Welcome to the Maryland Distinguished Lectures in Algebra and Number Theory! This is an annual lecture series concerning recent developments in Algebra and Number Theory, delivered by the world's foremost experts. It is organized by Thomas Haines.

### 2024 Zhiwei Yun

### Theme: Higher theta series

* Abstract:* Theta series play an important role in the classical

theory of modular forms. In the modern language of automorphic

representations, they are constructed from a pair of groups $G$ and

$H$ (one orthogonal and one symplectic, or both unitary groups) and

the remarkable Weil representation of $G\times H$. Kudla introduced

an analogue of theta series in arithmetic geometry, by forming a

generating series of algebraic cycles on Shimura varieties. The

arithmetic theta series has since become a very active program.

In joint work with Tony Feng and Wei Zhang, we consider analogues of

arithmetic theta series over function fields, and try to go further

than what was done over number fields. Our work concentrated on

unitary groups. We defined a generating series of algebraic cycles on

the moduli stack of unitary Drinfeld Shtukas (called the higher theta

series). We made the Modularity Conjecture: the higher theta series

is an automorphic form valued in a certain Chow group. This is a

function field analogue of the special cycles generating series

defined by Kudla and Rapoport, but with an extra degree of freedom

namely the number of legs of the Shtukas.

One concrete formula we proved was a higher derivative version of the

Siegel-Weil formula. It is an equality between degrees of

0-dimensional special cycles on the moduli of unitary Shtukas and

higher derivatives of the Siegel-Eisenstein series of another unitary

group. More recently, we have obtained a proof of a weaker version of

the Modularity Conjecture, confirming that the cycle class of the

higher theta series (valued in the cohomology of the generic fiber) is

automorphic.

The series of talks will feature a colloquium-style introduction to

some representation-theoretic and geometric background (the second talk), the other

two being more technical talks in which I will explain some ingredients in

the proofs of the higher Siegel-Weil formula and the weak Modularity

Conjecture.

*Higher Theta series, Part I*

**Tuesday, February 27 at 3:30 pm**

University of Maryland - Kirwan Hall Rm 3206

University of Maryland - Kirwan Hall Rm 3206

*Higher Theta series, Part II (Colloquium)*

**Wednesday, February 28 at 3:15 pmUniversity of Maryland - Kirwan Hall Rm 3206**

*Higher Theta series, Part III*

**Thursday, February 29 at 3:30 pm**

University of Maryland - Kirwan Hall Rm 3206

University of Maryland - Kirwan Hall Rm 3206

### 2022 Mark Kisin

### Theme: Arithmetic of abelian varieties and their moduli

**Essential dimension and prismatic cohomology (Colloquium)**

**Essential dimension and prismatic cohomology (Colloquium)**

**Wednesday, December 7 at 3:15pmUniversity of Maryland - Kirwan Hall Rm 3206**

* Abstract:* The smallest number of parameters needed to define an algebraic covering space is called its essential dimension. Questions about this invariant go back to Klein, Kronecker and Hilbert and are related to Hilbert's 13th problem. In this talk, I will give a little history, and then explain a new approach which relies on recent developments in $p$-adic Hodge theory. This is joint work with Benson Farb and Jesse Wolfson.

**Frobenius conjugacy classes attached to abelian varieties**

**Frobenius conjugacy classes attached to abelian varieties**

**Thursday, December 8 at 2:00pmUniversity of Maryland - Kirwan Hall Rm 3206**

** Abstract:** The Mumford-Tate group of an abelian variety $A$ over the complex numbers is an algebraic group $G$, defined in terms of the complex geometry of $A$, more specifically its Hodge structure. If $A$ is defined over a number field $K$, then a remarkable result of Deligne asserts that the ℓ -adic cohomology of $A$ gives rise to a $G$ -valued Galois representation ρ

_{ℓ}:Gal($\bar K/K$)→$G(\mathbb Q_\ell)$. We will show that for a place of good reduction v∤ℓ of A, the conjugacy class of Frobenius ρ

_{ℓ}(Frob

_{v}) does not depend on ℓ. This is joint work with Rong Zhou.

**Heights in the isogeny class of an abelian variety**

**Heights in the isogeny class of an abelian variety****Friday, December 9 at 2:00pmUniversity of Maryland - Kirwan Hall Rm 3206**

* Abstract:* Let $A$ be an abelian variety over $\bar{\mathbb Q}.$ In this talk I will consider the following conjecture of Mocz. Conjecture: Let $c > 0.$ In the isogeny class of $A,$ there are only finitely many isomorphism classes of abelian varieties of height $ c.$ I will sketch a proof of the conjecture when the Mumford-Tate conjecture - which is known in many cases - holds for $A.$ This result should be compared with Faltings' famous theorem, which is about finiteness for abelian varieties defined over a fixed number field. This is joint work with Lucia Mocz.

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**2021 Wei Zhang**

**2021 Wei Zhang**

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**2019 Ngô Bảo Châu**

**2019 Ngô Bảo Châu**

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