Automorphic Forms and Arithmetic

The classical theory of Gelfand pairs and its generalizations over the complex numbers has many applications to number theory and automorphic forms, such as the uniqueness of Whittaker models and the non-vanishing of the central value of a triple product L-function. With an eye towards similar applications in the modular setting, this talk presents an extension of the classical theory to representations of finite groups over algebraically closed fields whose characteristics possibly divide the orders of the groups.

Given a prime p, a finite extension L/Qp, a connected p-adic reductive group G/L, and a smooth irreducible representation $\pi$ of G(L), Fargues-Scholze recently attached a semisimple Weil parameter to such $\pi$, giving a general candidate for the local Langlands correspondence. It is natural to ask whether this construction is compatible with known instances of the correspondence after semisimplification. For G=GLn and its inner forms, Fargues-Scholze and Hansen-Kaletha-Weinstein show that the correspondence is compatible with the correspondence of Harris-Taylor/Henniart. We verify a similar compatibility for G=GSp4 and its unique non-split inner form G=GU2(D), where D is the quaternion division algebra over L, assuming that L/Qp is unramified and p>2. In this case, the local Langlands correspondence has been constructed by Gan-Takeda and Gan-Tantono. Analogous to the case of GLn and its inner forms, this compatibility is proven by describing the Weil group action on the cohomology of a local Shimura variety associated to GSp4, using basic uniformization of abelian type Shimura varieties due to Shen, combined with various global results of Kret-Shin and Sorensen on Galois representations in the cohomology of global Shimura varieties associated to inner forms of GSp4 over a totally real field. After showing the parameters are the same, we apply some ideas from the geometry of the Fargues-Scholze construction explored recently by Hansen, to give a more precise description of the cohomology of this local Shimura variety, verifying a strong form of the Kottwitz conjecture in the process.

Interesting moduli spaces don't have many integral points. More precisely, if $X$ is a variety over a number field, admitting a variation of Hodge structure whose associate period map is injective, then the number of $S$-integral points on $X$ of height at most $H$ grows more slowly than $H^{\epsilon}$, for any positive $\epsilon$. This is a sort of weak generalization of the Shafarevich conjecture; it is a consequence of a point-counting theorem of Broberg, and the largeness of the fundamental group of $X$. Joint with Ellenberg and Venkatesh.
arxiv:2109.01043

Kolyvagin's original Euler system (1990), based on Heegner points, complemented the height formula of Gross and Zagier to prove a key case of the Birch and Swinnerton-Dyer conjecture. I will introduce some new Euler systems. They are of a species theorized by Jetchev--Nekovar--Skinner, and pertain to those representations of the Galois group of a CM field that are automorphic, carry a conjugate-symplectic form, and have the simplest Hodge--Tate type.
The construction is based on Kudla's special cycles on unitary Shimura varieties, under an assumption of modularity for their generating series. Together with a recent height formula by Li--Liu and the forthcoming theory of JNS, this reduces some cases of the Beilinson--Bloch--Kato conjecture to the injectivity of Abel--Jacobi maps.

(Joint work with Frank Calegari and Vesselin Dimitrov.) The unbounded denominators conjecture, first raised by Atkin and Swinnerton-Dyer, asserts that a modular form for a finite index subgroup of SL_2(Z) whose Fourier coefficients have bounded denominators must be a modular form for some congruence subgroup. In this talk, we will give a sketch of the proof of this conjecture based on a new arithmetic algebraization theorem.

In 1987, Elkies proved that an elliptic curve defined over the field of rational numbers has infinitely many primes of supersingular reduction. I will discuss a generalization of this result to the case of special cyclic covers of the projective line ramified at 4 points. This talk is based on joint work in progress with Wanlin Li, Rachel Pries and Yunqing Tang.

Buyukboduk and Sakamoto in 2019 proposed a precise conjectural formula relating the leading coefficient at the trivial zero s=0 of the cyclotomic Katz p-adic L-functions associated with ray class characters of a CM field K to suitable L-invariants/regulators of K. They were able to prove this formula in most cases when K is an imaginary quadratic field thanks to the existence of the Euler system of elliptic units/Rubin-Stark elements. In this talk, we will present a formula relating the first derivative of the cyclotomic Katz p-adic L-functions for general CM fields attached to ring class characters to the product of the L-invariant and the value of the improved Katz p-adic L-function at s=0. In particular, when the trivial zero occurs at s=0, we prove that the Katz p-adic L-function has a simple zero at s=0 if certain L-invariant is non-vanishing. Our method uses the congruence of Hilbert CM forms and does reply on the existence of the conjectural Rubin-Stark elements. This is a joint work with Adel Betina.

Shimura varieties are moduli spaces of abelian varieties with extra structures. Over the decades, various mathematicians (e.g. Rapoport, Kottwitz, etc.) have constructed integral models of Shimura varieties. In this talk, I will discuss some motivic aspects of integral models of Hodge type constructed by Kisin (resp. Kisin-Pappas). I will talk about recent work on removing the normalization step in the construction of such integral models, which gives closed embeddings of Hodge type integral models into Siegel integral models under some assumption. I will explain how this question is related to the Grothendieck Standard Conjecture D for abelian varieties, and sketch a proof of this type of questions if time permits. I will also mention an application to toroidal compactifications of Hodge type integral models.

The supersingular locus of a Unitary (2,m-2) Shimura variety parametrizes supersingular abelian varieties of dimension m, with an action of a quadratic imaginary field meeting the "signature (2,m-2)" condition. In some cases, for example when m=3 or m=4, every irreducible component of the supersingular locus is isomorphic to a Deligne-Lusztig variety, and the intersection combinatorics are governed by a Bruhat-Tits building. We'll consider these cases for motivation, and then see how the structure of the supersingular locus becomes very different for m>4. (The new result in this talk is joint with Naoki Imai.)

Let $p$ be a fixed odd prime and $K$ an imaginary quadratic field where $p$ is inert. Let $f$ be an elliptic modular form with good ordinary reduction at $p$. We discuss how the cyclotomic Iwasawa theory of the Rankin-Selberg product of $f$ and a $p$-non-ordinary CM form allows us to study the Iwasawa theory of $f$ over the $\mathbf{Z}_p^2$-extension of $K$. We make use of the plus and minus theory of Kobayashi and Pollack as well as Euler systems built out of Beilinson--Flach elements. This is joint work with Kazim Buyukboduk.

We determine the behavior of automorphic Green functions along the boundary components of toroidal compactifications of orthogonal Shimura varieties. We use this analysis to define boundary components of special divisors and prove that the generating series of the resulting special divisors on a toroidal compactification is modular. This is joint work with Jan Bruinier.