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
Higher Theta series, Part II (Colloquium)
Wednesday, February 28 at 3:15 pm
University 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
2022 Mark Kisin
Theme: Arithmetic of abelian varieties and their moduli
Essential dimension and prismatic cohomology (Colloquium)
Wednesday, December 7 at 3:15pm
University 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
Thursday, December 8 at 2:00pm
University 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 ρℓ: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 ρℓ(Frobv) does not depend on ℓ. This is joint work with Rong Zhou.
Heights in the isogeny class of an abelian variety
Friday, December 9 at 2:00pm
University 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.
2021 Wei Zhang
2019 Ngô Bảo Châu