Special Program on $p$-adic $L$-functions and eigenvarieties

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Conference July 18-22, 2022


2pmLaiWilliamsLangParticipant questions

Titles and Abstracts

John Bergdall $p$-adic distributions for modular forms

Distributions of Hecke eigenvalues for modular eigenforms have been studied in two ways. The Sato-Tate conjecture makes predictions for a fixed eigenform. Theorems of Serre and Conrey-Duke-Farmer establish the distribution of a fixed eigenvalue as eigenforms vary in their weight. In both cases, measurements are made using Archimedean sizes. In addition, generalizations beyond modular eigenforms have been established.
Less is known about non-Archimedean questions. This will focus on $p$-adic distributions for eigenforms in the weight aspect. We first introduce a unresolved conjecture of Gouvêa on the $p$-th Hecke eigenvalues in level $\operatorname{SL}(2,\mathbb{Z})$. Then, we explain new predictions in level $\Gamma_0(p)$ for $\mathscr{L}$-invariants. These invariants have no Archimedean analogue and, so, I will end by discussing a framework tying the $p$-adic questions together. This is part of joint projects with Robert Pollack.

Daniel Barrera Branching laws and $p$-adic deformation

In this talk we will try to explain the role played by branching laws in the construction and deformation of objects which are relevant in arithmetic. More precisely, firstly we will consider $p$-adic deformation of $p$-adic $L$-functions for $\operatorname{GL}(2n)$ via the use of branching laws. Secondly, in the context of $\operatorname{GL}(2)$ over totally real number fields we will explain their use to study Galois cohomology classes.

Shilin Lai Title: Diagonal cycle on the product of unitary groups

The arithmetic Gan-Gross-Prasad conjecture relates the height of a cycle to the derivative of an $L$-function. We will interpolate the $p$-adic étale realization of this cycle as the automorphic form varies in a Hida family, generalizing Howard's big Heegner point. The construction is based on Hida theory developed using Emerton's completed cohomology. We will also indicate how this should be related to several different Iwasawa main conjectures.

Kazim Büyükboduk Arithmetic of critical $p$-adic $L$-functions

In joint work with Denis Benois, we give an étale construction of Bellaïche's $p$-adic $L$-functions about $\theta$-critical points on the Coleman–Mazur eigencurve. I will discuss applications of this construction towards leading term formulae in terms of $p$-adic regulators on what we call the thick Selmer groups, which come attached to the infinitesimal deformation at the said $\theta$-critical point along the eigencurve, and an exotic ($\Lambda$-adic) $\mathscr{L}$-invariant. Besides our interpolation of the Beilinson–Kato elements about this point, the key input to prove the interpolative properties of this $p$-adic $L$-function is a new $p$-adic Hodge-theoretic "eigenspace-transition via differentiation" principle.

Mladen Dimitrov Uniform irreducibility of Galois action on the $p$-primary part of Abelian 3-folds of Picard type

Half a century ago Manin proved a uniform version of Serre’s celebrated result on the openness of the Galois image in the automorphisms of the $p$-adic Tate module of any non-CM elliptic curve over a given number field. In this talk we will present a generalization to principally polarized semi-stable abelian 3-folds of Picard type, having no CM quotients. In the course of the proof we are led to determine precise levels for certain non-tempered, non-generic representations of $U(3)$, which we succeed by tapping into known cases of the Gan-Gross-Prasad Conjectures. This is a joint work with Dinakar Ramakrishnan.

Chi-Yun Hsu On partially classical Hilbert modular forms

When studying modular forms, we often regard them as in the larger space of $p$-adic overconvergent modular forms, so that $p$-adic analytic techniques can be applied, but the finite-slope condition also has to be imposed. In the situation of Hilbert modular forms associated with a totally real field $F$ of degree $g$ over $\mathbb{Q}$, we can consider $I$-classical overconvergent forms, for each subset $I$ of the $g$ real embeddings of $F$. When $I$ is the empty set, this is the usual notion of overconvergence. Then the finite-slope condition can be removed outside of $I$. To study partially classical modular forms, we establish a partial classicality theorem: when an overconvergent form satisfies a small slope condition depending on $I$, it is automatically $I$-classical. It is also expected that the Galois representation of a partially classical eigenform is partially de Rham.

Jaclyn Lang A modular construction of unramified $p$-extensions of $\mathbb{Q}(N^{1/p})$

In his 1976 proof of the converse of Herbrand’s theorem, Ribet used Eisenstein-cuspidal congruences to produce unramified degree $p$ extensions of the $p$-th cyclotomic field when $p$ is an odd prime. After reviewing Ribet’s strategy, we will discuss recent work with Preston Wake in which we apply similar techniques to produce unramified degree-$p$ extensions of $\mathbb{Q}(N^{1/p})$ when $N$ is a prime that is congruent to $-1\pmod{p}$. This answers a question posed on Frank Calegari’s blog.

Antonio Lei Tate--Shafarevich groups of $p$-supersingular elliptic curves and BDP Selmer groups over anticyclotomic extensions

Let $p\ge 5$ be a prime number and $E/\mathbb{Q}$ an elliptic curve with good supersingular reduction at $p$. Let $K$ be an imaginary quadratic field where $p$ splits. Under the generalized Heegner hypothesis, we relate the BDP Selmer groups to the Tate--Shafarevich groups of $E$ and study how they grow inside the anticyclotomic $\mathbb{Z}_p$-extension of $K$. In particular, we prove formulae that resemble the classical formula of Iwasawa on class groups. splits. This is joint work with Meng Fai Lim and Katharina Mueller. If time permits, we may also discuss related ongoing joint work with Jeffrey Hatley and Luochen Zhao on $\mu$-invariants of BDP Selmer groups.

Zheng Liu $p$-adic $L$-functions for $\operatorname{GSp}(4)\times \operatorname{GL}(2)$

For a cuspidal automorphic representation $\Pi$ of $\operatorname{GSp}(4)$ and a cuspidal automorphic representation $\pi$ of $\operatorname{GL}(2)$, Furusawa's formula can be used to study the special values of the degree-eight $p$-adic $L$-function $L(s,\Pi\times\pi)$. In this talk, I will explain a construction of the $p$-adic $L$-function for $\Pi\times\pi$ by using Furusawa's formula and a family of Eisenstein series. The construction includes choosing local test sections at $p$ and computing the corresponding local zeta integrals.

Alexandre Maksoud On the arithmetic of certain modular Artin motives

Given an ordinary modular form $f$ of weight one, a conjecture of Darmon, Lauder and Rotger (proven by Rivero and Rotger) relates special values of the Hida-Rankin $p$-adic $L$-function attached to f with the arithmetic of the adjoint motive of f. Their formula involves a certain $p$-adic regulator which turns out to be an $\mathscr{L}$-invariant in the sense of Benois. In this talk we first discuss the compatibility between Darmon-Lauder-Rotger's formula with a trivial zero conjecture à la Benois. We then formulate a more general trivial zero conjecture for Artin motives and present theoretical evidence for it.

Anna Medvedovsky Counting modular forms with fixed mod-$p$ Galois representation and Atkin-Lehner-at-$p$ eigenvalue

For $N$ prime to $p$, we count the number of classical modular forms of level $Np$ and weight $k$ with fixed mod-$p$ Galois representation and Atkin-Lehner-at-$p$ sign, generalizing both recent results of Martin generalizing work of Wakatsuki and Yamauchi (no rhobar constraint) and the $\overline{\rho}$-dimension-counting formulas of Bergdall-Pollack and Jochnowitz. Working with the Atkin-Lehner involution typically requires inverting $p$, which naturally complicates investigations modulo $p$. To resolve this tension, we use the trace formula to establish up-to-semisimplifcation isomorphisms between certain mod-$p$ Hecke modules (namely, refinements of Jochnowitz's weight-filtration graded pieces $W_k$) by exhibiting ever-deeper congruences between traces of prime-power Hecke operators acting on characteristic-zero Hecke modules. This last technique, relying on our refinement of the Brauer-Nesbitt theorem in characteristic $p$ for a single operator, is new, purely algebraic/combinatorial, and may well have applications in other contexts. Joint with Samuele Anni and Alexandru Ghitza.

Kartik Prasanna Modular forms of weight one, motivic cohomology and the Jacquet-Langlands correspondence.

This is a report on joint work (in progress) with Ichino. In a previous paper, we showed that the Jacquet-Langlands correspondence for Hilbert modular forms, all of whose weights are at least two, preserves rational Hodge structures -- as predicted by the Tate conjecture. In this talk, I will discuss a related result in the case of weight one forms. Since weight one forms are not cohomological, the Tate conjecture does not apply and thus it is not at all obvious what the content of such a result should be. I will motivate and explain the statement, which is suggested by another recent development, namely the conjectural connection between motivic cohomology and the cohomology of locally symmetric spaces, and outline a proof.

Sheng-Chi Shih On Iwasawa invariants of modular forms with reducible and non-$p$-distinguished residual Galois representations

In this talk, we will report a joint work with Jun Wang on the $p$-adic $L$-functions and the (strict) Selmer groups of the $p$-adic weight one cusp forms $f$, obtained via the $p$-stabilization of weight one Eisenstein series $E_1(\chi,1)$, under the assumption that a certain Eisenstein component of the ordinary universal cuspidal Hecke algebra is Gorenstein. Here $\chi$ is an odd primitive Dirichlet character with $\chi(p)=1$. As an application, we compute the Iwasawa invariants of ordinary modular forms of weight $k\geq 2$ with the same residual Galois representations as the one of $f$, which in our setting, is reducible and non-$p$-distinguished. Combining this with a result of Kato, we prove the Iwasawa main conjecture for these forms.

Carl Wang-Erickson $p$-adic families of critical overconvergent modular forms

Coleman proved that the dimension of spaces of overconvergent modular forms of critical slope is locally constant as a function of the weight with respect to the standard $p$-adic topology on weight space. While this suggests that critical slope overconvergent forms could appear in $p$-adic families parameterized by weight, as in the ordinary case, straightforward topological restrictions prevent a family of critical slope overconvergent modular forms from existing. In this talk we will explain a construction of $p$-adic families of critical slope overconvergent modular forms along with applications.

Chris Williams $p$-adic $L$-functions for $\operatorname{GL}(3) $

Let $\pi$ be a $p$-ordinary cohomological cuspidal automorphic representation of $\operatorname{GL}(n,\mathbb{A}_{\mathbb{Q}})$. For $n = 1,2$ existence of a $p$-adic $L$-function for $\pi$ goes back decades, but for $n > 2$ our understanding remains frustratingly incomplete: in all previously known constructions of $p$-adic $L$-functions, $\pi$ is assumed to be (at least) a functorial transfer via a proper subgroup of $\operatorname{GL}(n)$ (e.g. symmetric squares, Rankin--Selberg transfers, $\pi$ admitting Shalika models, etc).
In this talk, I will describe recent joint work with David Loeffler, where we construct a $p$-adic $L$-function when $n=3$, without any transfer/self-duality assumptions. Our method uses a `Betti Euler system', where we construct a tower of integral Eisenstein classes in the Betti cohomology for $\operatorname{GL}(3)$. If time permits, I'll discuss a cute application: we also prove the (automorphic realisation of) Deligne's algebraicity conjecture in this case, strengthening a result of Mahnkopf (by proving the expected compatibility of periods at infinity).

Lynelle Ye Geometry of Hilbert modular eigenvarieties

It is a longstanding question, first asked by Coleman and Mazur in 1998, whether eigenvarieties satisfy the valuative criterion for properness over weight space. We will show that eigenvarieties parametrizing $p$-adic overconvergent cuspidal Hilbert modular eigenforms for a totally real field $F$ are proper over integer weights in this sense, generalizing a result of Hattori for the case when the residue degrees of $p$ are at most $2$ (itself a generalization of Buzzard-Calegari's original work for the Coleman-Mazur eigencurve). This requires extending overconvergent Hilbert eigenforms farther than they have been constructed in the literature (by Andreatta-Iovita-Pilloni). We will also discuss conditional results for non-integer weights when $p$ is totally split.