbb{Q}_p)$ sending geometric Frobenius to $\lambda \in \br{\mathbb{F}}_p^{*}$. Our main theorem is the following result.

Let us state the main result in the paper \cite{CG23} with Anand Chitrao.
Let $p \geq 5$ be a prime and $E$ be a finite extension of $\qp$ containing $\sqrt{p}$.
We describe completely the semi-simplification of the reduction mod $p$ of
the irreducible two-dimensional semi-stable representation $V_{k, \cL}$ of $\mathrm{Gal}(\brqp/\qp)$
over $E$ with Hodge-Tate weights $(0,k-1)$ for $k \in [3, p + 1]$ and
$\cL$-invariant $\cL \in E$. To do this
we need a little bit more notation. Let as usual $r = k-2$ so that $1 \leq r \leq p-1$.
 Let $v_-$ and $v_+$ be
 the largest and smallest integers, respectively, such that $v_- < r/2 < v_+$ for $r \geq 1$.
 For $n \geq 1$, let $H_n = \sum_{i = 1}^{n}\frac{1}{i}$ be the $n$-th partial harmonic sum.
 Set $H_0 = 0$ and $H_{\pm} = H_{v_{\pm}}$. Let $v_p$ be the $p$-adic valuation on $\brqp$
 normalized such that $v_p(p) = 1$.
 Let $$\nu = v_p(\sL - H_{-} - H_{+})$$
 be the $p$-adic valuation of $\sL$ shifted by the
 partial harmonic sums $H_{-}$ and $H_{+}$.
 Everything hinges on the size of the parameter $\nu$. Let $\omega$ be the fundamental character of
 $\mathrm{Gal}(\br{{\mathbb Q}}_p/\qp)$ of
 level $1$. Similarly, for an integer $c$ (with
 $p + 1 \nmid c$), let $\omega_2^c$ be an extension from the inertia subgroup $I_{\qp}$ to $\mathrm{Gal}(\brqp/\bq_{p^2})$
 of the $c$-th power of the
 fundamental character $\omega_2$ of level $2$ chosen
 so that the (irreducible) representation $\ind (\omega_2^c)$ obtained
 by inducing this character from $\mathrm{Gal}(\brqp/\bq_{p^2})$ to $\mathrm{Gal}(\brqp/\qp)$ has
 determinant $\omega^c$. Let $\mu_{\lambda}$ be the unramified character of $\mathrm{Gal}(\brqp/\qp)$ sending geometric Frobenius to $\lambda \in \br{\mathbb{F}}_p^{*}$. Our main theorem is the following result.

\subsection{Idea of proof of Theorem~\ref{Main theorem in the second part of my thesis}}
A picture is worth a thousand words, and so we draw one to explain the proof.
Let $B_{k,\sL} = \tilde{B}(k,\sL)$.
The following diagram commutes:

\begin{theorem}[Iwahori mod $p$ LLC]\label{Iwahori mod p LLC}
 For $r \in \{0, \ldots, p - 1\}$, $\lambda \in \brFp$ and $\eta: \Qp^* \to \brFp^*$ a smooth
 character, we have the following correspondence between mod
 $p$ representations of $G_{{\mathbb Q}_p}$ and certain smooth mod $p$ representations of $G = \mathrm{GL}_2(\qp)$.
\begin{itemize}
 \item If $\lambda = 0$:
 \[
 (\ind \omega_2^{r + 1}) \otimes \eta \>\> \longmapsto
 \>\> \pi(r,0,\eta) \quad \qquad \qquad \qquad \qquad \quad
 \]
 \item If $\lambda \neq 0$:
 \begin{eqnarray*}
 \> \> \> \> \> (\mu_{\lambda}\omega^{r + 1} \oplus
\mu_{\lambda^{-1}})\otimes \eta & \longmapsto &
 \pi(r,\lambda,\eta)^{\rmss} \oplus \pi([p-3-r],\lambda^{-1},\eta \omega^{r+1})^{\rmss},
\\
 \end{eqnarray*}
where $[a] \in \{0,\ldots,p-2\}$ represents the class of $a$ modulo $(p-1)$.
 \end{itemize}
\end{theorem}

\begin{remark}
For all $i,j \geq 0$, set
$$\alpha_{i,j} = \frac{1}{p^{l(i)}} \sum_{\zeta \in \mu_{p^{l(i)}}}\zeta^{-i} (\zeta - 1)^j$$
\vspace{1cm}
Using some algebraic number theoretic arguments (see \cite[Lemma I.3.5]{Col10a}), it can be shown that
$$\begin{cases}
 \alpha_{i,j} =0 & \text{if } j < i \\
 \alpha_{i,j} = 1 & \text{if } j = i \\
 v_p(\alpha_{i,j}) \geq \left\lfloor \dfrac{j - p^{l(i) -1}}{p^{l(i)} - p^{l(i) -1}} \right\rfloor & \text{if } j > i.
 \end{cases}
 $$
Now
$${\mathbbm 1}_{i+p^{l(i)}\zp}(x) = \frac{1}{p^{l(i)}} \sum_{\zeta \in \mu_{p^{l(i)}}} \zeta^{x-i}
 = \sum_{n = 0}^\infty {x \choose n} \sum_{\zeta \in \mu_{p^{l(i)}}} \frac{1}{p^{l(i)}} \zeta^{-i} (\zeta -1)^n$$
 so its Mahler coefficients $a_n({\mathbbm 1}_{i+p^{l(i)}\zp})$ equal $\alpha_{i,n}$, which clearly tend to $0$ as $n \to \infty$.
 This gives another proof of the surjectivity of the map in Theorem~\ref{mahler}, ii). Indeed considering
 that map to be taking values in the bigger space $l_\infty({\mathbb N}_{\geq 0},E)$, we note that the pre-image $B$
 of the closed subspace $l_\infty^0({\mathbb N}_{\geq 0}, E)$ is closed and, by the above remarks and
 Proposition~\ref{LC basis}, part ii), contains
 $\mathrm{LC}(\zp,E)$. Then $B = \overline{B} \supset \overline{\mathrm{LC}(\zp,E)} = \sC^0(\zp, E)$,
 by Lemma~\ref{LC dense in C^0}. So $B = \sC^0(\zp, E)$.
\end{remark}

\begin{theorem}{\cite[Theorem 9.7]{CG23}} \label{Final theorem for leq}
 Let $i = 1, \> 2, \> \ldots, \> \lceil r/2 \rceil - 1$. If $\nu = i - r/2$, then the map $\IZind a^i d^{r - i} \twoheadrightarrow F_{2i, \> 2i + 1}$ factors as
 \[
 \IZind a^i d^{r - i} \twoheadrightarrow \frac{\IZind a^i d^{r - i}}{\im(T_{1, 2} - \lambda_i)} \twoheadrightarrow F_{2i, \> 2i + 1},
 \]
 where $$\lambda_i = (-1)^i i {r - i + 1 \choose i}p^{r/2 - i}\cL.$$ Moreover, the second map in the display above induces a surjection
 \[
 \pi(r - 2i, \lambda_i, \omega^i) \twoheadrightarrow F_{2i, \> 2i + 1}.
 \]
 \end{theorem}
 \begin{proof}
 All congruences in this proof are in the space $\br{\latticeL{k}}$ modulo the image of the subspace $\IZind \oplus_{j < r - i}\Fq X^{r - j}Y^j$ under $\IZind \SymF{k - 2} \twoheadrightarrow \br{\latticeL{k}}$.

\subsection{Reduction mod $p$ of $\latticeL{k}$}
 \label{Section containing the proof of the main theorem}
 In this section, we summarize all the results proved in \cite{CG23} (such as Theorem~\ref{Final theorem for leq} above)
 and mention how they are used to prove Theorem \ref{Main theorem in the second part of my thesis}.
 Recall that $$\nu = v_p(\cL - H_{-} - H_{+}).$$
\begin{theorem}{\cite[Theorem 12.1]{CG23}}\label{Main final theorem in the second part of my thesis}
 For $3 \leq k \leq p + 1$ and $p \geq 5$, the semi-simplification of the reduction $\br{V}_{k,\sL}$ of
 the semi-stable representation $V_{k,\sL}$ of $G_{\Qp}$ of Hodge-Tate weights $(0,k-1)$ and $\sL$-invariant $\sL$
 satisfies:
 \[
 \br{V}_{k, \cL} \sim
 \begin{cases}
 \ind (\omega_2^{r+1 + (i-1)(p-1)}), & \text{ if $(i-1) - r/2 < \nu < i -r/2$} \\
 \mu_{\lambda_i}\omega^{r+1-i} \oplus \mu_{\lambda_i^{-1}}\omega^{i}, & \text{ if $\nu = i -r/2$},
 \end{cases}
 \]
 where $1 \leq i \leq \dfrac{r+1}{2}$ if $r$ is odd and $1 \leq i \leq \dfrac{r+2}{2}$ if $r$ is even. The constants $\lambda_i$ are determined by
 \begin{eqnarray*}
 \lambda_i & = & \br{(-1)^i \> i {r+1-i \choose i}p^{r/2-i}(\cL - H_{-} - H_{+})}, \quad \text{ if } 1 \leq i < \dfrac{r + 1}{2} \\
 \lambda_{i} + \lambda_i^{-1} & = & \br{(-1)^i \> i{r + 1 - i \choose i}p^{r/2 - i}(\cL - H_{-} - H_{+})}, \quad \text{ if } i = \dfrac{r + 1}{2} \text{ and } r \text{ is odd}.
 \end{eqnarray*}
 We follow the conventions stated in the Introduction.
\end{theorem}
\begin{proof}
 We collect the necessary results proved in \cite{CG23} here.
 We first state the common results for odd and even weights (the fourth of which was
 sketched in Theorem \ref{Final theorem for leq}):
 \begin{enumerate}
 \item For $i = 1, 2, \ldots, \lceil r/2 \rceil - 1$ if $\nu > i - r/2$, then $F_{2i - 2, \> 2i - 1} = 0$.
 \item For $i = 1, 2, \ldots, \lceil r/2 \rceil - 1$ if $\nu = i - r/2$, then $\pi([2i - 2 - r], \lambda_i^{-1}, \omega^{r - i + 1}) \twoheadrightarrow F_{2i - 2, \> 2i - 1}$.
 \item For $i = 1, 2, \ldots, \lceil r/2 \rceil - 1$ if $\nu < i - r/2$, then $F_{2i, \> 2i + 1} = 0$.
 \item For $i = 1, 2, \ldots, \lceil r/2 \rceil - 1$ if $\nu = i - r/2$, then $\pi(r - 2i, \lambda_i, \omega^{i}) \twoheadrightarrow F_{2i, \> 2i + 1}$.
 \end{enumerate}
 Next, we state the extra results for odd weights around the point $\nu = \frac{1}{2}$.
 \begin{enumerate}
 \item[5.] If $\nu \geq 0.5$, then $\pi(p - 2, \lambda_{\frac{r + 1}{2}}, \omega^{\frac{r + 1}{2}}) \oplus \pi(p - 2, \lambda_{\frac{r + 1}{2}}^{-1}, \omega^{\frac{r + 1}{2}}) \twoheadrightarrow F_{r - 1, \> r}$.
 \item[6.] If $-0.5 < \nu < 0.5$, then
 $\dfrac{\IZind a^{\frac{r - 1}{2}}d^{\frac{r + 1}{2}}}{\im T_{-1, 0}} \twoheadrightarrow F_{r-1, \> r}$.

\noindent We remark that we have adopted
  the following conventions in the statement of the theorem:
   \begin{itemize}
    \item The first interval $- r/2 < \nu < 1 - r/2$ is interpreted as $\nu < 1 - r/2$.
    \item If $r$ is odd, then the last case $\nu = 1/2$ should be interpreted as $\nu \geq 1/2$. If $r$ is even,
          then the interval $0 < \nu < 1$ should be interpreted as $\nu > 0$ and we drop the case $\nu = 1$.
    \end{itemize}
In any case, the theorem says that the reduction $\overline{V}_{k,\mathcal{L}}$ varies through an alternating sequence of
irreducible and reducible mod $p$
representations as $\nu$ varies through finitely many marked points.

\subsection{Idea of proof of Theorem~\ref{Main theorem in the second part of my thesis}}
A picture is worth a thousand words, and so we draw one to explain the proof.
Let $B_{k,\mathcal{L}} = \tilde{B}(k,\mathcal{L})$.
The following diagram commutes:

 \[
\begin{tikzcd}
  V_{k,\mathcal{L}} \arrow[rrr, mapsto, "p-\text{adic LLC}"]  \arrow[dd, mapsto] & & & B_{k,\mathcal{L}} \arrow[dd, mapsto] \\
   & &\\
  \overline{V}_{k,\mathcal{L}} \arrow[rrr, mapsto, "\text{Iwahori mod $p$ LLC}"]  &  & & \overline{B}_{k,\mathcal{L}}
\end{tikzcd}
\]
where the vertical maps are (semi-simplifications of) reductions of lattices in the corresponding spaces in the
top row.
One is trying to compute the left vertical map. But one computes instead the  right vertical
map since the bottom map is injective.

We now give a broad outline of the remaining text in terms of this picture.
The paper is broken into four further sections.
The bottom map is explained in $\S 2$. The top right corner is explained in $\S 4$ based on
the foundational material described in $\S 3$. The computation of the right vertical map is then
explained in $\S 5$.

\subsection{Notation}
\begin{itemize}
        \item $p \geq 5$ is a prime.
        \item $E$ is a finite extension of $\mathbb{Q}_p$ containing $\sqrt{p}$ and $\mathcal{L}$.
              $\co_E$ is the ring of integers in $E$ with a uniformizer $\pi = \pi_E$ and residue field
$\mathbb{F}_q$. Note $\sqrt{p} \equiv 0 \mod \pi$.
        \item $k$ denotes the weight of a semi-stable
representation and $r = k - 2$.
        \item $v_-$, $v_+$ are the largest, smallest integers,
respectively, such that $v_- < r/2 < v_+$ for $r \geq 1$.
        \item For $n \geq 1$, $H_n = \sum\limits_{i = 1}^{n}\frac{1}{i}$
and $H_0 = 0, H_{\pm} = H_{v_{\pm}}$.
        \item $v_p$ is the $p$-adic valuation normalized such that $v_p(p)
= 1$.
        \item $\nu = v_p(\mathcal{L} - H_{-} - H_{+})$ is the valuation
of $\mathcal{L}$ shifted by the partial
harmonic sums $H_{-}$ and $H_{+}$.
        \item $I_{\mathbb{Q}_p}$ is the inertia subgroup of $\mathrm{Gal}(\brqp/\mathbb{Q}_p)$.
        \item $\omega$ is the fundamental character of $I_{\mathbb{Q}_p}$ of level
$1$;
        it has a canonical extension to $\mathrm{Gal}(\br{{\mathbb
Q}}_p/\mathbb{Q}_p)$.
        \item $\omega_2$ is the fundamental character of $I_{\mathbb{Q}_p}$ of
level $2$; for an integer $c$ with $p
+ 1 \nmid c$,
        choose an extension of $\omega_2^c$ to
$\mathrm{Gal}(\brqp/\bq_{p^2})$ so that the irreducible representation
        $\mathrm{ind}(\omega_2^c)$ obtained
        by inducing this extension from $\mathrm{Gal}(\brqp/\bq_{p^2})$ to
$\mathrm{Gal}(\brqp/\mathbb{Q}_p)$ has determinant $\omega^c$.
        \item $G$ is the group $\mathrm{GL}_2(\mathbb{Q}_p)$.
        \item $K$ is the maximal compact subgroup $\mathrm{GL}_2(\mathbb{Z}_p)$ of $G$.
        \item $I$ is the Iwahori subgroup of $G$ consisting of matrices in $K$
that are upper triangular $\!\!\!\!\mod p$.
        \item $B$ is the Borel subgroup of $G$ consisting of upper
triangular matrices.
        \item $\alpha = \begin{pmatrix}1 & 0 \\ 0 & p\end{pmatrix}$,
$\beta = \begin{pmatrix}0 & 1 \\ p & 0\end{pmatrix}$ and $w =
\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}$. Note that $\beta =
\alpha w$.
        \item $I_n = \{[a_0] + [a_1]p + \cdots + [a_{n - 1}]p^{n - 1}
\> \vert \> a_i \in \mathbb{F}_p\}$ for $n \geq 1$, where $[\quad]$ denotes Teichm\"uller representative. $I_0 = \{0\}$.
        \item $V_{r} = \mathrm{Sym}^{r}\Fq^2$ and         $\SymE{k - 2} := \vert \det \vert^{\frac{k - 2}{2}} \otimes
\mathrm{Sym}^{k - 2}E^2$ for $k \geq 2$.
        \item $d^r$ for an integer $r$ denotes the character $IZ \to
\mathbb{F}_p^*$ which sends $\begin{pmatrix}a & b \\ c & d\end{pmatrix} \in
I$ to $d^r \!\!\! \mod p$ and which is trivial on the scalar
matrix $p$.
        \item For a representation $(\rho, V)$ of $IZ$ over $E$ or $\mathbb{F}_q$,
let $\mathrm{ind}_{IZ}^G\> \rho$ be the vector space of functions $f : G \to V$
that are compactly supported modulo $IZ$ such that $f(hg) =
\rho(h)
        \cdot f(g)$, for all $h \in IZ$ and $g \in G$. The vector space
$\mathrm{ind}_{IZ}^G\> \rho$ has a $G$ action:
        $g \cdot f(g') = f(g' g)$, for all $g, g' \in G$ and $f \in \mathrm{ind}_{IZ}^G\>
\rho$. For $g \in G$ and $v \in V$, define the function $\llbracket g, v \rrbracket
\in \mathrm{ind}_{IZ}^G\> \rho$ by
    \[
        \llbracket g, v\rrbracket(g') =
        \begin{cases}
            \rho(g'g) \cdot v, & \text{ if }g'g \in IZ \\
            0, & \text{ otherwise}.
        \end{cases}
    \]
        \item Let $(\mathrm{ind}_B^G \> E)^{\mathrm{smooth}}$ be the $E$-vector space of locally
constant functions from $G$ to $E$, with the action of $G$ given
by $g \cdot f(g') = f(g'g)$ for any $g, g' \in G$ and $f \in
(\mathrm{ind}_B^G \> E)^{\mathrm{smooth}}$.
        \item Let $\mathrm{St}$ be the Steinberg representation of $G$ over $E$,
i.e., $\mathrm{St}$ is the vector space of all locally constant functions
$H : \mathbb{P}^1(\bQ_p) \to E$ modulo constant functions with the action
of $G$ given by $\left(\begin{pmatrix} a & b \\ c & d
\end{pmatrix}\cdot H\right)(z) = H\left(\dfrac{az + c}{bz +
d}\right)$.
        \item $[a] \in \{0, \ldots, p - 2\}$ denotes the class of $a \!\!
\mod p - 1$.
        \item $\delta_{a, b} = 1$ if $a = b$ and is $0$ otherwise.
\end{itemize}

To prove Theorem~\ref{Main theorem in the second part of my thesis}, the text explains it via a commutative diagram
\[
\begin{tikzcd}
  V_{k,\mathcal{L}} \arrow[rrr, mapsto, "p-\text{adic LLC}"]  \arrow[dd, mapsto] & & & B_{k,\mathcal{L}} \arrow[dd, mapsto] \\
   & &\\
  \overline{V}_{k,\mathcal{L}} \arrow[rrr, mapsto, "\text{Iwahori mod $p$ LLC}"]  &  & & \overline{B}_{k,\mathcal{L}}
\end{tikzcd}
\]
where “the vertical maps are (semi-simplifications of) reductions of lattices in the corresponding spaces in the top row.” The goal is “to compute the left vertical map,” i.e. the reduction of $V_{k,\mathcal{L}}$, but the strategy is to “compute instead the right vertical map since the bottom map is injective.” Concretely, the outline given is organized by sections: “The bottom map is explained in \S 2. The top right corner is explained in \S 4 based on the foundational material described in \S 3. The computation of the right vertical map is then explained in \S 5.”

To prove the main theorem, the text explains it via a commutative diagram
\[
\begin{tikzcd}
  V_{k,\mathcal{L}} \arrow[rrr, mapsto, "p-\text{adic LLC}"]  \arrow[dd, mapsto] & & & B_{k,\mathcal{L}} \arrow[dd, mapsto] \\
   & &\\
  \overline{V}_{k,\mathcal{L}} \arrow[rrr, mapsto, "\text{Iwahori mod $p$ LLC}"]  &  & & \overline{B}_{k,\mathcal{L}}
\end{tikzcd}
\]
where “the vertical maps are (semi-simplifications of) reductions of lattices in the corresponding spaces in the top row.” The goal is “to compute the left vertical map,” i.e. the reduction of $V_{k,\mathcal{L}}$, but the strategy is to “compute instead the right vertical map since the bottom map is injective.” Concretely, the outline is organized as follows: first the bottom map is explained; next the top right corner is explained based on earlier foundational material; finally, the computation of the right vertical map is carried out.

Question: Let p ≥ 5 and 3 ≤ k ≤ p + 1, and set r = k − 2. Let V_{k,𝓛} be the irreducible two-dimensional semi-stable representation of Gal(Q̄_p/Q_p) with Hodge–Tate weights (0, k−1) and 𝓛-invariant 𝓛. Let v₋ and v₊ be the adjacent integers satisfying v₋ < r/2 < v₊, define H_n = ∑_{j=1}^n 1/j for n ≥ 1 with H_0 = 0, and put H₋ = H_{v₋}, H₊ = H_{v₊}, and ν = v_p(𝓛 − H₋ − H₊). Here ω is the mod p cyclotomic character, ω₂ is the fundamental character of niveau 2, μ_λ is the unramified character sending Frobenius to λ, ind(ω₂^m) denotes the corresponding irreducible induced two-dimensional representation, and an overline on a scalar denotes reduction mod p. Which explicit formula describes the semisimplification of the mod p reduction V̄_{k,𝓛}?

For 1 ≤ i ≤ (r+1)/2 if r is odd, and for 1 ≤ i ≤ (r+2)/2 if r is even, one has
V̄_{k,𝓛}^ss ∼ ind(ω₂^{r+1+(i−1)(p−1)}) if (i−1) − r/2 < ν < i − r/2,
and
V̄_{k,𝓛}^ss ∼ μ_{λ_i}ω^{r+1−i} ⊕ μ_{λ_i^{-1}}ω^i if ν = i − r/2,
where the mod p constants λ_i are determined by
λ_i = overline{(-1)^i · i · binom(r+1−i, i) · (𝓛 − H₋ − H₊)/p^{i−r/2}} for 1 ≤ i < (r+1)/2,
and, when i = (r+1)/2 and r is odd,
λ_i + λ_i^{-1} = overline{(-1)^i · i · binom(r+1−i, i) · (𝓛 − H₋ − H₊)/p^{i−r/2}}.

Choices:
B.
For 1 \leq i \leq (r+1)/2 if r is odd, and for 1 \leq i \leq (r+2)/2 if r is even, one has
\overline{V}_{k,\mathcal{L}}^{ss} \sim \operatorname{ind}(\omega_2^{r+1+i(p-1)}) if (i-1)-r/2 < \nu < i-r/2,
and
\overline{V}_{k,\mathcal{L}}^{ss} \sim \mu_{\lambda_i}\omega^{r+1-i} \oplus \mu_{\lambda_i^{-1}}\omega^i if \nu = i-r/2,
where the mod p constants \lambda_i are determined by
\lambda_i = \overline{(-1)^i \cdot i \cdot \binom{r+1-i}{i} \cdot (\mathcal{L}-H_- -H_+)/p^{i-r/2}} for 1 \le i < (r+1)/2,
and, when i=(r+1)/2 and r is odd,
\lambda_i+\lambda_i^{-1}=\overline{(-1)^i \cdot i \cdot \binom{r+1-i}{i} \cdot (\mathcal{L}-H_- -H_+)/p^{i-r/2}}.
C.
For 1 \leq i \leq (r+1)/2 if r is odd, and for 1 \leq i \leq (r+2)/2 if r is even, whenever \nu = i-r/2 one has
\overline{V}_{k,\mathcal{L}}^{ss} \sim \mu_{\lambda_i}\omega^{r+1-i} \oplus \mu_{\lambda_i^{-1}}\omega^i,
where the mod p constants \lambda_i are determined by
\lambda_i = \overline{(-1)^i \cdot i \cdot \binom{r+1-i}{i} \cdot (\mathcal{L}-H_- -H_+)/p^{i-r/2}} for 1 \le i < (r+1)/2,
and, when i=(r+1)/2 and r is odd,
\lambda_i+\lambda_i^{-1}=\overline{(-1)^i \cdot i \cdot \binom{r+1-i}{i} \cdot (\mathcal{L}-H_- -H_+)/p^{i-r/2}}.
D.
For 1 \leq i \leq (r+1)/2 if r is odd, and for 1 \leq i \leq (r+2)/2 if r is even, one has
\overline{V}_{k,\mathcal{L}}^{ss} \sim \operatorname{ind}(\omega_2^{r+1+(i-1)(p-1)}) if (i-1)-r/2 \le \nu \le i-r/2,
and
\overline{V}_{k,\mathcal{L}}^{ss} \sim \mu_{\lambda_i}\omega^{r+1-i} \oplus \mu_{\lambda_i^{-1}}\omega^i if \nu = i-r/2,
where the mod p constants \lambda_i are determined by
\lambda_i = \overline{(-1)^i \cdot i \cdot \binom{r+1-i}{i} \cdot (\mathcal{L}-H_- -H_+)/p^{i-r/2}} for 1 \le i < (r+1)/2,
and, when i=(r+1)/2 and r is odd,
\lambda_i+\lambda_i^{-1}=\overline{(-1)^i \cdot i \cdot \binom{r+1-i}{i} \cdot (\mathcal{L}-H_- -H_+)/p^{i-r/2}}.
E.
For 1 \leq i \leq (r+1)/2 if r is odd, and for 1 \leq i \leq (r+2)/2 if r is even, one has
\overline{V}_{k,\mathcal{L}}^{ss} \sim \operatorname{ind}(\omega_2^{r+1+(i-1)(p-1)}) if (i-1)-r/2 < \nu < i-r/2,
and
\overline{V}_{k,\mathcal{L}}^{ss} \sim \mu_{\lambda_i}\omega^{r+1-i} \oplus \mu_{\lambda_i^{-1}}\omega^i if \nu = i-r/2,
where the mod p constants \lambda_i are determined uniformly for all i by
\lambda_i+\lambda_i^{-1}=\overline{(-1)^i \cdot i \cdot \binom{r+1-i}{i} \cdot (\mathcal{L}-H_- -H_+)/p^{i-r/2}}.

dropped the irreducible-interval case for strict inequalities in \nu

strict boundary separation between interval and equality cases

special odd midpoint formula uses \lambda_i+\lambda_i^{-1} only at i=(r+1)/2

The stem sets notation and asks for the explicit reduction formula, but it does not explicitly reveal the correct interval/equality case split, exponent pattern, or the special midpoint condition on the \(\lambda_i\).

This is essentially a verbatim theorem-recall item: the student is asked to identify the exact explicit formula of a stated result rather than derive or apply it in a new situation.

There is some reasoning needed to distinguish subtle variants (shifted exponent, weak omission of cases, boundary inequalities, midpoint formula), but the task is still mainly exact recognition/recall rather than genuine generative mathematical reasoning.

The distractors are highly plausible and target realistic failure modes: index shifts, boundary-condition mistakes, omission of the strict-inequality case, and confusion about when the \(\lambda_i+\lambda_i^{-1}\) condition applies.

Technically well-constructed with strong distractors and little answer leakage, but fundamentally a theorem-recall/restatement question rather than a non-tautological reasoning problem.