[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 $p \ge 5$ be a prime, let $k$ be an integer with $3 \le k \le p+1$, and set $r=k-2$ (so $1 \le r \le p-1$). Let $V_{k,\mathcal L}$ be the two-dimensional semi-stable representation of $G_{\mathbb Q_p}=\operatorname{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p)$ with Hodge--Tate weights $(0,k-1)$ and $\mathcal L$-invariant $\mathcal L$. Define $v_-$ and $v_+$ to be the largest and smallest integers, respectively, such that $v_-<r/2<v_+$; for $n\ge1$ let $H_n=\sum_{j=1}^n \frac1j$, set $H_0=0$, and put $H_-=H_{v_-}$ and $H_+=H_{v_+}$. Let $\nu=v_p(\mathcal L-H_- -H_+)$, where $v_p(p)=1$. Here $\omega$ denotes the mod $p$ cyclotomic character, $\omega_2$ a fundamental character of niveau $2$, $\mu_\lambda$ the unramified character sending Frobenius to $\lambda$, $\operatorname{ind}(\omega_2^m)$ the corresponding irreducible induced two-dimensional mod $p$ representation, and $\overline{x}$ means reduction mod $p$. Which statement holds for the semisimplified mod $p$ reduction of every such $V_{k,\mathcal L}$?

The semisimplified reduction satisfies
\[
\overline{V}_{k,\mathcal L}\sim
\begin{cases}
\operatorname{ind}(\omega_2^{\,r+1+(i-1)(p-1)}), & \text{if } (i-1)-r/2<\nu<i-r/2,\\[4pt]
\mu_{\lambda_i}\omega^{r+1-i}\oplus \mu_{\lambda_i^{-1}}\omega^{i}, & \text{if } \nu=i-r/2,
\end{cases}
\]
where
\[
1\le i\le \frac{r+1}{2}\ \text{if $r$ is odd},\qquad 1\le i\le \frac{r+2}{2}\ \text{if $r$ is even},
\]
and the mod $p$ constants $\lambda_i$ are given by
\[
\lambda_i=\overline{(-1)^i\, i\binom{r+1-i}{i}\frac{\mathcal L-H_- -H_+}{p^{\,i-r/2}}},\qquad \text{if }1\le i<\frac{r+1}{2},
\]
while, if $r$ is odd and $i=\frac{r+1}{2}$,
\[
\lambda_i+\lambda_i^{-1}=\overline{(-1)^i\, i\binom{r+1-i}{i}\frac{\mathcal L-H_- -H_+}{p^{\,i-r/2}}}.
\]

Choices:
B.
The semisimplified reduction satisfies
\[
\overline{V}_{k,\mathcal L}\sim
\begin{cases}
\operatorname{ind}(\omega_2^{\,r+1+i(p-1)}), & \text{if } (i-1)-r/2<\nu<i-r/2,\\[4pt]
\mu_{\lambda_i}\omega^{r+1-i}\oplus \mu_{\lambda_i^{-1}}\omega^{i}, & \text{if } \nu=i-r/2,
\end{cases}
\]
where
\[
1\le i\le \frac{r+1}{2}\ \text{if $r$ is odd},\qquad 1\le i\le \frac{r+2}{2}\ \text{if $r$ is even},
\]
and the mod $p$ constants $\lambda_i$ are given by
\[
\lambda_i=\overline{(-1)^i\, i\binom{r+1-i}{i}\frac{\mathcal L-H_- -H_+}{p^{\,i-r/2}}},\qquad \text{if }1\le i<\frac{r+1}{2},
\]
while, if $r$ is odd and $i=\frac{r+1}{2}$,
\[
\lambda_i+\lambda_i^{-1}=\overline{(-1)^i\, i\binom{r+1-i}{i}\frac{\mathcal L-H_- -H_+}{p^{\,i-r/2}}}.
\]
C.
Whenever \(\nu=i-r/2\) for some index in the range
\[
1\le i\le \frac{r+1}{2}\ \text{if $r$ is odd},\qquad 1\le i\le \frac{r+2}{2}\ \text{if $r$ is even},
\]
one has a reducible semisimplified reduction of the form
\[
\overline{V}_{k,\mathcal L}\sim \mu_{\lambda_i}\omega^{r+1-i}\oplus \mu_{\lambda_i^{-1}}\omega^{i},
\]
with the endpoint case for odd \(r\) determined by
\[
\lambda_i+\lambda_i^{-1}=\overline{(-1)^i\, i\binom{r+1-i}{i}\frac{\mathcal L-H_- -H_+}{p^{\,i-r/2}}}.
\]
D.
The semisimplified reduction satisfies
\[
\overline{V}_{k,\mathcal L}\sim
\begin{cases}
\operatorname{ind}(\omega_2^{\,r+1+(i-1)(p-1)}), & \text{if } (i-1)-r/2\le \nu\le i-r/2,\\[4pt]
\mu_{\lambda_i}\omega^{r+1-i}\oplus \mu_{\lambda_i^{-1}}\omega^{i}, & \text{if } \nu=i-r/2,
\end{cases}
\]
where
\[
1\le i\le \frac{r+1}{2}\ \text{if $r$ is odd},\qquad 1\le i\le \frac{r+2}{2}\ \text{if $r$ is even},
\]
and the mod $p$ constants $\lambda_i$ are given by
\[
\lambda_i=\overline{(-1)^i\, i\binom{r+1-i}{i}\frac{\mathcal L-H_- -H_+}{p^{\,i-r/2}}},\qquad \text{if }1\le i<\frac{r+1}{2},
\]
while, if $r$ is odd and $i=\frac{r+1}{2}$,
\[
\lambda_i+\lambda_i^{-1}=\overline{(-1)^i\, i\binom{r+1-i}{i}\frac{\mathcal L-H_- -H_+}{p^{\,i-r/2}}}.
\]
E.
The semisimplified reduction satisfies
\[
\overline{V}_{k,\mathcal L}\sim
\begin{cases}
\operatorname{ind}(\omega_2^{\,r+1+(i-1)(p-1)}), & \text{if } (i-1)-r/2<\nu<i-r/2,\\[4pt]
\mu_{\lambda_i}\omega^{r+1-i}\oplus \mu_{\lambda_i^{-1}}\omega^{i}, & \text{if } \nu=i-r/2,
\end{cases}
\]
where
\[
1\le i\le \frac{r+1}{2}\ \text{if $r$ is odd},\qquad 1\le i\le \frac{r}{2}\ \text{if $r$ is even},
\]
and the mod $p$ constants $\lambda_i$ are given by
\[
\lambda_i=\overline{(-1)^i\, i\binom{r+1-i}{i}\frac{\mathcal L-H_- -H_+}{p^{\,i-r/2}}},\qquad \text{if }1\le i<\frac{r+1}{2},
\]
while, if $r$ is odd and $i=\frac{r+1}{2}$,
\[
\lambda_i+\lambda_i^{-1}=\overline{(-1)^i\, i\binom{r+1-i}{i}\frac{\mathcal L-H_- -H_+}{p^{\,i-r/2}}}.
\]

dropped the open-interval irreducible case and retained only the boundary reducible case

strict versus non-strict boundary conditions in the \(\nu\)-intervals

The stem provides definitions and notation but does not state or strongly hint at the correct case split, exponent indexing, or boundary conditions. The answer is not leaked.

This is essentially a theorem-recall item: the correct option is the full theorem statement, and the task is to identify the exact restatement among near-variants rather than derive a new conclusion.

There is some reasoning pressure because the options differ in subtle but mathematically meaningful ways (index shift, strict vs. non-strict inequalities, endpoint range, weaker true statement). However, success depends more on precise recall/recognition than on generating a conclusion from the premises.

The distractors are strong: they are plausible, distinct, and reflect realistic mathematical errors such as off-by-one indexing, boundary-condition mistakes, incomplete statements, and incorrect endpoint ranges.

A technically well-constructed but theorem-recall-heavy MCQ. It avoids answer leakage and has strong distractors, but it is largely a direct restatement rather than a genuinely generative reasoning task.


\begin{theorem}\label{main theorem}
 	Let $q \ge 3$ be fixed. Uniformly for
 $
 	\tau \le \log n + \log_2 n - \theta(n),
 	$
 	where $2\leq \theta(n)\ll \log_3 n$ is any function tending arbitrarily slowly to infinity as $n \to \infty$, we have
 	\[
 	\phi_n(\tau)
 	:= \frac{1}{|\mathcal{H}_n|}
 	\sum_{\substack{D \in \mathcal{H}_n \\ L(1, \chi_D) \ge e^{\gamma}\tau}} 1
 	= \exp\!\left(
 	-\,C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)
 	\,\frac{q^{\,\tau - C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)}}{\tau}
 	\bigl(1 + o_\theta(1)\bigr)
 	\right),
 	\]
 	where the constants $C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)$ and $C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)$ depend on $\tau$, $q$, and $\kappa(\tau)$, yet remain bounded as the argument varies between $1$ and $q$.
 	Further details on these quantities can be found in~\cite[Theorem~1.3]{Lumley}.
 \end{theorem}

\begin{lemma}\label{character sum over nonsquare}
	Let $\ell=\ell_1 \ell_2^2\neq \square\in \mathbb{F}_q[t]$ with $\ell_1$ square-free. For any $\varepsilon>0$, we have
	\[
	\sum_{D\in \mathcal{H}_n}\chi_D(\ell)\ll  q^{(1/2+\varepsilon)n}\exp\left((d(\ell_1))^{1-\varepsilon}\right)\prod_{P\mid \ell_2}\left(1+|P|^{-1/2-\varepsilon}\right).
	\]

\end{lemma}

One of the central problems in analytic number theory is to determine the value distribution and obtain sharp bounds for the maximal size of Dirichlet $L$-functions on the line $\operatorname{Re}(s)=1$. These quantities are deeply intertwined with fundamental arithmetic invariants such as class numbers of number fields, and are closely linked to estimates for character sums via Fourier analysis. In particular, the tail behavior of the value distribution plays a decisive role in determining the true order of magnitude of extreme values.

Over the past few decades, significant progress has been made toward understanding the tails of these distributions and the maximal size of $L$-functions on $\operatorname{Re}(s)=1$. This line of research encompasses the Riemann zeta function and Dirichlet $L$-functions in the modulus aspect (see~\cite{AMM, AMMP, BS3, DOKIC, GSzeta}), as well as the family of quadratic Dirichlet $L$-functions $L(1,\chi_d)$ (see~\cite{DM, GS, Lumley}), and more generally, various families of automorphic $L$-functions (see~\cite{Lamzouri}).

An important $\mathrm{GL}(1)$ family of symplectic type is formed by the quadratic Dirichlet $L$-functions $L(s,\chi_d)$, where $d$ ranges over fundamental discriminants. The analytic study of this family is notably challenging due to the lack of a natural orthogonality relation among the characters. The tail behavior of $L(1,\chi_d)$ is closely tied to the distribution of class numbers of quadratic number fields. In this direction, Granville and Soundararajan~\cite{GS} carried out a seminal analysis of these quantities, and following their approach, Dahl and Lamzouri~\cite{DL} investigated an interesting subfamily, known as the \emph{Chowla family}, corresponding to real quadratic fields.

\subsection{Distribution of $L$-functions in the hyperelliptic ensemble}\label{large values section}
We now move our discussion to the setup of $L$-functions over function fields, a topic of interest in modern research in number theory; see, e.g., \cite{AT, AK, BF, BF2, DFL, DL, FR, Floreanegative, Florea, Jung, Lumley2}.
This includes investigations of various statistical properties of these $L$-functions, such as moments, non-vanishing, and value distribution.

Before we enunciate the main theorem of this paper, we need to present
 some basic notations on function fields.
 Let $\mathbb{F}_{q}$ be a finite field of odd cardinality and $\mathbb{F}_{q}[t]$ be the polynomial ring over $\mathbb{F}_{q}$ in variable $t$. Let $D\in \mathbb{F}_q[t]$ be a monic square-free polynomial. The quadratic character $\chi_D$ attached to $D$ is defined using quadratic residue symbol for $\mathbb{F}_{q}[t]$ by $\chi_{D}(f)=\left(\frac{f}{D}\right)$ and the corresponding Dirichlet $L$-function is denoted by $L(s, \chi_D)$.
 Let $\mathcal{H}_n$
 be the family of all curves given in affine form by $C_P: y^2=D(t)$, where $D$ be a monic square-free polynomial of degree $n$. These curve are non-singular of genus $g$ given by \eqref{lambda} and \eqref{genus}.

In~\cite{Lumley}, Lumley studied the distribution of $L(1,\chi_D)$ as $D$ ranges over $\mathcal{H}_n$.
 The following theorem strengthens the range of uniformity for the tail of the distribution of $L(1,\chi_D)$ established in~\cite[Theorem~1.2]{Lumley}.
Throughout, $\log$ denotes the logarithm to base $q$, while $\ln$ is the natural logarithm, and $\log_j$ (respectively $\ln_j$) for the $j$-fold iterated logarithm.

\begin{align}\label{genus}
2g=d(D)-1-\lambda
\end{align}

\begin{align}\label{lambda}
\lambda=
\left\{
\begin{array}
[c]{ll}
1, & \;\text{if\, $d(D)$ even}, \\
0, &\; \text{if\, $d(D)$ odd},
\end{array}
\right.
\end{align}

Before we enunciate the main theorem of this paper, we need to present
 some basic notations on function fields.
 Let $\mathbb{F}_{q}$ be a finite field of odd cardinality and $\mathbb{F}_{q}[t]$ be the polynomial ring over $\mathbb{F}_{q}$ in variable $t$. Let $D\in \mathbb{F}_q[t]$ be a monic square-free polynomial. The quadratic character $\chi_D$ attached to $D$ is defined using quadratic residue symbol for $\mathbb{F}_{q}[t]$ by $\chi_{D}(f)=\left(\frac{f}{D}\right)$ and the corresponding Dirichlet $L$-function is denoted by $L(s, \chi_D)$.
 Let $\mathcal{H}_n$
 be the family of all curves given in affine form by $C_P: y^2=D(t)$, where $D$ be a monic square-free polynomial of degree $n$. These curve are non-singular of genus $g$ given by \eqref{lambda} and \eqref{genus}.

\subsection{Truncating the Euler product of $L(1, \chi_D)$}
We show that $L(1, \chi_D)$ can be well approximated by its short Euler product for almost all $D \in \mathcal{H}_n$. Let
\begin{align}\label{short euler product}
L(1,\chi_D; M):= \prod_{d(P)\le M}\left(1-\frac{\chi_D(P)}{|P|}\right)^{-1}=\sum_{\substack{f\in \mathcal{M}\\ P\mid f \implies d(P)\leq M}}\frac{\chi_D(f)}{|f|}.
\end{align}
 Let $M:=3\log n$. From \cite[Lemma 2.2]{Lumley}, for $D\in \mathcal{H}_n$, we have
 \begin{align}\label{asym for L(1)}
 L(1,\chi_{D})=L(1,\chi_D; M)\left(1+O\left(\frac{1}{n^{1/2}\log n}\right)\right).
 \end{align}
The following lemma provides a better approximation on the truncation length and the associated error chosen optimally for the proof of Theorem~\ref{main theorem}.
\begin{lemma}\label{full l to its truncation}
For sufficiently large $n$,
\[
L(1, \chi_D)=L(1, \chi_D; N)\left(1+O\left(\frac{1}{f(n)\log n}\right)\right)
\]
for all but at most $q^{\frac{7n}{10}}$ elements $D\in \mathcal{H}_n$, where $N=\log n+\log_2 n+3\log f(n)$, and $f(n) \to \infty$ arbitrarily slowly as $n \to \infty$, with $f(n) \ll \log n$.
 \end{lemma}
\begin{proof}
 By using \eqref{asym for L(1)}, we have
 \[
 L(1, \chi_D) =
 L(1, \chi_D; N)
 \exp \!\bigg(
 \sum_{\substack{N < d(P) \le M}}
 \bigg(\frac{\chi_D(P)}{|P|}
 + O\!\left( \frac{1}{|P|^2} \right)
 \bigg)\bigg)
 \bigg( 1 + O\!\bigg( \frac{1}{n^{\tfrac{1}{2}} \log n} \bigg) \bigg).
 \]
 Next, we identify a density zero subfamily of $\mathcal{H}_n$ for which
 \begin{align}\label{DP bound}
 \Bigg|\sum_{N < d(P) \le M} \frac{\chi_D(P)}{|P|}\Bigg| \ge \frac{1}{h(n)}
 \end{align}
 holds, where $
 \log n \ll h(n) \to \infty$ as $n \to \infty.$
 It is worth noting that $h(n)$ must grow faster than $\log n$ chosen appropriately later in order for the saddle point analysis to yield an asymptotic formula exhibiting double exponential decay (see Section~\ref{spa}).

The additional factor $1/\sqrt{\log n}$ in the choice of $c$ is essential for proving Theorem \ref{theorem on resonance method}.
Hence, using \eqref{asym for L(1)}, we can that there exists a $D\in \mathcal{H}_n$ such that
\[
L(1,\chi_D)\ge e^{\gamma}\left(\log n +\log_2 n+C_2(q)-\beta+o(1)\right).
\]
We use \eqref{asym for L(1)}, \eqref{ratio final}, \eqref{constant c} to obtain
\begin{align*}
&\bigg(\tau_{\beta, n}+\frac{1}{2\sqrt{\log n}}\bigg)\sum_{D\in \mathcal{H}_n}R_D^2\leq \sum_{D\in \mathcal{H}_n}L(1, \chi_D)R_D^2\\
&= \sum_{\substack{D\in \mathcal{H}_n\\ L(1, \chi_D)< \tau_{\beta, n}}}L(1, \chi_D)R_D^2+\sum_{\substack{D\in \mathcal{H}_n\\ L(1, \chi_D)\geq \tau_{\beta, n}}}L(1, \chi_D)R_D^2< \tau_{\beta, n} \sum_{D\in \mathcal{H}_n}R_D^2+ \sum_{\substack{D\in \mathcal{H}_n\\ L(1, \chi_D)\geq \tau_{\beta, n}}}L(1, \chi_D)R_D^2.
\end{align*}
From the above computation of $S_2$ together with \eqref{scx} gives us
\[
\sum_{\substack{D\in \mathcal{H}_n\\ L(1, \chi_D)\geq \tau_{\beta, n}}}L(1, \chi_D)R_D^2> \frac{e^{\gamma}}{2 \sqrt{\log n}}\sum_{D\in \mathcal{H}_n}R_D^2> \frac{e^{\gamma}}{2}\frac{q^{(1+c')n\left(1+O(\log_2 n/\log n)\right)}}{\sqrt{\log n}}.
\]
On the other hand, using \cite[Proposition 1.4]{Lumley} that
\[
\max_{D\in \mathcal{H}_n}L(1, \chi_D)\leq 2e^{\gamma}\log n+O_q(1),
\]
we have from \eqref{RD},
\begin{align*}
\sum_{\substack{D\in \mathcal{H}_n\\ L(1, \chi_D)\geq \tau_{\beta, n}}}L(1, \chi_D)R_D^2&\leq |\mathcal{H}_n| \left(\max_{D\in \mathcal{H}_n} R_D^2 \cdot \max_{D\in \mathcal{H}_n}L(1, \chi_D)\right)\phi_n(\beta)\\
&\leq \left(2e^{\gamma}\log n\right) q^{\left(1+\frac{2c\zeta_{\mathbb{A}}(2)}{\ln q}\right)\left(1+O\left(\frac{\log_2 n}{\log n}\right)\right)n} \phi_n(\beta).
\end{align*}
Comparing the above lower and upper bound, and using $c$ as expressed in \eqref{constant c}, we finally conclude that
\[
\phi_n(\beta)> \frac{1}{4(\log n)^{3/2}}q^{-\frac{c(3\ln 2-\pi/2)\zeta_{\mathbb{A}}(2)}{\ln q}\left(1+O\left(\log_2 n/\log n\right)\right)}\geq e^{-\frac{q^{-\beta}\left(1+O\left(1/\sqrt{\log n}\right)\right)\ln q}{2}}.
\]
\end{proof}

From the estimation of $\mathcal{S}_{c, N}$ and $\mathcal{R}_{c, N}$, we finally obtain
\begin{align}\label{ratio}
\frac{S_1}{S_2}\ge \frac{|\mathcal{H}_n| \mathcal{E}(N) \mathcal{E}_1(N) \prod_{d(P)< N} (1-|P|^{-1}) + O\left(q^{\left(\frac12+\frac{2c\zeta_{\mathbb{A}}(2)}{\ln q}+\frac{11\varepsilon}{6}\right)n}\right)}{|\mathcal{H}_n| \mathcal{E}(N) \mathcal{E}_2(N)\prod_{d(P)<N}h(P)^{-1} + O\left( (q^{\left(\frac12+\frac{2c\zeta_{\mathbb{A}}(2)}{\ln q}+2\varepsilon\right)n}\right)}.
\end{align}
 Noting $N=\log n+\log_2 n +\log c$, we first optimize $c$ from the ratio to the right-hand side of \eqref{ratio} by comparing the error terms. To do this, we first calculate the size of $\mathcal{E}(N)=\prod_{d(P)<N}\left(1-r_P^2 \right)^{-2} \left(1+r_P^2\right)$. Applying Abel's summation formula,
\begin{align*}
&\ln \mathcal{E}(N)=-2\sum_{n\leq N}\Pi_{q}(n)\ln\left(\tfrac{2q^n}{q^N}-\tfrac{q^{2n}}{q^{2N}}\right)
+\sum_{n\leq N}\Pi_{q}(n)\ln\left(2-\tfrac{2q^n}{q^N}+\tfrac{q^{2n}}{q^{2N}}\right)\\
&= \frac{\zeta_{\mathbb{A}}(2)}{\ln q}\left(2(2-\ln 4)+(\ln 2+\pi/2-2)\right)\frac{q^N}{N}\left(1+O\left(\frac{\log N}{N}\right)\right),
\end{align*}
where the first and second integrals are referenced from \cite[page $841$]{AMMP} and \cite[page $20$]{DM} respectively by using the prime polynomial theorem and change of variable $q^t=u$. So, from the choice of $N$, we have
\begin{align*}
\mathcal{E}(N)=q^{\widetilde{c}\left(1+O\left(\frac{\log_2 n}{\log n}\right)\right)n},
\end{align*}
where $\widetilde{c}=\left(c(2+\pi/2-3\ln 2)\zeta_{\mathbb{A}}(2)\right)/\ln q$.
Noticing that $\mathcal{E}_2(N)\asymp 1/n$, from \eqref{ratio}, we compare the first term of the denominator to the both error terms to restrict ourself
\begin{align}\label{optimzed constant}
\frac12+\frac{2c\zeta_{\mathbb{A}}(2)}{\ln q}< 1+\frac{c\left(2+\pi/2-3\ln 2\right)\zeta_{\mathbb{A}}(2)}{\ln q}\implies c<\frac{\ln q}{2(3\ln 2-\pi/2)\zeta_{\mathbb{A}}(2)}.
\end{align}
Also, from the above computations, we obtain
\begin{align}\label{scx}
\mathcal{S}_{c, N}=q^{n\widetilde{c}\left(1+O(\log_2 n/\log n)\right)}.
\end{align}
To extract the main term from the ratio on the right-hand side of \eqref{ratio}, it remains to estimate $\mathcal{E}_1(N)$, $\prod_{d(P)\le N} \left(1-|P|^{-1}\right)$, and $\mathcal{E}_2(N)\prod_{d(P)\le N} h(P)^{-1} $.
Using the classical estimate using the Euler--Maclaurin summation formula,
\[
\sum_{n\le N} \frac{1}{n} = \ln N + \gamma + \frac{1}{2N} + O\Bigl(\frac{1}{N^2}\Bigr),
\]
one readily obtains
\[
\prod_{d(P)\le N} \left(1-\frac{1}{|P|}\right)^{-1} = e^{\gamma} N \left(1 + \frac{1}{2N} + O\Bigl(\frac{1}{N^2}\Bigr)\right).
\]
Similarly, following the approach used above for evaluating Euler products, we find
\[
\mathcal{E}_1(N) = q^{\left( \left(c_2 - c_3 + O\bigl(\frac{\log N}{N}\bigr)\right) \frac{\zeta_{\mathbb{A}}(2)}{N \ln q} \right)},
\]
where the constants $c_2$ and $c_3$ are as in \cite[page 21]{DM}, and
\[
\mathcal{E}_2(N)\prod_{d(P)\le N} h(P)^{-1} = \prod_{d(P)\le N} \left( 1 + \frac{(1-r_P^2)^2}{|P|(1+r_P^2)} \right)
= q^{\left( \left(c_2 + O\bigl(\frac{\log N}{N}\bigr)\right) \frac{\zeta_{\mathbb{A}}(2)}{N \ln q} \right)}.
\]

We emphasize that Theorem~\ref{main theorem} subsequently improves the uniformity range of~\cite[Theorem~1.2]{Lumley} from
 $\log n - 2\log_2 n - \log_3 n$ to $\log n + \log_2 n - \theta(n)$, where $\theta(n)$ grows arbitrarily slowly to infinity as $n \to \infty$. A similar range of uniformity as in Lumley’s work is also found in the work of Dahl and Lamzouri~\cite[Theorem~1.2]{DL}.
 For a discussion on the significance of these uniformity ranges, see~\cite{MV}.
Motivated by the work of Granville and Soundararajan~\cite{GS}, Lumley conjectured that as $n\to \infty$,
 \[
 \max_{D\in \mathcal{H}_{n}} L(1,\chi_D) = \log n + \log_2 n + O(1).
 \]
 It is plausible that if the asymptotic formula in Theorem~\ref{main theorem} were to hold up to the boundary of the admissible range, then the above conjecture would follow.
 Our result, however, falls slightly short of this limit, requiring that $\theta(n)$ remain bounded as stated in Theorem~\ref{main theorem}.
 Nevertheless, this constitutes the first unconditional result approaching the conjectural range, comparing upon the works of~\cite{DDLL, DL, GS, GSzeta, Lamzouri, Lumley}.
 Conditionally on the Generalized Riemann Hypothesis (GRH), the corresponding result was established by Granville and Soundararajan~\cite[Theorem~4]{GS} in the number field case.

Finally, by employing a long resonator method, we provide further evidence that the double-exponential decay may persists up to the conjectured level, where $\theta(n)$ transitions from an arbitrarily slowly unbounded function to a bounded one.

\begin{theorem}\label{theorem on resonance method}
	Let $q \ge 3$ and for every $\beta > 0$, define $\tau_{\beta, n}
	= e^{\gamma}\!\left(\log n + \log_2 n + C_2(q) - \beta\right),$ where
	\[
	C_2(q)
	= \frac{1}{2}
	- \left(\frac{\pi}{4} - \frac{\ln 2}{2}\right)\frac{q}{q-1}
	+ \log\!\left(
	\frac{(q-1)\ln q}{2q(3\ln 2 - \pi/2)}
	\right).
	\]
	Then we have
	\[
\phi_n(\beta)
:= \frac{1}{|\mathcal{H}_n|}
\sum_{\substack{D \in \mathcal{H}_n \\ L(1, \chi_D) \ge \tau_{\beta, n}}} 1
	\ge \exp\!\left(
	-\frac{q^{-\beta}\ln q}{2}
	\left(1 + o(1)\right)
	\right).
	\]
\end{theorem}
The best previously known unconditional result in this direction over number fields was due to Granville and Soundararajan~\cite[Theorem~5b]{GS}, while the conditional analogue under the GRH was obtained by the author and Maiti~\cite[Theorem~2]{DM}.

During the proof of Theorem~\ref{theorem on resonance method}, we also observed that
\[
\max_{D\in \mathcal{H}_n} L(1,\chi_D)
\ge e^{\gamma}\Bigl(\log n + \log_2 n + C_2(q) + o(1)\Bigr),
\]
where \(C_2(q)\) is as defined in the statement of the theorem. This result refines~\cite[Theorem~1.6]{Lumley} by providing an explicit value for the constant \(C_2(q)\). More strikingly, one has \(C_2(q) > 0\) for \(q > 10\); for instance, \(C_2(17) \approx 0.04\). This is the first time a positive constant is observed, showing that the maximum of \(L(1,\chi_D)\) can exceed the range \(\log n + \log_2 n\) (see~\cite[Conjecture~2]{MV}).

\subsubsection{Applications} For a monic square-free polynomial $D \in \mathbb{F}_q[t]$, define
\[
h_D = \lvert \mathrm{Pic}(\mathcal{O}_D) \rvert,
\]
where $\mathrm{Pic}(\mathcal{O}_D)$ denotes the Picard group of the ring of integers $\mathcal{O}_D \subset \mathbb{F}_q[t](\sqrt{D(t)})$.
Artin~\cite{Artin} established a class number formula over the hyperelliptic ensemble that connects $h_D$ and $L(1, \chi_D)$:
\begin{equation*}
L(1, \chi_D) = \frac{\sqrt{q}}{\sqrt{|D|}}\, h_D = q^{-g} h_D
\quad \text{for } D \in \mathcal{H}_{2g+1}.
\end{equation*}
From Theorem~\ref{main theorem}, we directly extend the range of uniformity in \cite[Corollary~1.8]{Lumley}, showing that the tail of the distribution of large values of $h_D$ over $\mathcal{H}_{2g+1}$ decays doubly exponentially. An analysis towards maximal value for $h_D$ follows from Theorem~\ref{theorem on resonance method}.

For $n = 2g + 2$ and $D \in \mathcal{H}_{2g+2}$, Artin also proved that
\begin{equation*}
L(1, \chi_D) = \frac{q - 1}{\sqrt{|D|}}\, h_D R_D,
\end{equation*}
where $R_D$ denotes the regulator of $\mathcal{O}_D$ (see \cite[Chapter 14]{ROS}).
Analogous results hold for the tail of the distribution of $h_D R_D$ as $D$ varies over $\mathcal{H}_{2g+2}$, improving~\cite[Corollary~1.9]{Lumley} in a similar manner.
 \subsection{Essence of the paper} We employ two distinct methods to prove Theorems~\ref{main theorem} and~\ref{theorem on resonance method}.
 The study of the distribution of $L(1, \chi_d)$ over number fields through an underlying probabilistic model was initiated by Granville and Soundararajan~\cite{GS}.
 Lumley~\cite{Lumley} studied the tail of the distribution for $L(1, \chi_D)$ over function fields from the corresponding probabilistic model (see \cite[eq.~(1.7)]{Lumley}) and computing large moments og $L(1, \chi_D)$ over the family.
 To extend the range of uniformity stated in Theorem~\ref{main theorem}, we able to compute much higher moments for the short Euler product $L(1, \chi_D; N)$ and then utilize the saddle-point method.
 The Lemma~\ref{full l to its truncation} provides an optimal way to approximate $L(1, \chi_D)$ by $L(1, \chi_D; N)$, allowing us to taken a shorter truncation length $N$.
 This step relies crucially on a large sieve type estimate together with Chebyshev’s inequality.

 To prove Theorem~\ref{theorem on resonance method}, we employ the long resonance method for quadratic Dirichlet $L$-functions over number fields, as developed by the author and Maiti in~\cite{DM}.
 The character sum estimate in Lemma~\ref{character sum over nonsquare} plays a central role in both results: it extends the range of uniformity as in Theorem \ref{main theorem} via computing larger moments of $L(1, \chi_D; N)$ over the family, and it enables the control of long resonators in Theorem~\ref{theorem on resonance method}. These ideas appear difficult to implement for higher degree $L$-functions but may still be applicable to families of $\mathrm{GL}(1)$ $L$-functions associated with higher order characters.

To extend the range of uniformity stated in Theorem~\ref{main theorem}, we are able to compute much higher moments for the short Euler product $L(1, \chi_D; N)$ and then utilize the saddle-point method. The Lemma~\ref{full l to its truncation} provides an optimal way to approximate $L(1, \chi_D)$ by $L(1, \chi_D; N)$, allowing us to taken a shorter truncation length $N$. This step relies crucially on a large sieve type estimate together with Chebyshev’s inequality. The character sum estimate in Lemma~\ref{character sum over nonsquare} plays a central role: it extends the range of uniformity as in Theorem~\ref{main theorem} via computing larger moments of $L(1, \chi_D; N)$ over the family.

To extend the range of uniformity stated in the main theorem, we are able to compute much higher moments for the short Euler product $L(1, \chi_D; N)$ and then utilize the saddle-point method.

For sufficiently large $n$,
\[
L(1, \chi_D)=L(1, \chi_D; N)\left(1+O\left(\frac{1}{f(n)\log n}\right)\right)
\]
for all but at most $q^{\frac{7n}{10}}$ elements $D\in \mathcal{H}_n$, where $N=\log n+\log_2 n+3\log f(n)$, and $f(n) \to \infty$ arbitrarily slowly as $n \to \infty$, with $f(n) \ll \log n$. This provides an optimal way to approximate $L(1, \chi_D)$ by $L(1, \chi_D; N)$, allowing us to taken a shorter truncation length $N$. This step relies crucially on a large sieve type estimate together with Chebyshev’s inequality.

Let $\ell=\ell_1 \ell_2^2\neq \square\in \mathbb{F}_q[t]$ with $\ell_1$ square-free. For any $\varepsilon>0$, we have
\[
\sum_{D\in \mathcal{H}_n}\chi_D(\ell)\ll  q^{(1/2+\varepsilon)n}\exp\left((d(\ell_1))^{1-\varepsilon}\right)\prod_{P\mid \ell_2}\left(1+|P|^{-1/2-\varepsilon}\right).
\]
This character sum estimate plays a central role: it extends the range of uniformity in establishing the main theorem via computing larger moments of $L(1, \chi_D; N)$ over the family.

Question:
Let \(\mathbb F_q\) be a finite field of odd cardinality with fixed \(q\ge 3\), and let \(\mathcal H_n\) denote the set of monic square-free polynomials \(D\in \mathbb F_q[t]\) of degree \(n\) (equivalently, the hyperelliptic ensemble \(y^2=D(t)\)). For each \(D\in\mathcal H_n\), let \(\chi_D(f)=\left(\frac{f}{D}\right)\) be the quadratic character modulo \(D\), and let \(L(s,\chi_D)\) be the associated Dirichlet \(L\)-function. Define the tail distribution
\[
\phi_n(\tau):=\frac{1}{|\mathcal H_n|}\sum_{\substack{D\in\mathcal H_n\\ L(1,\chi_D)\ge e^{\gamma}\tau}}1,
\]
where \(\gamma\) is Euler's constant, \(\log\) denotes logarithm base \(q\), \(\log_j\) denotes the \(j\)-fold iterated base-\(q\) logarithm, and \(\{x\}\) denotes the fractional part of \(x\). For any function \(\theta(n)\) with \(2\le \theta(n)\ll \log_3 n\) and \(\theta(n)\to\infty\) arbitrarily slowly as \(n\to\infty\), which statement holds uniformly for all \(\tau\le \log n+\log_2 n-\theta(n)\)?

\[
\phi_n(\tau)=\exp\!\left(-\,C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)\,\frac{q^{\,\tau-C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)}}{\tau}\bigl(1+o_\theta(1)\bigr)\right),
\]
where \(C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)\) and \(C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)\) may depend on \(\tau\), \(q\), and \(\kappa(\tau)\), but remain bounded as their argument varies between \(1\) and \(q\).

Choices:
B.
\[
\phi_n(\tau)=\exp\!\left(-\,C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)\,\frac{q^{\,\tau-C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)}}{\tau}\bigl(1+o_\theta(1)\bigr)\right),
\]
where \(C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)\) and \(C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)\) depend only on \(q\) and remain bounded as their argument varies between \(1\) and \(q\).
C.
\[
\phi_n(\tau)=\exp\!\left(-\,C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)\,\frac{q^{\,\tau-C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)}}{\tau}\,(1+o(1))\right),
\]
for some bounded functions \(C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)\) and \(C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)\) as their argument varies between \(1\) and \(q\).
D.
\[
\phi_n(\tau)=\exp\!\left(-\,C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)\,\frac{q^{\,\tau-C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)}}{\tau}\bigl(1+o_\theta(1)\bigr)\right),
\]
uniformly for all \(\tau\le \log n+\log_2 n+\theta(n)\), where \(C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)\) and \(C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)\) may depend on \(\tau\), \(q\), and \(\kappa(\tau)\), but remain bounded as their argument varies between \(1\) and \(q\).
E.
\[
\phi_n(\tau)=\exp\!\left(-\,C_1\!\left(q^{\log \kappa(\tau)}\right)\,\frac{q^{\,\tau-C_0\!\left(q^{\log \kappa(\tau)}\right)}}{\tau}\bigl(1+o_\theta(1)\bigr)\right),
\]
where \(C_0\!\left(q^{\log \kappa(\tau)}\right)\) and \(C_1\!\left(q^{\log \kappa(\tau)}\right)\) may depend on \(\tau\), \(q\), and \(\kappa(\tau)\), yet remain bounded as the argument varies between \(1\) and \(q\).

dependence of bounded constants on \(\tau\) and \(\kappa(\tau)\)

dropped the specific \(o_\theta(1)\) uniformity/dependence information while keeping the same main asymptotic shape

upper range of uniformity in \(\tau\) beyond \(\log n+\log_2 n-\theta(n)\)

periodic dependence through fractional part \(\{\log \kappa(\tau)\}\)

The stem does not explicitly reveal the correct option. It sets up the theorem context and asks which asymptotic holds, but the exact dependence on the fractional part, the error term, and the range of uniformity are not given away.

This is very close to asking for the precise statement of a known theorem, so it is partly a restatement task. However, it is not completely tautological because the options differ in subtle but meaningful ways involving uniformity, parameter dependence, and periodicity.

Some reasoning is needed to distinguish the exact theorem statement from weaker or false variants, especially regarding the range in tau and the role of {log kappa(tau)}. Still, the item mainly tests recall/recognition of a technical result rather than substantial generative mathematical reasoning.

The distractors are strong: they are mathematically close to the correct statement, target realistic failure modes (weakened error term, incorrect dependence of constants, overextended uniformity range, and loss of periodic fractional-part structure), and are clearly distinct.

A technically well-constructed theorem-recognition MCQ with strong distractors and little answer leakage, but it primarily tests precise recall of a result rather than deeper generative reasoning.