RTG Seminar

Week 10 (April 21): Near-center derivatives and arithmetic 1-cycles

Ryan Chen (MIT)

Abstract:

Theta series for lattices count lattice vectors of fixed norm. Such theta series give some of the first examples of automorphic forms.

It is possible to form "theta series" in other geometric contexts, e.g. for counting problems involving abelian varieties.
It is expected that these theta series again have additional automorphic symmetry.

I will explain some “near-central” instances of an arithmetic Siegel--Weil formula from Kudla’s program. These "geometrize" the classical Siegel--Weil formulas, on lattice and lattice vector counting via Eisenstein series.
At these near-central points of functional symmetry, we observe that both the "leading" special value (complex volumes) and the "subleading" first derivative (arithmetic volume) simultaneously have geometric meaning.

The key input is a new "limit phenomenon" relating positive characteristic intersection numbers and heights in mixed characteristic, as well as its automorphic counterpart.

Week 11 (April 28): Arithmetic Siegel-Weil

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Week 12 (April 28): Arithmetic inner product formula

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Week 1 (February 3): Introduction

Kartik Prasanna

Abstract:

This semester the RTG seminar will focus on the Kudla program. I will give an introduction to this program, which seeks to understand the relation between arithmetic intersection theory on Shimura varieties and L-functions. While there are close connections to the body of conjectures relating algebraic cycles to L-functions (such as the Bloch-Beilinson conjectures), the Kudla program has some additional mysterious features. For instance, in certain cases, it involves not the leading Taylor coefficient of an L-function but rather the subleading term. Other topics that it is related to include the theory of the theta correspondence, Kudla-Millson theory and Borcherds lifts. This talk will explain some of the history of the subject and how some of these ideas tie together.

Notes

Week 2 (February 10): Classical Theta correspondence

Yu-Sheng Lee

Abstract:

We review the definition of Weil representations for a dual reductive pair and the theta correspondence between their representations. To motivate the Kudla-Milson program introduced in the previous talk, we discuss the Shimura correspondence between modular forms of integral and half-integral weights, as well as the period relations between them. In particular, I will explain how these period relations depend on the auxiliary choices that define the Weil representation.

Week 3 (February 17): Siegel-Weil formula and Rallis inner product formula

Jialiang Zou

Abstract:

In the first part , I will introduce the classical Siegel-Weil formula, which expresses an Eisenstein series as a weighted average of theta series associated with lattices in a genus. I will then explore how to interpret it within the framework of the theta lifts. In the second part, I will discuss the Rallis inner product formula, which relates the Petersson inner product of theta lifts to the special value of an L-function. I will explain how this result is established using a see-saw argument and the Siegel-Weil formula.

Notes

Week 4 (February 24): Theta dichotomy

Elad Zelingher

We discuss a local condition for the local theta lift not to vanish in the equal rank case of even unitary groups. This relation is expressed in terms of the root number associated to the representation in question. We use this relation to explain results about the non-vanishing of the global theta lift.

Notes

Week 5 (March 17): Unitary Shimura varieties

Calvin Yost-Wolff

This talk introduces the setting of the Kulda program we will study throughout the rest of the semester: unitary Shimura varieties. Geometrically, unitary Shimura varieties are arithmetic quotients of unit balls which are realizable as the complex points of a quasi-projective algebraic variety. Arithmetically, this variety descends to a variety over a totally real number field. Integrally, certain unions of these varieties represent a moduli problem which has an integral structure.

Week 6 (March 24): RTG: Special cycles and Kudla's generating function

Sean Cotner

We give a definition of special cycles on certain unitary Shimura varieties and use these to define Kudla's generating function, the subject of Kudla's modularity conjecture.

Week 7 (March 31): Geometric modularity, geometric theta lifting and Kudla-Millson theory

Kartik Prasanna

Week 8 (April 7): Geometric Siegel-Weil and geometric inner product formula

Miao (Pam) Gu

In this talk we first introduce the geometric Siegel-Weil formula, which relates the geometric theta series to Eisenstein series. We then give a geometric analogue of the Petersson inner product. By combining the geometric Siegel-Weil formula and the Rallis inner product formula we obtain the geometric inner product formula.

Week 9 (April 14): Arithmetic modularity and special cycles on integral models

Alex Bauman

Abstract:

We study the analogue of Kudla-Millson's work in the arithmetic setting. The arithmetic modularity conjecture states that Kudla's generating series defines a holomorphic automorphic form valued in the Chow group of the unitary Shimura variety. We also discuss known progress towards this conjecture. We next define the arithmetic Chow group, a refinement of the usual Chow group of an algebraic variety defined by Gillet-Soule in the spirit of Arakelov theory, which will be useful for defining further refinements of Kudla's conjecture.