ENG 
ESP Session S21 - Galois representations and automorphic forms
Talks
Tuesday, July 13, 12:00 ~ 12:40 UTC-3
Flag varieties and representations of p-adic groups
Charlotte Chan
MIT, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
The intimate connection between algebraic geometry and representation theory has led to many deep and fruitful discoveries over the last century. I will illustrate some historical instances of this relationship. I will then survey recent advances in understanding the role of geometry in the representation theory of p-adic groups and the Langlands program.
Tuesday, July 13, 12:50 ~ 13:30 UTC-3
p-adic equidistribution of CM points
Ricardo Menares
Pontificia Universidad Católica de Chile, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.
A fundamental result by Duke, later extended by Clozel and Ullmo, establishes that CM points of growing discriminant equidistribute on the complex modular curve, with respect to the hyperbolic measure.
In this talk we will describe, for any prime p, the distribution of CM points on the p-adic space associated to the modular curve. In stark contrast with the complex case, there is a countable, infinite collection of accumulation measures describing the distribution of CM points.
Joint work with Sebastián Herrero (Pontificia Universidad Católica de Chile) and Juan Rivera-Letelier (University of Rochester).
Tuesday, July 13, 13:40 ~ 14:20 UTC-3
Poincare duality for modular representations of $p$-adic groups and Hecke algebras
Karol Koziol
University of Michigan, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
The mod-$p$ representations theory of $p$-adic reductive groups (such as $\textrm{GL}_2(\mathbb{Q}_p)$) is one of the foundations of the rapidly developing mod-$p$ local Langlands program. However, many constructions from the case of complex coefficients are quite poorly behaved in the mod-$p$ setting, and it becomes necessary to use derived functors. In this talk, I'll describe how this situation looks for the functor of smooth duality on mod-$p$ representations, and discuss the construction of a Poincare duality spectral sequence relating Kohlhaase's functors of higher smooth duals with modules over the (pro-$p$) Iwahori-Hecke algebra.
Tuesday, July 13, 14:30 ~ 15:10 UTC-3
Drinfeld's lemma for $F$-isocrystals
Kiran S. Kedlaya
University of California San Diego, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
Drinfeld's lemma on the fundamental groups of schemes in characteristic $p>0$ plays a fundamental role in the construction of the "automorphic to Galois" Langlands correspondence in positive characteristic, as in the work of V. Lafforgue. We describe a corresponding statement in which the roles of lisse etale sheaves and constructible sheaves are instead played by overconvergent $F$-isocrystals and arithmetic $\mathcal{D}$-modules, which is needed in order to transpose Lafforgue's argument to $p$-adic coefficients.
Joint work with Daxin Xu (Morningside Center of Mathematics).
Tuesday, July 13, 15:20 ~ 16:00 UTC-3
Density questions on arithmetic equivalence
Guillermo Mantilla Soler
Universidad Konrad Lorenz and Aalto University, Colombia - This email address is being protected from spambots. You need JavaScript enabled to view it.
It is a classic result that two number fields have equal Dedekind zeta functions if and only if the arithmetic type of a prime p is the same in both fields for almost all prime p. Here, almost all means with the possible exception of a set of Dirichlet density zero. In this talk we'll show that the condition density zero can be improved to a specific positive density that depends solely on the degree of the fields. More specifically, for every positive n we exhibit a positive constant c_{n} such that any two degree n number fields K and L are arithmetically equivalent if and only if the set of primes p such that the arithmetic type of p in K and L is not the same has Dirichlet density at most c_n. We also show that to check whether or not two number fields are arithmetically equivalent it is enough to check equality between finitely many coefficients of their zeta functions, and we give an upper bound for such a number.
Thursday, July 15, 12:00 ~ 12:40 UTC-3
Iwasawa theory for ${\rm GL}_2\times{\rm GL}_2$ and diagonal cycles
Francesc Castella
University of California, Santa Barbara, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk I will explain the construction, in joint work with Raul Alonso Rodriguez and Oscar Rivero, of an anticyclotomic Euler system for the tensor product of the Galois representations attached to two modular forms arising from generalized Gross--Kudla--Schoen diagonal cycles and their variation in $p$-adic families. As applications of this construction, we prove new cases of the Bloch--Kato conjecture in analytic rank zero and a divisibility towards an Iwasawa main conjecture.
Thursday, July 15, 13:15 ~ 13:55 UTC-3
On a formula of Gross-Zagier, and a real quadratic analogue.
Henri Darmon
McGill, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.
Joint work with Jan Vonk.
Thursday, July 15, 14:05 ~ 14:45 UTC-3
p-adic families of Yoshida lifts
Zheng Liu
University of California, Santa Barbara, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
We construct a Hida family of Yoshida lifts for two given Hida families of modular forms, and compute the Petersson inner products of its specializations. The key step in the construction is to choose suitable Schwartz functions at $p$. The computation of the Petersson inner products can be viewed as a generalization of the computation in the works by Bocherer--Dummigan--Schulze-Pillot and Hsieh--Namikawa. Our computation makes use of an equivariant property of the chosen Schwartz functions at $p$ for the action of $U_p$ operators. This is an ongoing joint work with Ming-Lun Hsieh.
Joint work with Ming-Lun Hsieh (Academia Sinica).
Thursday, July 15, 15:20 ~ 16:00 UTC-3
Overconvergent Eichler–Shimura morphism for families of Siegel modular forms
Giovanni Rosso
Concordia University, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.
Classical results of Eichler and Shimura decompose the cohomology of certain local systems on the modular curve in terms of holomorphic and anti-holomorphic modular forms. A similar result has been proved by Faltings' for the étale cohomology of the modular curve and Falting's result has been partly generalised to Coleman families by Andreatta–Iovita–Stevens. In this talk, based on joint work with Hansheng Diao and Ju-Feng Wu, I will explain how one constructs a morphism from the overconvergent cohomology of $\mathrm{GSp}_{2g}$ to the space of families of Siegel modular forms. This can be seen as a first step in an Eichler–Shimura decomposition for overconvergent cohomology and involves a new definition of the sheaf of overconvergent Siegel modular forms using the Hodge–Tate map at infinite level.
Joint work with Hansheng Diao (Tsinghua University, Yau Mathematical Sciences Center, China) and Ju-Feng Wu (Concordia University, Canada).
Wednesday, July 21, 16:00 ~ 16:40 UTC-3
Ramification of supercuspidal parameters
Michael Harris
Columbia University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
Let $G$ be a reductive group over a local field $F$ of characteristic $p$. Genestier and V. Lafforgue have constructed a semi-simple local Langlands parametrization for irreducible admissible representations of $G$, with values in the $\ell$-adic points of the $L$-group of $G$; the local parametrization is compatible with Lafforgue's global parametrization of cuspidal automorphic representations. Using this parametrization and the theory of Frobenius weights, we can define what it means for a representation of $G$ to be {\it pure} or {\it mixed}.
Assume $G$ is split semisimple. In work in progress with Gan and Sawin, we have shown that a pure supercuspidal representation has ramified local parameter, provided the field of constants in $F$ has at least $3$ elements and has order prime to the order of the Weyl group of $G$. The last hypothesis allows us to use Fintzen's result that the representation is obtained by compact induction. In particular, if the parameter of a pure representation $\pi$ is unramified then $\pi$ is a constituent of an unramified principal series. We are also able to prove in some cases that the ramification is wild, and we have some results on mixed supercuspidals as well. The method is specific to local fields of positive characteristic.
Joint work with Wee Teck Gan and Will Sawin.
Wednesday, July 21, 16:50 ~ 17:30 UTC-3
Orthogonal modular forms, paramodular forms, and congruences
Gonzalo Tornaría
Universidad de la República, Uruguay - This email address is being protected from spambots. You need JavaScript enabled to view it.
The goal of this talk is to explain how one can use orthogonal modular forms to find and prove congruences between paramodular forms.
In the first part of the talk I will give a brief review of orthogonal modular forms and how the case of SO(5) can be used to compute paramodular forms, based on recent work of Rama-T, Rösner-Weissauer, Dummigan-Pacetti-Rama-T.
In the second part of the talk I will explain how we use orthogonal modular forms to prove bi-congruences between paramodular forms as predicted by Golyshev. A key ingredient for this is the unexpected appearance of orthogonal eigenforms which /do not/ correspond to paramodular forms (see Rama-T in ANTS 2020).
Joint work with Gustavo Rama (Universidad de la República, Uruguay).
Wednesday, July 21, 17:40 ~ 18:20 UTC-3
Generalized theta series and the central values of $L$-functions of Hilbert modular forms
Nicolás Sirolli
CONICET / Universidad de Buenos Aires, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.
The central values of twisted $L$-functions of a modular form, after the work of Waldspurger, are known to be related to Fourier coefficients of half-integral weight modular forms.
A solution to the problem of computing effectively these half-integral weight modular forms was proposed by Gross, using ternary theta series attached to quaternion algebras. This construction imposed restrictions on the levels of the modular forms involved, as well as quadratic restrictions on the twisting discriminants.
In this work we show how to remove these restrictions by constructing families of generalized theta series giving the central values. Furthermore, our results hold over arbitrary totally real fields.
Joint work with Gonzalo Tornaría (Universidad de la República, Uruguay).
Wednesday, July 21, 18:30 ~ 19:10 UTC-3
Applying the Langlands program to the inverse Galois problem
Adrián Zenteno
Universidad Nacional Autónoma de México, México - This email address is being protected from spambots. You need JavaScript enabled to view it.
In the last decade, the study of the images of the Galois representations associated to regular algebraic cuspidal automorphic representations of $GL_n(\mathbb{A}_{\mathbb{Q}})$, via global Langlands correspondence, with prescribed local conditions has been an effective strategy to address the inverse Galois problem for finite groups of Lie type. In this talk we will explain how, by combining this strategy with Langlands functoriality and globalization of supercuspidal representations, we can construct residual Galois representations with controlled image and obtain new families of finite groups of type $B_m$, $C_m$ and $D_m$ arising as Galois groups over $\mathbb{Q}$.
Wednesday, July 21, 19:20 ~ 20:00 UTC-3
Harmonic analysis on certain spherical varieties
Jayce Getz
Duke University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
Braverman and Kazhdan proposed a conjecture, later refined by Ngo and partially set in the framework of spherical varieties by Sakellaridis, that asserts that affine spherical varieties admit Schwartz spaces, Fourier transforms, and Poisson summation formulae. If proven, the conjecture would imply Langlands functoriality in great generality. In this talk I will describe explicit formulae for the Fourier transform on Braverman-Kazhdan spaces (also known as pre-flag varieties) and for certain spherical varieties that are built out of triples of quadratic spaces.
Joint work with Chun-Hsien Hsu (Duke University) and Spencer Leslie (Duke University).
Posters
Langlands Functoriality Conjecture for $SO_{2n}^*$ in positive characteristic.
Héctor del Castillo
Pontificia Universidad Católica de Valparaíso , Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.
In [1,2] Cogdell, Kim, Piateski-Shapiro and Shahidi prove Langlands functioriality conjecture for globally generic cuspidal automorphic representations from the split classical groups, unitary groups or even quasi-split special orthogonal groups to the general linear groups. They have done this in the context of characteristic zero. Later, Lomelí in [4,5] extends this result to split classical groups and unitary groups in positive characteristic. In this poster we will discuss functoriality conjecture in the globally generic case for the even quasi-split non-split special orthogonal groups in positive characteristic and some of its applications [3].
$\textbf{References}$.
[1] James W. Cogdell, Ilya I. Piatetski-Shapiro and Freydoon Shahidi. $\textit{Functoriality for the classical groups}$. Publications Mathématiques de l'IHÉS, Volume 99, pp. 163-233 (2004).
[2] James W. Cogdell, Ilya I. Piatetski-Shapiro and Freydoon Shahidi. $\textit{Functoriality for the Quasisplit}$ $ \textit{Classical Groups}$. On Certain L-Functions. Clay Mathematics Proceedings Vol. 13, pp. 117-140 (2011).
[3] Héctor del Castillo. $\textit{Langlands Functoriality Conjecture for $SO_{2n}^*$ in positive characteristic}$. Ph.D. thesis.
[4] Luis Lomelí. $\textit{Functoriality for the Classical Groups over Function Fields}$. International Mathematics Research Notices, Volume 2009, Issue 22, pp. 4271–4335 (2009).
[5] Luis Lomelí. $\textit{Rationality and holomorphy of Langlands–Shahidi L-functions over function fields}$. Mathematische Zeitschrift 291 (1-2), pp. 711-739 (2019).
On the mod $p$ cohomology of unipotent groups
Daniel Kongsgaard
University of California, San Diego, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
Let $\mathcal{G}$ be a split and connected reductive $\mathbb{Z}_p$-group and let $\mathcal{N}$ be the unipotent radical of a Borel subgroup. In this poster we study the cohomology with trivial $\mathbb{F}_p$-coefficients of the nilpotent pro-$p$-group $N = \mathcal{N}(\mathbb{Z}_p)$ and the Lie algebra $\mathfrak{n} = \operatorname{Lie}(\mathcal{N}_{\mathbb{F}_p})$. We proceed by arguing that $N$ is a $p$-valued group using ideas of Schneider (unpublished) and Zábrádi, which by a result of Sørensen gives us a spectral sequence $E_1^{s,t} = H^{s,t}(\mathfrak{g},\mathbb{F}_p) \Longrightarrow H^{s+t}(N,\mathbb{F}_p)$, where $\mathfrak{g} = \mathbb{F}_p \otimes_{\mathbb{F}_p[\pi]} \operatorname{gr} N$ is the graded Lie $\mathbb{F}_p$-algebra attached to $N$ as in Lazards work. We then argue that $\mathfrak{g} \cong \mathfrak{n}$ by looking at the Chevalley constants, and, using results of Polo and Tilouine and ideas from Große-Klönne, we show that the dimensions of the $\mathbb{F}_p$-cohomology of $\mathfrak{n}$ and $N$ agree, which allows us to conclude that the spectral sequence collapses on the first page.
On the $p$-adic $L$-function of Bianchi modular forms
Luis Santiago Palacios
Universidad de Santiago de Chile, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.
Fix $p$ a rational prime, in this poster we present two works about the $p$-adic $L$-function for Bianchi modular forms, that is, the case of automorphic forms for $\mathrm{GL}_2$ over an imaginary quadratic field $K$.
(1) In the first work, when $K$ has class number 1, we obtain the functional equation of: (a) the $p$-adic $L$-function constructed in [Wil17], for a small slope $p$-stabilisation of a cuspidal Bianchi modular form and (b) the $p$-adic $L$-function constructed in [BSWWE18], for a critical slope $p$-stabilisation of a Base change cuspidal Bianchi modular form that is $\Sigma$-smooth. To treat case (b) we use $p$-adic families of Bianchi modular forms.
(2) In the second work we construct the $p$-adic $L$-function of a small slope Eisenstein Bianchi modular form introducing the notions of $C$-cuspidality and partial Bianchi modular symbols, both inspired in [BD15], and modifying accordingly the construction in [Wil17]. We also provide an example where our construction applies: let $\psi$ be a Hecke character of $K$ and consider the base change to $K$ of the theta series of $\psi$, this base change is a non-cuspidal Bianchi modular form that we can $p$-stabilise to have small slope, then we obtain by our methods its $p$-adic $L$-function, even more, we can relate such $p$-adic $L$-function with the Katz $p$-adic $L$-function of $\psi$ introduced in [Kat78]. This is a work in progress.
References
[BD15] Joël Bellaïche and Samit Dasgupta, The $p$-adic $L$-functions of evil Eisenstein series, Compositio Mathematica151(2015), no. 6, 999–1040.
[BSWWE18] Daniel Barrera Salazar, Chris Williams, and Carl Wang-Erickson, Families of Bianchi modular symbols: critical base-change $p$-adic $L$-functions and $p$-adic Artin formalism, arXiv preprint arXiv:1808.09750v6 (2018).
[Kat78] Nicholas M Katz, $p$-adic $L$-functions for CM fields, Inventiones mathematicae 49 (1978), no. 3-4, 199–297.
[Wil17] Chris Williams, $p$-adic $L$-functions of Bianchi modular forms, Proceedings of the London Mathematical Society 114 (2017), no. 4, 614–656.
Using $\mathbb{Q}$-curves and Hecke characters to solve Fermat-type equations
Lucas Villagra Torcomian
Universidad Nacional de Córdoba, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this poster we wil see how can we adapt the classical modular method using $\mathbb{Q}$-curves and Hecke characters in order to solve the diophantine equations $x^4+dy^2=z^p$ and $x^2+dy^6=z^p$. The results obtained are in [1] and [2].
[1] Ariel Pacetti and Lucas Villagra Torcomian. $\mathbb{Q}$-curves, hecke characters and some diophantine equations, 2020.
[2] Ariel Pacetti and Lucas Villagra Torcomian. $\mathbb{Q}$-curves, hecke characters and some diophantine equations II, 2021.
Joint work with Ariel Pacetti (Center for Research and Development in Mathematics and Applications).