Abstract: Diagonals of multivariate rational functions are an important class of functions arising in number theory, algebraic geometry, combinatorics, and physics. For instance, many hypergeometric functions are diagonals as well as the generating function for Apery's sequence. A natural question is to determine the diagonal grade of a function, i.e., the minimum number of variables one needs to express a given function as a diagonal. The diagonal grade gives the ring of diagonals a filtration. In this talk we study the notion of diagonal grade and the related notion of Hadamard grade (writing functions as the Hadamard product of algebraic functions), resolving questions of Allouche-Mendes France, Melczer, and proving half of a conjecture recently posed by a group of physicists. This work is joint with Andrew Harder.

Abstract: The Kudla-Millson lift is a map from the homology in degree q of a locally symmetric space associated to SO(p,q), to modular forms of weight (p+q)/2. It maps a cycle C to a modular form whose Fourier coefficients are intersection numbers between C and a family of special cycles on the locally symmetric space. I will present a similar construction for a locally symmetric space associated to SL_N, which is also related to recent work of Bergeron-Charollois-Garcia. Passcode: &YB&uY=9Q3

Abstract: Strongly divisible modules are semi-linear algebra objects that correspond to Galois stable lattices in semi-stable representations with Hodge–Tate weights in the Fontaine–Laffaille range. In this talk, we reduce the construction of strongly divisible modules of rank 2 to solving systems of equations and inequalities. We expect that this method provides at least one Galois stable lattice in each 2-dimensional semi-stable non-crystalline representation of the absolute Galois group over an unramified extension, with Hodge–Tate weights in the Fontaine–Laffaille range. Moreover, when the mod-$p$ reduction is an extension of two distinct characters, we further expect that this method exhausts the two non-homothetic lattices. This is joint work with Seongjae Han. Passcode: ?1y26$yU9p

Abstract: By using the degenerate Whittaker functions, we study the Fourier expansion of the Gan-Gurevich lifts which are Hecke eigen quaternionic cusp forms of weight k (k>1, even) on the split exceptional group G_2 which come from elliptic newforms of weight 2k without supercuspidal local components. In particular, our results give a partial answer to Gross' conjecture. This is a joint work with Takuya Yamauchi.

Abstract: Any construction of a quantum computer requires finding a good set of universal quantum logic gates: abstractly, a finite set of matrices in U(2^n) such that short products of them can efficiently approximate arbitrary unitary transformations. The 2-qubit case n=2 is of particular practical interest. I will present the first construction of an optimal, so-called "golden" set of 2-qubit gates. 

The modern theory of automorphic representations on unitary groups---in particular, the endoscopic classification and higher-rank versions of the Ramanujan bound---will play a crucial role in proving the necessary analytic estimate: specifically, a weight-aspect variant of a density hypothesis first considered by Sarnak and Xue. 

Abstract: The Poisson summation conjecture of Braverman-Kazhdan, Lafforgue, Ngo and Sakellaridis is an ambitious proposal to prove analytic properties of quite general Langlands L-functions using vast generalizations of the Poisson summation formula. In this talk, we present the construction of a generalized Whittaker induction such that the associated L-function is the product of the triple product L-function and L-functions whose analytic properties are understood. We then formulate an extension of the Poisson summation conjecture and prove that it implies the expected analytic properties of triple product L-functions. Finally, we use the fiber bundle method to reduce this extended Poisson summation conjecture to a case of the Poisson summation conjecture in which spectral methods can be employed together with certain local compatibility statements. This is joint work with Jayce Getz, Chun-Hsien Hsu, and Spencer Leslie.

Abstract: In this talk, I will explain how to give a simple characterization of determining whether a profinite etale cover of an abelian variety is perfectoid. I will also explain how this result is motivated by (the geometric) Sen theory and confirms a conjecture of Rodríguez Camargo on perfectoidness of p-adic Lie torsors for abelian varieties. This is joint work with Rebecca Bellovin and Sean Howe. - Passcode: +*Y.26a6wK

Abstract: Grothendieck envisioned that proving his standard conjectures would lead to the construction of a Tannakian category of motives, but the standard conjectures are still open in general. One of them, the Hodge standard conjecture, holds in characteristic 0 due to Hodge theory, and another one, the Kunneth type standard conjecture, follows in characteristic p from the Riemann hypothesis part of the Weil conjectures. We will combine these notions to unconditionally construct a Tannakian category of "motives for almost-algebraic cycles" in characteristic 0 that agrees with Grothendieck's construction if the Tate and Hodge conjectures hold.

Abstract: By qualifying weakly divisible locally and understanding the underlying structure, we define Pseudo and Sudo maximal Rings. We use these rings to give a weighted lower bound for number-fields with bounded discriminant and fixed degree, developing a cool sieve, that might be applicable in more generality.

Abstract: Let K be a number field of degree n = [K:Q] and absolute discriminant D = |disc(K)|.  For an integer l >= 2, the size of the l-torsion of the class group of K is trivially at most the size of the class group of K, which can be estimated in terms of D and n by  a classical result of Landau.  Improvements over this trivial bound, both conditional and unconditional, have generated significant interest in many cases depending on the integer l, the degree n, and the subfield structure of K. In this talk, I will discuss an unconditional log-power savings improvement over this trivial bound for all integers l and all number fields K. This is joint work with Robert Lemke Oliver.

Abstract: We discuss some of the difficulties in enumerating number fields and addressing one such difficulty in a minor way to get an improvement on the lower bound for number of number fields with fixed degree and bounded discriminants. Passcode: e77$74u@uG

Abstract: Eisenstein proved, in 1852, that if a function f(z) is algebraic, then its Taylor expansion at a point has coefficients lying in some finitely-generated Z-algebra. I will explain ongoing joint work with Josh Lam which studies the extent to which the converse of this theorem holds. Namely, we conjecture that if f(z) satisfies a (possibly non-linear!) algebraic ODE, non-singular at 0, and its Taylor expansion has coefficients lying in a finitely-generated Z-algebra, then f is algebraic. For linear ODE, we prove this conjecture when (A) f(z) satisfies a Picard-Fuchs equation, with initial conditions the class of an algebraic cycle, or (B) f(z) is a modular form, as well as in a number of other cases. For non-linear ODE, we prove it when f(z) satisfies an "isomonodromy" ODE with "Picard-Fuchs" initial conditions.

Abstract: Iwasawa theory is a branch of algebraic number theory which studies how arithmetic objects grow in infinite towers of number fields. In the 1990s, Ralph Greenberg formulated a striking conjecture about how Tate-Shafarevich groups of elliptic curves grow in infinite towers. In this talk, I'll introduce this conjecture and discuss some recent results surrounding it. Passcode: 57FU.hT5QI

Abstract: The classical Satake transform gives an isomorphism between the complex spherical Hecke algebra of a p-adic reductive group G, and the Weyl-invariants of the complex spherical Hecke algebra of a maximal torus of G. This provides a way for understanding the K-invariant vectors in smooth irreducible complex representations of G (where K is a maximal compact subgroup of G), and allows one to construct instances of unramified Langlands correspondences. In this talk, I'll present joint work with Cédric Pépin in which we attempt to understand the analogous situation with mod p coefficients, and working at the level of the derived category of smooth G-representations. 

Abstract: Let p be a prime number and K be a number field. For a given mod p Galois representation \rho:{\rm Gal}(\overline{K}/K)\longrightarrow GL_n(\overline{\mathbb{F}}_p), we may expect the existence of an automorphic representation of G(\mathbb{A}) for some reductive group G which give rise to \rho via conjectural global Langlands correspondence and its reduction. This is called as 

Serre's (mod p automorphy) conjecture for (K,\rho,G).

Many people have tried to reformulate this vague statement to specify 

possibly corresponding automorphic representations from data which (K,\rho,G) inherits.

In this talk, we give an example-based survey around this topic including speaker's works for Serre's conjecture for (\Q,\rho,GSp_4) where \rho takes the values in GSp_4.

Abstract: Unexpected and striking oscillations in the average a_p values of sets of elliptic curves, dubbed murmurations, were recently discovered using techniques from data science. Since then, similar patterns have been discovered for many other types of arithmetic objects. In this talk we use results from random matrix theory to prove the existence of murmurations in many cases, subject to standard conjectures in random matrix theory, and in two cases subject to GRH only. In particular, we handle the case of elliptic curves, about which essentially nothing was known hitherto, even heuristically, and which does not appear amenable to the techniques which have been used up to now in the study of murmurations.

To handle the case of elliptic curves an additional ingredient is needed: an estimation of the distribution of conductors in a family of elliptic curves ordered by height. This is of independent interest. We determine said distribution using elementary but technical methods in the spirit of the well-known Brumer-McGuinness-Watkins heuristics, and our result can be viewed as partial progress towards a proof of those heuristics.

The preprints associated to this talk are https://arxiv.org/abs/2408.12723 and https://arxiv.org/abs/2408.09745 .

Passcode: #^35#N3Lrq

Abstract: Let $K/k$ be a Galois extension of number fields with Galois group $G$. For a conjugacy class $C$ of $G$, the least unramified prime with Frobenius element in $C$ is known to be at most a fixed absolute power of the discriminant of $K$ due to the celebrated work of Lagarias, Montgomery, and Odlyzko. This theorem has been extensively studied with the primary method exploiting statistics of  zeros of L-functions. The current record for the exponent is 16 due to Kadiri, Ng, and Wong. For $G = S_n$, we will describe a method based on detecting sign changes  that improves this exponent to decay exponentially with $n$ as $n \to \infty$.  The ideas also apply to other groups $G$ and conjugacy invariant subsets $C$. This talk is based on joint work with Peter Cho and Robert Lemke Oliver. 

Abstract: The root number of an L-function captures important arithmetic information, such as, conjecturally, the parity of the rank in the case of elliptic curves. As such, statistics of root numbers can tell us about the typical behaviour of arithmetic objects. In joint work with Rahul Dalal, we prove an equidistribution result for root numbers of self-dual automorphic representations of GL_N as the weight varies. This is done in the framework of endoscopy and the stable trace formula.  Passcode: ^6W1nnr.JK

Abstract: The celebrated Serre--Tate theorem says that deformations of an abelian variety are naturally parameterized in terms of deformation of the abelian variety's Barsotti--Tate group. In particular, this says that the natural functor from Mumford's moduli spaces of principally polarized abelian varieties to the moduli stack of Barsotti--Tate groups is formally étale. In this talk I will discuss joint work with Naoki Imai and Hiroki Kato which shows a similar result holds true for integral canonical models of arbitrary Shimura varieties of abelian type (at hyperspecial level), and how this uniquely characterizes such models (at individual level). This involves the construction of a 'syntomic realization functor' on such integral canonical models. (passcode: 9^Pq5YDHVS)

Abstract: (Joint with Ben Bakker) Given an Elliptic curve E with complex multiplication, it is known that E has (potentially) good reduction everywhere. Concretely, this means that the j-invariant of E is an algebraic integer. The generalization of this result to Abelian-Varieties follows from the Neron-Ogg-Shafarevich criterion for good reduction.

We generalize this result to Exceptional Shimura varieties S. Concretely, we show that there exists some integral model S_0 of S such that all special points of S extend to (algebraically) integral points of S_0. To prove this we establish a Neron-Ogg-Shafarevich criterion in this setting. Our methods are general and apply, in particular, to arbitrary variations of hodge structures with an immersive Kodaira-Spencer map. 

We will explain the proof (which is largely in the realm of birational p-adic geometry) and the open questions that remain.

Abstract: We say that an integral model of an algebraic variety has the Hilbert Property if its set of integral points is not thin, which can be seen as a version of the Hilbert’s irreducibility theorem for this variety. A conjecture of Corvaja and Zannier predicts that a smooth simply connected variety with a Zariski dense set of S-integral points has the Hilbert Property for S-integral points, possibly after a finite enlargement of the base field and the set of primes S. In this talk, we will discuss the history and motivation for this problem, and we will explain how to prove this conjecture for certain log-K3 surfaces, with particular attention to the case of affine cubic surfaces. (passcode: 6BsMdd^y2Z)

Abstract: Bruhat-Tits' parahoric group schemes, denoted by P, are certain non-reductive integral O_K​-models of reductive groups G, defined over p-adic local fields K, and are associated with maximal compact subgroups of G(K). Extending the work of Balaji–Seshadri and Pappas–Rapoport, we demonstrate that these group schemes are reductive up to a finite, generically-Galois cover of O_K​. Leveraging this result, we show that generically trivial P-torsors over O are trivial in several important special cases. This is joint work with Balaji, Česnavičius, Elmanto and Youcis. (passcode: 8JPUhya+yv)

Abstract: Eisenstein proved, in 1852, that if a function f(z) is algebraic, then its Taylor expansion at a point has coefficients lying in some finitely-generated Z-algebra. I will explain ongoing joint work with Josh Lam which studies the extent to which the converse of this theorem holds. Namely, we conjecture that if f(z) satisfies a (possibly non-linear!) algebraic ODE, non-singular at 0, and its Taylor expansion has coefficients lying in a finitely-generated Z-algebra, then f is algebraic. For linear ODE, we prove this conjecture when (A) f(z) satisfies a Picard-Fuchs equation, with initial conditions the class of an algebraic cycle, or (B) f(z) is a modular form, as well as in a number of other cases. For non-linear ODE, we prove it when f(z) satisfies an "isomonodromy" ODE with "Picard-Fuchs" initial conditions.

Abstract: We introduce some aspects of Modular Representation Theory of finite groups. We will show how fusion systems arise

in the context of blocks, discuss conjectural connections and how those systems tie up with other aspects of the theory. (Passcode: WaJNG2A#s*)

Abstract: The geometric Langlands correspondence is a complex geometry analogue of the Langlands correspondence in number theory. I will explain the relationship and indicate some of the ingredients in tne proof. This is joint work with Arinkin, Beraldo, Campbell, Chen, Faergeman, Gaitsgory, Lin, and Raskin.

Abstract: The Quillen-Lichtenbaum conjecture asserts that the ratio of even and odd K-groups correspond to special values of zeta-functions in the number fields context. This was settled by Rost-Voevodsky modulo Iwasawa theory. I will give a number theorist-friendly introduction to this circle of ideas and explain a refinement of this conjecture obtained in joint work with Zhang where we expressed the ratio of even and odd equivariant K-groups as special values of Artin L-functions of certain Galois characters. (passcode: iiHr%#YA94)