## Algebraic Number Theory and Arithmetic Geometry

##### Session Organizers

Ellen Eischen, University of Oregon

Sug Woo Shin, University of California - Berkeley

Liang Xiao, Peking University

### 5 December, 2022

##### Abstract

For A an abelian variety defined over a number field, a conjecture of Serre predicts that the set S of primes of ordinary reduction for A has positive density.  If A is an elliptic curve without CM, Serre proved that S has density 1, and Elkies showed that, if L admits a real embedding, then the complement of S, that is the set of supersingular primes, is infinite.

In this talk, I will discuss generalizations of both Serre’s and Elkies’s theorems  to abelian varieties of type IV, that is with multiplication by a CM field, which are parametrized by unitary Shimura curves.

This talk is based on joint work in progress with Victoria Cantoral-Farfan, Wanlin Li, Rachel Pries, and Yunqing Tang.

##### Abstract

We study the dimensions of the eigenspaces for the Atkin-Lehner involutions acting on spaces of modular forms $M_k(\Gamma_0(Np))$, with the additional constraint imposed by fixing the mod $p$ Galois representation attached to the eigenforms. For this purpose we establish isomorphisms up to semisimplification between certain Hecke modules in characteristic $p$, generalizing Serre’s work relating modular forms mod $p$ to quaternion algebras. These isomorphisms are in turn obtained via a delicate study of congruences in the trace formulas for Hecke and Atkin-Lehner operators.

##### Abstract

We discuss applications of (a part of) the Iwasawa main conjecture to the non-triviality of Kato’s Kolyvagin systems and the structure of Selmer groups of elliptic curves over the rationals without any rank restriction.

##### Abstract

The Rapoport-Zink space for $\mathrm{GSp}(4)$ is a local counterpart of the Siegel threefold. Its $\ell$-adic etale cohomology is equipped with an action of the product of three groups: $\mathrm{GSp}_4(\mathbb{Q}_p)$, its non-trivial inner form $J(\mathbb{Q}_p)$, and the Weil group of $\mathbb{Q}_p$. This action is expected to be strongly related with the local Langlands correspondence for $\mathrm{GSp}_4$ and $J$. In this talk, I will explain how the $\mathrm{GSp}_4(\mathbb{Q}_p)$-supercuspidal part of the cohomology is described by the local Langlands correspondence. If time permits, I will also give some observations on the $\mathrm{GSp}(6)$ case.

### 6 December, 2022

##### Abstract

The Ichino-Ikeda conjecture is an explicit relation between the central L-value and squares of a certain period of automorphic forms.
This conjecture has been established by Beuzart-Plessis, Yifeng Liu, Wei Zhang, Xinwen Zhu, Chaudouard and Zydor for unitary groups. I will report on a joint work in progress with Michael Harris and Ming-Lun Hsieh on the construction of $p$-adic L-functions for $U(3)\times U(2)$ via the Ichino-Ikeda conjecture.

##### Abstract

In this talk, we will present our recent results on the local multiplicity formula of strongly tempered spherical subgroups. In particular, we formulate the epsilon dichotomy conjecture for those spherical subgroups and prove this conjecture for the tempered representations of endoscopy type. Globally, the corresponding automorphic periods are related to the central values of certain L-functions of symplectic type. The conjectural Ikeda–Ichino type formulas are given. This is a joint project with Chen Wan at Rutgers University, Newark.

##### Abstract

Hassett–Tschinkel and Benoist–Wittenberg recently introduced a new rationality obstruction that refines the classical the Clemens–Griffiths intermediate Jacobian obstruction to rationality, and exhibited its strength by showing that this new obstruction characterizes rationality for intersections of two quadrics. We show that this phenomenon does not extend to all geometrically rational threefolds. We construct examples of conic bundle threefolds over $\mathbb{P}^2$ that have no refined intermediate Jacobian obstruction to rationality, yet fail to be rational. This is joint work with S. Frei, L. Ji, S. Sankar, and I. Vogt.