Building Bridges: 6th EU/US Summer School & Workshop on Automorphic Forms and Related Topics

There was Building Bridges: 6th EU/US Summer School & Workshop on Automorphic Forms and Related Topics which is an annual 2-weeks conference for the topics related to automorphic forms. I heard about the conference when I was a senior undergraduate student (6 years ago), which I couldn’t make because I attended other REU program. Now I had a chance to attend this conference, and I want to thank organizers for their works; it was fun and I could met a lot of new people! `Also I learned a lot of recent progress on other area that I haven’t checked for a while, and got some new ideas from them.

Summer school

On the first week, there was a summer school on three different topics: subconvexity of $L$-functions, mock modular forms and quantum modular forms, and automorphic forms and representations. Each of the topics are covered by two lecturers with four lectures on the morning, and we spent most of the evenings for doing exercises.

Subconvexity of $L$-functions by Philippe Michel and Paul Nelson

Among three series of the talks, this is the most unfamiliar topic for me. Hence I will try to write things that I learned from the lecture.

We all know about the infamous Riemann hypothetis (RH). Lindelöf hypothesis (LH) is a weaker statement that the values of zeta functions on the critical line $\Re s = \frac{1}{2}$ satisfies

\[\left|\zeta\left(\frac{1}{2} + it\right)\right| = O(t^{\epsilon})\]

for any $\epsilon > 0$ (it is not trivial why RH implies LH, which is proven by Backlund). Like RH, Lindelöf hypothesis is still wide open, and many people try hard to decrease the exponent. Now, for $\sigma \in \mathbb{R}$ define $\mu(\sigma)$ as an infimum of $a$ satisfying $|\zeta(\sigma + it)| = O(t^a)$. Then clearly $\mu(\sigma) = 0$ for $\sigma > 1$, since $\zeta(s)$ converges on the region $\Re s \ge \sigma > 1$. By the functional equation of $\zeta(s)$, we have $\mu(\sigma) = \mu(1 - \sigma) - \sigma + 1/2$ and this gives $\mu(\sigma) = -\sigma + 1/2$ for $\sigma < 0$. The most interesting region is when $0 < \sigma < 1$, and Phargmén-Lindelöf principle shows that $\mu(\sigma)$ is a convex function, hence $\mu(\sigma) \le \frac{1}{2}(1 - \sigma)$ for all $0 < \sigma < 1$. Especially, this gives $\mu(\frac{1}{2}) \le \frac{1}{4}$, and we call $\frac{1}{4}$ as a convex bound. In general, we have a convex bound

\[\left|L\left(\pi, \frac{1}{2} + it\right)\right| = O(Q(\pi, t)^{\frac{1}{4} + \epsilon})\]

for any $\epsilon > 0$, where $Q(\pi, t)$ is the analytic conductor of $\pi$. More precisely, let $q_\pi$ be the arithmetic conductor of $\pi$ (generalization of the modulus of Dirichlet character or level of automorphic forms). Also, assume that the local factor of $L(\pi, s)$ at $\infty$ is

\[L_\infty(\pi, s) = \prod_{i=1}^{d} \Gamma_\mathbb{R}(s - \mu_{\pi, i})\]

Then we define $Q(\pi, t)$ as

\[Q(\pi, t) := q_\pi \prod_{i=1}^{d} (1 + |\mu_{\pi, i} - it|).\]

Now, any bounds better than these are called subconvex bounds, and this is the main theme of the lectures. The first subconvex bound for $\zeta(s)$ was obtained by Hardy and Littlewood, which is

\[\left|\zeta\left(\frac{1}{2} + it\right)\right| = O(t^{\frac{1}{6} + \epsilon})\]

by Weyl’s method. The current best record is $\frac{13}{84}$ by Bourgain. For general automorphic $L$-functions $L(\pi, s)$, we can ask more relaxed question by considering certain families of $\pi$. For example, we can consider a family of modular forms of fixed weight, but varying level ($q$-aspect), or vice versa ($t$-aspect). Michel and Venkatesh proved subconvexity bound in general for $\mathrm{GL}_2$, and recently Nelson proved $t$-aspect result of $\mathrm{GL}_n$ $L$-functions for all $n \ge 1$. Using the refined GGP formula for unitary groups and microanalysis, Nelson and Venkatesh also confirmed the case of $\mathrm{U}_{n} \times \mathrm{U}_{n+1}$.

There are several applications of subconvexity, including

• equidistribution of integer points on spheres
• “beat” Strum’s bound
• equidistribution of mod $\mathfrak{p}$ reduction of CM elliptic curves
• quantum unique ergodicity.

The bourbaki paper by Michel is a good source for these. The second half of the talks (by Nelson) was about the methods for proving subconvexity, including amplification and moment methods. But this part was a bit technical and hard to follow for me.

What I found later during the workshop is that Paul Nelson was live-TeXing most of the talks, which was incredibly fast. He might using neovim (but with some custom settings including autocompletion) and put relevant paper of the top, which seems to be a very efficient setup for note taking. (I wondef if he also took a note of my talk or not.)

Mock modular forms and quantum modular forms by Amanda Folsom and Holly Swisher

These were the second familiar talk for me. They introduced the foundational work on mock modular forms by Zwegers, harmonic Maass forms by Bruinier and Funke, and quantum modular forms by Zagier. I could learn many interesting recent progress on this subject, and also I liked the exercises they provided. Especially, the combinatorial applications of the talk were almost new to me, since I have never studied on the topic before. The only thing I new was the connection between partition numbers and Dedekind eta function (which is a weight $\frac{1}{2}$ modular form):

\[\sum_{n \ge 1} p(n)q^n = \prod_{n \ge 1} \frac{1}{1 - q^n} = q^{1/24}\eta(q)^{-1}.\]

I personally thank to Amanda Folsom because she recognized my name and my master’s thesis work. After attending their lectures and learning about some recent developments, I got a new research idea that might extend my old work. Hope I have time to work on it later.

Automorphic forms and $L$-functions by Ralf Schmidt and Abhishek Saha

For me, these lectures were the most familiar talk since I’m working on the subject. They covered the basic theory of automorphic forms and representations, including classical automorphic forms (holomorphic modular forms and Maass forms), automorphic representations, their $L$-functions, Langlands functoriality, etc.

In fact, the most satisfying part of the talk was not the talk itself, rather an answer to my question by Saha (although talks were great too!). Recently, I asked a question on MO few days ago, and still no answer. I had a chance to ask the same question to Saha, and the answer was Hecke algebra. We can definitly lift modular forms and Maass forms as functions on $\mathrm{SL}_2(\mathbb{R})$ or $\mathrm{SL}_2(\mathbb{A})$, but then we lost a lot of Hecke operators, compared to $\mathrm{GL}_2$, and this is one of the main reason why working with $\mathrm{GL}_2$ is easier. In case of half-integeral weight modular forms, Shimura proved that only $T_{p^2}$ can be defined, but not $T_p$. So we already lost a lot of Hecke operators, and there are no difference between $\widetilde{\mathrm{GL}}_{2}$ and $\widetilde{\mathrm{SL}}_{2} = \mathrm{Mp}_{2}$ in this point of view.

Workshop

The second week was workshop, mostly 20 minute talks. Unfortunately, I had to return early due to my teaching duties, but I can attend all the Monday and Tuesday talks with the first two Wednesday talks, and here are some of them based on my own understandings - so it could be totally wrong! Let me know if there are things to be fixed.

Jolanta Marzec-ballesteros - Doubling method for self-dual linear codes

Doubling method by Garett and Piatetski-Shapiro-Rallis is an important tool in the study of automorphic representations. For example, let $G, H$ are reductive groups with $G \times G \hookrightarrow H$ and $E(h, s)$ is an Eisenstein series attached to a certain parabolic subgroup $P \subset H$. If $f$ is an automorphic form of $G$, then we have

\[\langle E((g, g'), s), f(g) \rangle = \int_G E((g, g'), s)f(g) \mathrm{d}g = \cdots = L(f, s) f(g').\]

The important step is to find coset representations of $P \backslash H / (G \times G)$.

Jolanta Marzec-ballesteros’ work is about analogous construction for self-dual linear codes. Linear codes (of length $2n$) are nothing but subspaces $C$ of $\mathbb{F}_{2}^{2n}$. With natural inner product $\langle -,-\rangle : \mathbb{F}_{2}^{2n} \times \mathbb{F}_{2}^{2n} \to \mathbb{F}_{2}$, we call $C$ self-dual if $C^\perp = C$ (which implies $\dim C = n$). We define weight of $c \in C$ as the Hamming weight: the number of nonzero coordinates of $c$, i.e.

\[\mathrm{wt}(c) := \# \{i: c_i \ne 0\}.\]

Then the genus 1 weight enumerator (or just weight enumerator) is defined as a two-variable homogeneous polynomial

\[W_1(C, (x_0, x_1)) = \sum_{c \in C} x_0^{2n - \mathrm{wt}(c)} x_1^{\mathrm{wt}(c)}.\]

More generally, we define genus $g \ge 1$ weight enumerator as

\[W_g(C, \mathbf{x}) = \sum_{(c^j) \in C^g} \prod_{\alpha \in \mathbb{F}_{2}^{g}} \mathbf{x}_{\alpha}^{w_\alpha(c^1, \dots, c^g)}\]

where $w_{\alpha}(c^1, \dots, c^g)$ counts the number of rows of the matrix $(c^1, \dots, c^g) \in \mathbb{F}_{2}^{2n \times g}$ equal to $\alpha$, which is a homogeneous polynomial in $2^g$ variable of degree $2n$. Then these functions are the analogue of modular forms of genus $g$ (i.e. Siegel modular forms), and its expansion $\sum_A b_A \cdot (A)$ corresponds to a Fourier expansion of modular forms, where the term $(2n) = \mathbf{x}_{(1, \dots, 1)}^{2n}$ corresponds to a constant term. It is also known that, for a Clifford-Weil group $\mathcal{C}_{g}$ acting on $\mathbb{C}[\mathbf{x}_{\alpha}: \alpha \in \mathbb{F}_{2}^{g}]$, the invariant space is generated by the weight enumerators of self-dual codes. Hence the action can be considered as an analogoue of the “modular transformation law”. Now, the “doubling method” for linear codes says that, for a Siegel-type Eisenstein series $M_{2g}$ (defined in terms of $\mathcal{C}_{2g}$ and its parabolic subgroup), we can compute inner product of it with a function $f(\mathbf{x})$, which becomes a constant multiple of $f(\mathbf{x})$.

Petru Constantinescu - Non-vanishing of geodesic periods of automorphic forms

Joint work with Asbjørn Christian Nordentoft, paper is here.

For a fundamental discriminant $D < 0$, it is well-known that there is a bijectin between the class group $\mathrm{Cl}_K$ of $K = \mathbb{Q}(\sqrt{D})$, the $\Gamma = \mathrm{PSL}_2(\mathbb{Z})$-equivalence class of primitive quadratic forms of discriminant $D$, and Heegner points on $\Gamma \backslash \mathbb{H}$, where the second bijection is given by

\[ax^2 + bxy + cy^2 \leftrightarrow z_A = \frac{-b + i \sqrt{|D|}}{2a}.\]

When $D > 0$, we have a bijection between the narrow class group $\mathrm{Cl}_K^+$ and closed geodesics of $\Gamma \backslash \mathbb{H}$, which maps $ax^2 + bxy + cy^2$ to a semicircle with endpoints $\frac{-b \pm \sqrt{D}}{2a}$ (denote by $C_A$). In both cases, Duke proved that Heegner points / closed geodesics are equidistributed under the natural measure on $\mathbb{H}$ as $|D| \to \infty$.

Now, let $f$ be a nonzero Maass form and $\chi$ be a character of the class group $\mathrm{Cl}_K$. There exists a weight 1 and level $\Gamma_0(D)$ modular form $\theta_\chi$ of character $\chi$. Waldspurger proved the following formula on the central $L$-value of $f \times \theta_\chi$ when $D < 0$:

\[L\left(f \times \theta_\chi, \frac{1}{2}\right) = \frac{C_f}{|D|^{1/2}} \left|\sum_{A \in \mathrm{Cl}_K} \chi(A) f(z_A) \right|^2.\]

For $D > 0$, we have a similar formula by Zhang and Popa, where the sum of the right hand side is replaced by

\[\sum_{A \in \mathrm{Cl}_K^{+}} \chi(A) \int_{C_A} f(z) \frac{|\mathrm{d}z|}{y}.\]

When $D < 0$, using Waldspurger’s formula, Duke’s equidistribution result, and subconvexity, Michel and Venkatesh proved that the number of characters of $\mathrm{Cl}_{K}$ is greater than $D^{\delta}$ for $\delta = 1 / 2700$ as $|D| \to \infty$. Especially, this proves existence of $\chi \in \widehat{\mathrm{Cl}_{K}}$ with $L(f \times \theta_\chi, \frac{1}{2}) \ne 0$ for sufficiently large $|D|$. The main result of Petru and Asbjørn is the analogous result when $D > 0$, answering Michel’s question. Especially, they proved that 100% of the geodesic periods are nonvanishing: for $C(X) := \{ C \subset \Gamma \backslash \mathbb{H} : \ell(C) \le X \}$, prime geodesic theorem gives $|C(X)| \sim \frac{e^X}{X}$. Now, they proved:

\[\left|\left\{C \in C(X) : \int_{C} f(z) \frac{|\mathrm{d}z|}{y} = 0 \right\}\right| \ll \frac{e^X}{X^{5/4}}.\]

Similar result holds for general Fuschian groups. Combined with the result of Raulf on the size of $h(D)$, they proved that, for the positive proportion of $D$, there exists $\chi \in \widehat{\mathrm{Cl}_K^+}$ such that $L(f \times \theta_\chi, \frac{1}{2}) \ne 0$.

Zoe Batterman, Akash Narayanan, Christopher Yao - An excised orthogonal model for families of cusp forms

Joint work with many other people at SMALL REU, here is a relevant paper. I found that this survey paper is also helpful.

There’s a program relating the zeros of $L$-functions and eigenvalues of random matrices of a certain compact Lie group, which was initially conjectured for the Riemann zeta function by Montgomery and Dyson. They consider a family of a quadratic twists of a fixed newform, i.e. $L(f \otimes \psi_{d}, s)$ for varying $d$, and matched the density of the lowest-lying zeros with the eigenvalues of certain Lie groups:

Case	Group
$\chi_f$ principal, even twists	$\mathrm{SO}(2N)$
$\chi_f$ principal, odd twists	$\mathrm{SO}(2N+1)$
$\chi_f$ non-principal, $f = \bar{f}$	$\mathrm{USp}(2N)$
$\chi_f$ non-principal, $f \ne \bar{f}$	$\mathrm{U}(N)$

Here $\chi_f$ is a Nebentypus of $f$ and $\bar{f}$ is a dual form of $f$. I was not able to fully understand their papers, but it seems that there’s a “repulsion phenomena” observed by Miller in this paper, and one needs “excised model” to fit the distribution more precisely. There’s a paper that proposed excised random matrix models for the first time, and BBJMNSY’s work generalizes their work for arbitrary weight, odd prime level newforms.

Seewoo Lee - Algebraic proof of modular form inequalities for optimal sphere packings

ME! You can find the presentation file here and paper here (and also blog), which is currently under review. The most hard part is to squeeze my original talk (at Berkeley and POSTECH) as 1/3, which were both one hour. Especially, there were more things to say since the work is now “complete” in the sense that I was able to finish the proof of the third inequality for the dimension 24 (during the previous two talks, that part was work in progress). I was rushing, but succeed to finish the talk in 20 minutes and got some questions after, mostly on the possible generalizations to other dimensions. Of course, I was thinking about it for a while, and I hope I can tell you something more in a future.

Samuele Anni - Counting modular forms mod $p$ satisfying constraints at $p$

This work is about counting mod $p$ Galois representation associated to mod $p$ modular forms. Let $f \in S_k := S_k(\Gamma_0(Np), \overline{\mathbb{Q}_{p}})$ be a modular form of level $Np$ with $p \nmid N$. We call $f$ as a $p$-new form if it does not coming from level $N$ modular forms. We have a Atkin-Lehner involution $W_p$ on $S_k$, and this decomposes $S_k$ into a direct sum $S_k = S_k^+ \oplus S_k^-$ of $(+1)$ and $(-1)$-eigenspaces. Since we can easily compute the dimension $s_k := \mathrm{dim} S_k$, to compute dimensions $s_k^{\pm} := \mathrm{dim} S_k^{\pm}$, it is enough to compute $d_k := s_k^+ - s_k^-$. Based on computations (it seems that one can compute Galois representation associated to $f$ in a “polynomial time” - I forgot whose result is this from) and there’s an obvious pattern you can find: $|d_k|$ only depends on $p$ and $N$, and sign of $d_k$ equals $(-1)^{k/2}$. Especially, we have (again I forgot who proved this): if we put $d_k^\ast = d_k - 1$ for $k = 2$ and $d_k$ for $k \ge 4$,

\[d_k^\ast = (-1)^{\frac{k}{2}} \frac{\# \mathrm{FP}}{2}\]

where $\mathrm{FP}$ is the set of fixed points of $W_p$ on $X_0(Np)$, which equals

\[\#\mathrm{FP} = c_p \cdot h(-p) \cdot \prod_{q \mid N,\,q \text{ prime}} \left(1 + \left(\frac{-4p}{q}\right)\right)\]

for some $c_p$.

The main result of Samuele Anni (and collaborators) is the refinement of the formula, in terms of the associated Galois representations: compute dimensions of

\[\begin{align*} S_k &= \bigoplus_{\bar{\rho}} S_{k, \bar{\rho}} \\ S_{k, \bar{\rho}} &= S_{k, \bar{\rho}}^+ \oplus S_{k, \bar{\rho}}^-. \end{align*}\]

Again, once we know how to compute $s_{k, \bar{\rho}} := \mathrm{dim} S_{k, \bar{\rho}}$, we only need to compute the differences $d_{k, \bar{\rho}} := s_{k, \bar{\rho}}^+ - s_{k, \bar{\rho}}^-$. The modified $d_{k, \bar{\rho}}^\ast$ is alternating:

\[d_{k+2, \bar{\rho}[1]}^\ast = - d_{k, \bar{\rho}}^\ast\]

where $\bar{\rho}[1]$ is the Tate twist of $\bar{\rho}$, and the proof is based on the trace formula and refined Brauer-Nesbitt theorem.

Jan-Willem Van Ittersum - Integer partitions detect the primes

Here’s a paper, joint work with Will Craig and Ken Ono. MacMahon’s partition function $M_a(n)$ is given by

\[M_a(n) := \sum_{\substack{0 < s_1 < \cdots < s_a \\ m_1 s_1 + \cdots + m_a s_a = n}} m_1 m_2 \cdots m_a,\]

i.e. sum of the multiplicities of partitions of $n$ with $a$-parts. It has a nice generating function

\[\mathcal{U}_{a}(q) := \sum_{n \ge 1} M_a(n) q^n = \sum_{0 < s_1 < s_2 < \cdots < s_a} \frac{q^{s_1 + s_2 + \cdots + s_a}}{(1 - q^{s_1})^2 \cdots (1 - q^{s_a})^2}.\]

The main theorem is the following: one can construct prime-detecting functions using the MacMahon’s partition function. A function $f: \mathbb{Z}_{\ge 1} \to \mathbb{Z}$ is called prime-detecting if $f(n) \ge 0$ for all $n$ and $f(n) = 0$ if and only if $n$ is a prime, which is closed under the addition and multiplication by any $g$ with $g(n) > 0$ for all $n$. They proved that

\[\psi(n) := (n^2 + 3n + 2) M_1(n) - 8 M_2(n)\]

is a prime detecting function, and there are infinitely many of them. We also have a vector version of it (MacMahonesque partition function): for $\vec{\ell} \in \mathbb{Z}_{\ge 0}^{a}$, we define $M_{\vec{\ell}}(n)$ as

\[M_{\vec{\ell}}(n) = \sum_{\substack{0 < s_1 < \cdots < s_a \\ m_1 s_1 + \cdots + m_a s_a = n}} m_1^{\ell_1} m_2^{\ell_2} \cdots m_a^{\ell_a}\]

and prime-detecting functions can be made out of these functions. Proofs are based on the theory of multiple $q$-zeta values by Bachmann and Kühn, and quasimodular forms. It is easy to construct (non-homogeneous) quasimodular forms that detect primes: for example, if $G_k$ is an Eisenstein series nofrmalized by $a_{G_k}(1) = 1$, then

\[f_{k, \ell} := (D^{\ell} + 1) G_{k + 1} - (D^{k} + 1) G_{\ell + 1}\]

is prime-detecting: for $n \ge 2$, the $n$-th Fourier coefficient vanishes if and only if $n$ is prime, and always nonnegative. More generally, they defined

\[H_k := \begin{cases} \frac{1}{6} ((D^2 - D + 1) G_2 - G_4) & k = 6 \\ \frac{1}{24} (-D^2 G_{k-6} + (D^2 + 1) G_{k-4} - G_{k-2}) & k \ge 8 \end{cases}\]

and proved that every prime-detecting quasimodular forms can be expressed as a combination of derivatives of $H_k$’s.

Cristina Ballantine - Congruences for the number of 3- and 6-regular partitions and quadratic forms

Part of this work can be found on this paper, joint with Mircea Merca and Cristian-Silviu Radu.

Another talk about partition numbers and Ramanujan congruences. For an integer $r \ge 2$, $r$-regular partition is defined as a partition where all the parts are not divisible by $r$. We denote $b_r(n)$ for the number of $r$-regular partitions of $n$. For example, $b_2(n)$ counts the number of partitions of $n$ into odd numbers. Now we wonder if we have Ramanujan-type congruences for $b_r(n)$, i.e. congruences of the form $b_r(An + B) \equiv 0\,(\mathrm{mod}\,M)$ for some $A, B, M$. The most interesting part of the talk is that, when $M$ is cubic-free, then generating series are congruent to “theta functions” rather than other modular forms and Galois representations attached to them. For example, we have

\[\begin{align*} \sum_{n \ge 1} b_2(n) q^n &= \prod_{n \ge 1} (1 + q^n) \\ &= \prod_{n \ge 1} (1 - q^n)\quad (\mathrm{mod}\,2) \\ &= \sum_{j \ge 0} (-1)^j q^{j(3j-1)/2} \end{align*}\]

where the last equality is the Euler’s pentagonal number theorem. They also proved congruences of $b_3(n)$ and $b_6(n)$ modulo 2, e.g.

\[b_3(2(p^2 n + p \alpha - 24^{-1})) \equiv 0\,(\mathrm{mod}\,2)\]

for all $0 \le \alpha \le p - 1$ with $\alpha \ne \lfloor p/24 \rfloor$ and prime $p$, using the theory of quadratic forms and theta functions. Unfortunately, the proof uses Strum’s bound and they can only prove for finitely many $p$’s (until the computer crashes).

Qihang Sun - Exact formulae for ranks of partitions

Rademacher’s famous formula gives an “exact formula” of the partition function $p(n)$ in terms of Kloosterman sums and Bessel functions, generalizing Hardy-Ramanujan’s famous asymptotic of $p(n)$. Qihang obtained a similar formula for the rank of partition function. For a given partition $\lambda$ of $n$, rank of it is defined as (the largest part) - (number of parts). For example, the rank of $4 + 3 + 1 + 1 + 1 = 10$ is $4 - 5 = -1$. If we put $N(m, n)$ for the number of partitions of $n$ with rank $m$, then the two-variable generating function of it can be written as

\[R(w, q) = 1 + \sum_{n \ge 1} \sum_{m \in \mathbb{Z}} N(m, n) w^m q^n = 1 + \sum_{n \ge 1} \frac{q^{n^2}}{(wq;q)_n (w^{-1}q;q)_n}.\]

Especially, when $w = \zeta_b^a$ is a root of unity, we write

\[R(\zeta_b^a, q) = 1 + \sum_{n \ge 1} A\left(\frac{a}{b}; n\right) q^n.\]

The main result is the formula for $A(\ell / p; n)$ for prime $p$, e.g. when $p = 3$, we have

\[A\left(\frac{1}{3}; n\right) = \frac{2 \pi \zeta_{8}^{-1}}{(24n - 1)^{1/4}} \sum_{3 | c > 0} \frac{S(0,n, c, \left(\frac{\cdot}{3}\right)\overline{\nu_\eta})}{c} I_{\frac{1}{2}} \left(\frac{\pi \sqrt{24n - 1}}{6c}\right),\]

which generalizes Bringmann’s result on the aymptotic of $A(\frac{1}{3}; n)$ and Bringmann-Ono’s exact formula for $A(\frac{1}{2}; n)$.

Proof uses Maass-Poincare series and harmonic Maass forms. Especially, it requires some analytic continuation, which was obtained via certain estimation of Kloosterman sums.

Sheng-Yang Kevin Ho - The rational torsion subgroups of the Drinfeld modular Jacobians for prime-power levels

Parf of the talk is included in this paper.

Let $X_0(N)$ be the modular curve for $\Gamma_0(N)$ and $J_0(N)$ be its Jacobian. By Mordell-Weil theorem, we know that $J_0(N)(\mathbb{Q})$ is a finitely generated abelian group, and our question is: what is $\mathcal{T}(N) := J_0(N)(\mathbb{Q})_{\mathrm{tors}}$? The conjecture by Ogg is that it equals $\mathcal{C}(N)$, the rational cuspidal divisor class group of $\mathcal{T}(N)$, which is easier to compute than $\mathcal{T}(N)$. There are several progress on the problem, including the most recent result by H. Yoo for squarefree $N$’s. Also, the structure of $\mathcal{C}(N)$ is completely known for all $N$ by Yoo (see this paper). Usual approach to attack the problem is via Eisenstein ideals and $\Delta$ quotients.

Now, Shen-Yang’s work answers the same questions, but over a function field $K = \mathbb{F}_{q}(t)$. In this case, we consider function field analogus of the ingredients we mentioned above, e.g. modular forms. Drinfeld’s modular form is a function field analogue defined on the Drinfeld’s upper half plane $\Omega = \mathbb{C}_{\infty} \backslash K_{\infty}$ (here $K_\infty = \mathbb{F}_q((\frac{1}{T}))$ and $\mathbb{C}_{\infty} = \widehat{\overline{K_\infty}}$). Drinfeld’s (rank 2) $A$-modules (where $A = \mathbb{F}_q[T]$) is an analogue of an elliptic curve for function fields. For an ideal $\mathfrak{n}$ of $A$, we can define a congruence subgroup $\Gamma_0(\mathfrak{n})$, modular curve $X_0(\mathfrak{n})$, and its Jacobian $J_0(\mathfrak{n})$ similarly, and the torsion subgroup $\mathcal{T}(\mathfrak{n}) := J_0(\mathfrak{n})(K)_{\mathrm{tors}}$ is known to be finite by Lang and Neron. Now, the same question can be asked with $\mathcal{C}(\mathfrak{n})$, and still we believe that the Ogg’s conjecture holds for the function field case. The conjecture is known for prime $\mathfrak{n}$, and $\ell$-parts for square-free $\mathfrak{n}$ (i.e. we have $\mathcal{C}(\mathfrak{n})_{\ell} = \mathcal{T}(\mathfrak{n})_{\ell}$) for some $\ell$. Shen-Yang extend the result to prime powers $\mathfrak{n} = \mathfrak{p}^r$: he explicitly computed $\mathcal{C}(\mathfrak{p}^{r})$ and proved $\mathcal{C}(\mathfrak{p}^{r})_{\ell} = \mathcal{T}(\mathfrak{p}^{r})_{\ell}$ for $\ell \nmid q(q-1)$, which is

\[\mathcal{C}(\mathfrak{p}^r)_\ell = \mathcal{R}(\mathfrak{p}^r)_\ell \simeq \left(\frac{\mathbb{Z}_\ell}{M(\mathfrak{p})\mathbb{Z}_\ell}\right)^{r-1} \times \frac{\mathbb{Z}_\ell}{N(\mathfrak{p})\mathbb{Z}_\ell}\]

where

\[M(\mathfrak{p}) := \frac{|\mathfrak{p}|^2 - 1}{q^2 - 1}, \qquad N(\mathfrak{p}) := \begin{cases} \frac{|\mathfrak{p}| - 1}{q^2 - 1} & 2 \mid \deg(\mathfrak{p}) \\ \frac{|\mathfrak{p}| - 1}{q - 1} & 2 \nmid \deg(\mathfrak{p}) \end{cases}.\]

The proof mimics proofs for the case over $\mathbb{Q}$, but with Drinfeld’s stuff, and harmonic cochains on the Bruhat-Tits tree (don’t ask me what it is).

Nagarjuna Chary Addanki - On signs of eigenvalues of Ikeda lifts

The relevant paper can be found here.

Saito-Kurokawa lift $\mathrm{SK}_{k}: S_{2k}(\Gamma_1) \to S_{k+1}(\Gamma_2)$ takes a holomorphic modular form $f$ of weight $2k$ and gives a genus 2 Siegel modular form of weight $k+1$, where their $L$-functions are related as

\[L(\mathrm{SK}_{k}(f), s) = \zeta(s - k + 1) \zeta(s - k) L(f, s).\]

Ikeda generalized it to higher genus, which takes a modular form of weight $2k$ (of level 1) to a Siegel modular form of genus $2n$ and weight $k + n$. The signs of eigenvalues of Hecke eigenforms are intensively studied, and one of the main reason is because the signs determines the form uniquely (up to a constant). Especially, Kowalski-Lau-Soundrarajan-Wu proved that, if two normalized Hecke eigenforms $f, g \in S_k(\Gamma_1)$ satisfy $\mathrm{sign}(\lambda_f(p^r)) = \mathrm{sign}(\lambda_g(p^r))$ for almost all $p$ and $r$, then $f = g$. The result is generalized to the genus 2 cases by Gun-Kohnen-Paul.

Breulmann proved that a Siegel modular form $F \in S_k(\Gamma_2)$ is a Saito-Kurokawa lift if and only if $\lambda_F(m) > 0$ for all $m \ge 1$. The main result of Addanki’s work is to generalize Breulmann’s result to Ikeda lifts. Especially, they proved that all most all Hecke eigenvalues of Ikeda lifts are nonnegative.

Marios Voskou - Hyperbolic Counting Problems

This paper seems to be the most relevant.

There’s a famous Gauss circle problem, which asks to estimate the number of integer points inside a circle of radius $R$. By considering the area of the circle, it is easy to see that the number $N(R)$ is approximately $\pi R^2$, i.e. $N(R) / \pi R^2 \to 1$ as $R \to \infty$. Now, the hard part to estimate the error $E(R) := N(R) - \pi R^2$, which has order of growth smaller than or equal to $R$ (by considering circumference). But the conjecture is that we can do better; we expect

\[|E(R)| = O(R^{1/2 + \epsilon}).\]

Current best bounds for the exponent $t$ with $|E(R)| = O(R^{t})$ is

\[\frac{1}{2} < t < \frac{131}{208} = 0.6298...\]

but we still don’t know how to get the exponent $1/2$ ($+ \epsilon$).

We can think of analogous question on the hyperbolic space. Recall that the metric on the complex upper half plane $\mathbb{H}$ is given by

\[\mathrm{d}s = \frac{\sqrt{(\mathrm{d}x)^2 + (\mathrm{d}y)^2}}{y}\]

and the geodesics are vertical lines or half-circles centered at the real line. For the “integer points”, once we consider them as “lattice points” in $\mathbb{R}^{2}$, then the natural analogue for $\mathbb{H}$ would be the translations of a fixed point $z$ by elements in $\mathrm{SL}_2(\mathbb{Z})$, or more generally finite index subgroup $\Gamma \subset \mathrm{SL}_{2}(\mathbb{Z})$. So the question is to estimate

\[N(z, w; R) := \{\mathrm{dist}(\gamma z, w) \le R : \gamma \in \Gamma \}.\]

It worths to mention that the area and the circumference argument does not work anymore, since both has the same exponential growth $e^R$ in radius $R$. The hyperbolic lattice counting problem is also widely studied (Selberg, Günther, Good), and we have $O(e^{2R / 3})$ error bound result. Voskou considered an analogous problem for counting geodesics, and proved an error bound of the same order $O(e^{2R / 3})$. As an application, they also proved an asymptote of the following ideal counting function of $\mathbb{Z}[\sqrt{2}]$

\[\sum_{n \le X} \mathscr{N}(n) \mathscr{N}(pn \pm 1)\]

where $\mathscr{N}(n)$ is the number of ideals in $\mathbb{Z}[\sqrt{2}]$ with norm $n$, with error term $O(X^{2/3})$. (Note that we can replace $\mathbb{Z}[\sqrt{2}]$ with any ring of integers of a class number 1 real quadratic field.)

Concluding remarks

I really enjoyed the conference. Also, I had a chance to visit a beach near the campus, although the road to there was not easy enough for me.

I also went to the museum called “Mucem”, and I highly recommend to visit there if anyone reading this post is planning to visit Marseille.

I would end the post with a group photo:

Tags: math