## Neelam Saikia : Frobenius Trace Distributions for Gaussian Hypergeometric Functions

- Number Theory ( 326 Views )In the 1980’s, Greene defined hypergeometric functions over finite fields using Jacobi sums. The framework of his theory establishes that these functions possess many properties that are analogous to those of the classical hypergeometric series studied by Gauss and Kummer. These functions have played important roles in the study of Ap ́ery-style supercongruences, the Eichler-Selberg trace formula, Galois representations, and zeta-functions of arithmetic varieties. In this talk we discuss the distributions (over large finite fields) of natural families of these functions. For the 2F1 functions, the limiting distribution is semicircular, whereas the distribution for the 3F2 functions is the more exotic Batman distribution.

## Rahul Krishna : A New Approach to Waldspurgers Formula.

- Number Theory ( 305 Views )I will present a new trace formula approach to Waldspurger's formula for toric periods of automorphic forms on $PGL_2$. The method is motivated by interpreting Waldspurger's result as a period relation on $SO_2 \times SO_3$, which leads to a strange comparison of relative trace formulas. I will explain the local results needed to carry out this comparison, and discuss some small progress towards extending these results to high rank orthogonal groups.

## David Schwein : Recent progress on the formal degree conjecture

- Number Theory ( 288 Views )The local Langlands correspondence is a dictionary between representations of two kinds of groups: reductive p-adic groups (such as the general linear group) and the absolute Galois groups of p-adic fields. One entry in the dictionary is a conjectural formula of Hiraga, Ichino, and Ikeda for the size of a representation of a p-adic group, its "formal degree", in terms of the corresponding representation of a Galois group. In this talk, after reviewing the broad shape of p-adic representation theory, I'll explain why the conjecture is true for almost all supercuspidals, the fundamental building blocks of the subject.

## Jayce Robert Getz : Summation formula for spherical varieties

- Number Theory ( 267 Views )Braverman and Kazhdan, L. Lafforgue, Ngo, and Sakellaridis have pursued a set of conjectures asserting that analogues of the Poisson summation formula are valid for all spherical varieties. If proven, these conjectures imply the analytic continuation and functional equations of quite general Langlands L-functions (and thus, by converse theory, much of Langlands functoriality). I will explain techniques for proving the conjectures in special cases that include the first known case where the underlying spherical variety is not a generalized flag variety.

## Evangelia Gazaki : Torsion phenomena for zero-cycles on a product of curves over a number field

- Number Theory ( 252 Views )For a smooth projective variety X over an algebraic number field a conjecture of Bloch and Beilinson predicts that the kernel of the Abel-Jacobi map of X is a torsion group. When X is a curve, this follows by the Mordell-Weil theorem. In higher dimensions however there is hardly any evidence for this conjecture. In this talk I will focus on the case when X is a product of smooth projective curves and construct infinitely many nontrivial examples that satisfy a weaker form of the Bloch-Beilinson conjecture. This relies on a recent joint work with Jonathan Love.

## Manish Mishra : Self-dual cuspidal representations

- Number Theory ( 249 Views )Let F be a non-archimedean local field (such as ℚ_p). The Langlands philosophy says that the arithmetic of F is intimately related to the category R(G) of smooth complex representations of G(F) where G denotes a reductive F-group (for example the general linear group). The building blocks of R(G) are the "supercuspidal" representations of G(F). I will define this term in the talk. The category R(G) comes equipped with an involution - the "contragradient" or the "dual". The supercuspidal representations of G(F) which are self-dual are of considerable interest in the subject. In this talk, I will talk about a joint work with Jeff Adler about the existence of supercuspidals and self-dual supercuspidals. Specifically, we show that G(F) always admits supercuspidal representations. Under some mild hypotheses on G, we determine precisely when G(F) admits self-dual supercuspidal representations. These results are obtained from analogous results for finite reductive groups which I will also talk about.

## Junyan Xu : Bounds for certain families of character sums: how to obtain strong bounds with more exceptions from weak bounds with fewer exceptions

- Number Theory ( 231 Views )I will first introduce some generalities about exponential sums, in particular that square-root cancellation is expected for many algebraic character sums over the rational points of an algebraic variety over a finite field. I will then set the stage for my work: we consider a family of exponential sums, which in our case is parameterized by the rational points of a variety (the parameter space). Our task is to obtain a good bound on the number of exceptional ("bad") parameters for which square-root cancellations fail. Following an idea of Michael Larsen, we consider even moments of the family of exponential sums. If the summands are of certain product form, a transformation can be applied to produce another family of exponential sums (of the same type). If the summands are products of multiplicative characters composed with certain polynomial functions, a weak bound can then be applied to the character sums in this family (with few bad parameters), yielding bounds for the moments. We know from the theory of l-adic sheaves that the parameter space for the original family have a stratification by smooth varieties, which is uniform in some sense as long as the degrees of the characters and polynomials are bounded. Moreover, on each stratum the character sum behave in certain uniform way, so that we can talk about good and bad strata. The bounds on moments yield bounds on dimensions of bad strata, which in turn yield bounds on the number of bad parameters (in any box) of the original family. Though not optimal, the bounds already imply nontrivial Burgess bounds for forms, in joint work with Lillian Pierce.

## Yunqing Tang : Picard ranks of reductions of K3 surfaces over global fields

- Number Theory ( 231 Views )For a K3 surface X over a number field with potentially good reduction everywhere, we prove that there are infinitely many primes modulo which the reduction of X has larger geometric Picard rank than that of the generic fiber X. A similar statement still holds true for ordinary K3 surfaces over global function fields. In this talk, I will present the proofs via the intersection theory on GSpin Shimura varieties and also discuss various applications. These results are joint work with Ananth Shankar, Arul Shankar, and Salim Tayou and with Davesh Maulik and Ananth Shankar.

## Dihua Jiang : Fourier Coefficients and Endoscopy Correspondence for Automorphic Forms.

- Number Theory ( 230 Views )Fourier coefficients of automorphic forms are invariants which encode the analytic and arithmetic properties of automorphic forms. In this talk, we introduce the general notion of Fourier coefficients for automorphic representations of reductive groups and use them to construct explicit endoscopy correspondences, which construct via integral transforms with automorphic kernel functions members in global Arthur packets for classical groups. For instance, we will discuss with some details the recent work joint with Lei Zhang.

## William Sokurski : Fourier operators on GL(2) for odd Adjoint powers

- Number Theory ( 229 Views )Recently A. Braverman, D. Kazhdan, and L. Lafforgue have interpreted Langlands' functoriality in terms of a generalized harmonic analysis on reductive groups that requires the existence of new spaces of functions and an associated, generally non-linear, involutive Fourier transform. This talk will discuss some of these objects involved in the local p-adic situation, after introducing some ideas and basic constructions involved. Specifically, the local Fourier transforms have a nice interpretation in terms of their spectral decomposition giving the gamma factors that appear in functional equations of L functions, which, in the standard case allows one to write down the epsilon factors attached to supercuspidal representations as non-abelian Gauss sums. For G=GL(2), we use the local Langlands correspondence to provide L and epsilon factors for odd adjoint power transfers and use this to interpret the Adjoint power Fourier-transform such that its spectral decomposition on supercuspidal representations is given explicitly by certain non-abelian Kloosterman sums, which we use to give a form of the Fourier operator.

## Michael Harris : Chern classes of automorphic vector bundles

- Number Theory ( 226 Views )Holomorphic modular forms on the Shimura variety S(G) attached to the reductive group G can be interpreted naturally as sections of automorphic vector bundles: locally free sheaves that can be defined analytically by exploiting the structure of a Shimura variety as a quotient of a symmetric space. The construction can also be made algebraic, and in this way one gets a canonical functor from the tensor category of representations of a certain Levi subgroup K of G to the tensor category of vector bundles on S(G), and thus a homomorphism from the representation ring of K to K_0(S(G)). When S(G) is compact we determine how the image of this homomorphism behaves under Chern characters to Deligne cohomology and continuous l-adic cohomology. When S(G) is non-compact and of abelian type, we use perfectoid geometry to define Chern classes in the l-adic cohomology of the minimal compactification of S(G); these are analogous to the topological cohomology classes defined by Goresky and Pardon, using differential geometry. (Joint work with Helene Esnault.)

## Jürgen Klüners : The negative Pell equation and the Cohen-Lenstra heuristic

- Number Theory ( 218 Views )For a (squarefree) integer d the negative Pell equation is given by: X^2 - d Y^2 = -1. It is easy to see that this equation has no solution over the integers, if d is negative or d is congruent to 3 modulo 4. In this talk we would like to study the asymptotic behavior of integers d such that this equation is solvable. This question is related to the behavior of the class group of the quadratic field generated by a square root of d. The distribution of those class groups is described by the Cohen-Lenstra heuristics.

## Mahesh Kakde : Congruences between derivatives of geometric L-series

- Number Theory ( 205 Views )I will present a formulation of equivariant Tamagawa number conjecture for flat smooth sheaves on separated schemes of finite type over a finite field. After sketching a proof of this I will give application to Chinburg’s conjectures in Galois module theory and tower of fields conjecture. If time permits I will also give an application towards equivariant BSD for abelian varieties defined over global function fields. This is a joint work with David Burns.

## Samit Dasgupta : The Brumer-Stark Conjecture

- Number Theory ( 205 Views )I will give a very informal talk on some work I am doing now with Mahesh Kakde. We hope to make progress on the Brumer-Stark conjecture using the theory of group-ring families of modular forms. I will motivate and state the conjecture, and describe the flavor of our approach.

## Shuichiro Takeda : The Langlands quotient theorem for symmetric spaces

- Number Theory ( 203 Views )We will discuss how to generalize the Langlands quotient theorem to symmetric spaces. The key idea is to generalize so-called Casselmans criterion for temperedness to the context of symmetric spaces by using the work of Kato-Takano.

## Hiro-aki Narita : Special Bessel models with the local Maass relation and non-tempered automorphic forms on orthogonal groups

- Number Theory ( 201 Views )I will provide some general class of automorphic forms or representations on a general orthogonal group, having a non-tempered non-archimedean local component. We call them non-tempered automorphic forms or representations. It is a fundamental problem to find non-tempered cusp forms, which are nothing but counterexamples to the Ramanujan conjecture. The general class above includes the cusp forms given by the Oda-Rallis-Schiffmann lifting to O(2,m) and non-holomorphic lifting to O(1,8n+1) recently given by the joint work with Yingkun Li and Ameya Pitale. Such general class is given by means of the notion of the special Bessel model and the local Maass relation.

## Zhilin Luo : Bias of root numbers for Hilbert new forms of cubic level

- Number Theory ( 198 Views )We express the bias of global root numbers of Hilbert new forms of cubic level via special values of Dedekind L-functions attached to CM extensions determined by the level. In particular, our formula includes the case when weight 2 appears. We establish the formula by 1) a limit form of Jacquet-Zagier trace formula on PGL_2 associated to certain not necessarily integrable test functions at Archimedean places (when weight 2 occurs), and 2) showing the meromorphic continuation of certain Dirichlet series with coefficients given by special value of Dedekind L-functions via spectral side of the Jacquet-Zagier trace formula. This is a joint work with Q. Pi and H. Wu. arXiv: 2110.08310.

## Wei Zhang : Selmer groups and the indivisibility of Heegner points

- Number Theory ( 190 Views )We will discuss the conjecture of Kolyvagin on the indivisibility of Heegner points and its role in constructing rational points on elliptic curves over rational numbers, particularly in the proof of a recent result of this type: "the Selmer rank being one implies that the Mordell--Weil rank being one".

## W. Spencer Leslie : A new lifting via higher theta functions

- Number Theory ( 188 Views )Theta functions are automorphic forms on the double cover of symplectic groups and are important for constructing automorphic liftings. For higher-degree covers of symplectic groups, there are generalized theta representations and it is natural to ask if these ``higher'' theta functions play a similar role in the theory of metaplectic forms. In this talk, I will discuss new lifting of automorphic representations on the 4-fold cover of symplectic groups using such theta functions. A key feature is that this lift produces counterexamples of the generalized Ramanujan conjecture, which motivates a connection to the emerging ``Langlands program for covering groups'' by way of Arthur parameters. The crucial fact allowing this lift to work is that theta functions for the 4-fold cover still have few non-vanishing Fourier coefficients, which fails for higher-degree covers.

## Michal Zydor : Relative trace formula of Jacquet-Rallis, recent progress

- Number Theory ( 188 Views )I will discuss the relative trace formula approach to the global Gan-Gross-Prasad conjectures for unitary groups. The focus will be on the spectral side. I will present the various terms that appear in the spectral development of the relative trace formula and discuss what is still missing. This is a joint work with Pierre-Henri Chaudouard.

## Ma Luo : Algebraic de Rham theory for relative completion of $\mathrm{SL}_2(\mathbb{Z})$

- Number Theory ( 186 Views )In this talk, I will first review relative (unipotent) completions of discrete groups in general, and $\mathrm{SL}_2(\mathbb{Z})$ in particular. We then develop an explicit $\mathbb{Q}$-de Rham theory for the relative completion of $\mathrm{SL}_2(\mathbb{Z})$, which enables us to construct iterated integrals of modular forms of the second kind that provide its periods. Following Francis Brown, these periods are called `multiple modular values'. They contain periods of modular forms.

## Ken Ono : Zeta polynomials for modular forms

- Number Theory ( 182 Views )The speaker will discuss recent work on Manin's theory of zeta polynomials for modular forms. He will describe recent results which confirm Manin's speculation that there is such a theory which arises from periods of newforms. More precisely, for each even weight k>2 newform f, the speaker will describe a canonical polynomial Zf(s) which satisfies a functional equation of the form Zf(s)=Zf(1−s), and also satisfies the Riemann Hypothesis: if Zf(ρ)=0, then Re(ρ)=1/2. This zeta function is arithmetic in nature in that it encodes the moments of the critical values of L(f,s). This work builds on earlier results of many people on period polynomials of modular forms. This is joint work with Seokho Jin, Wenjun Ma, Larry Rolen, Kannan Soundararajan, and Florian Sprung.

## Jacques Hurtubise : Moduli and principal parts of a map into the flag manifold of a loop group

- Number Theory ( 181 Views )Rational maps from the Riemann sphere into itself can be described in terms of poles and principal parts; doing the same for maps into the flag manifold of a loop group gives insight into the topology of moduli of instantons and calorons. (joint work with Michael Murray)

## Michael Mossinghoff : Oscillation problems in number theory

- Number Theory ( 178 Views )The Liouville function λ(*n*) is the completely
multiplicative arithmetic function defined by λ(*p*) =
−1 for each prime *p*. Pólya investigated its summatory
function *L*(*x*) = Σ_{n≤x}
λ(*n*), and showed for instance that the Riemann hypothesis
would follow if *L*(*x*) never changed sign for large *x*.
While it has been known since the work of Haselgrove in 1958 that
*L*(*x*) changes sign infinitely often, oscillations in
*L*(*x*) and related functions remain of interest due
to their connections to the Riemann hypothesis and other questions in
number theory. We describe some connections between the zeta function and a
number of oscillation problems, including Pólya's question and some
of its weighted relatives, and, in joint work with T. Trudgian,
describe a method involving substantial computation that establishes new
lower bounds on the size of these oscillations.

## Benedict Morrissey : Regular quotients and Hitchin fibrations (joint work with Ngô B.-C.)

- Number Theory ( 178 Views )Orbital integrals for the Lie algebra can be analyzed using the Hitchin fibration. In turn the Hitchin fibration can be analyzed via the morphism g^{reg} ----> g//G from the regular elements of the Lie algebra, to the GIT quotient by the adjoint action. In trying to generalize this story by replacing the action of G on g by the action of G on some sufficiently nice variety M, we must replace the GIT quotient with what we call the regular quotient. This talk will look at the reasons for this, and the difference between the GIT and regular quotients in the case of G acting on G by conjugation (when the derived group of G is not simply connected), G acting on the commuting scheme, and G acting on the Vinberg monoid.

## Bruce Berndt : The Circle and Divisor Problems, Bessel Function Series, and Ramanujans Lost Notebook

- Number Theory ( 177 Views )A page in Ramanujan's lost notebook contains two identities for trigonometric sums in terms of doubly infinite series of Bessel functions. One is related to the famous ``circle problem'' and the other to the equally famous ``divisor problem.'' We discuss these classical unsolved problems. Each identity can be interpreted in three distinct ways. We discuss various methods that have been devised to prove the identities under these different interpretations. Weighted divisor sums naturally arise, and new methods for estimating trigonometric sums need to be developed. Trigonometric analogues and extensions of Ramanujan's identities to Riesz and logarithmic sums are discussed. The research to be described is joint work with Sun Kim and Alexandru Zaharescu.

## Ayla Gafni : Extremal primes for elliptic curves without complex multiplication

- Number Theory ( 174 Views )Fix an elliptic curve $E$ over $\mathbb{Q}$. An ''extremal prime'' for $E$ is a prime $p$ of good reduction such that the number of rational points on $E$ modulo $p$ is maximal or minimal in relation to the Hasse bound. In this talk, I will discuss what is known and conjectured about the number of extremal primes $p\le X$, and give the first non-trivial upper bound for the number of such primes when $E$ is a curve without complex multiplication. The result is conditional on the hypothesis that all the symmetric power $L$-functions associated to $E$ are automorphic and satisfy the Generalized Riemann Hypothesis. In order to obtain this bound, we use explicit equidistribution for the Sato-Tate measure as in recent work of Rouse and Thorner, and refine certain intermediate estimates taking advantage of the fact that extremal primes have a very small Sato-Tate measure.

## Raphael Beuzart-Plessis : Recent progress on the Gan-Gross-Prasad and Ichino-Ikeda conjectures for unitary groups

- Number Theory ( 172 Views )In the early 2000s Gan, Gross and Prasad made remarkable conjectures relating the non-vanishing of central values of certain Rankin-Selberg L-functions to the non-vanishing of certain explicit integrals of automorphic forms, called 'automorphic periods', on classical groups. They have been subsequently refined by Ichino-Ikeda and Neal Harris into precise conjectural identities relating these two invariants thus generalizing a famous result of Waldspurger for toric periods on GL(2). In the case of unitary groups, those have been established by Wei Zhang under some local restrictions. I will review the current state of the art on this and in particular how certain results in local harmonic analysis allow to remove almost all the local restrictions made by Zhang.

## Michael Harris : L-functions and the local Langlands correspondence

- Number Theory ( 172 Views )Henniart derived the following theorem from his numerical local Langlands correspondence: If $F$ is a non-archimedean local field and if $\pi$ is an irreducible representation of $GL(n,F)$, then, after a finite series of cyclic base changes, the image of $\pi$ contains a fixed vector under an Iwahori subgroup. This result was indispensable in all demonstrations of the local correspondence. Scholze gave a different proof, based on the analysis of nearby cycles in the cohomology of the Lubin-Tate tower (and this result also appears, in a somewhat different form, in proofs based on the global correspondence for function fields). An analogous theorem should be valid for every reductive group, but the known proofs only work for GL(n). I will sketch a different proof, based on properties of L-functions and assuming the existence of cyclic base change, that also applies to classical groups; I will also explain how the analogous result for a general reductive group is related to the local parametrization of Genestier-Lafforgue.