Arithmetic geometry, hermitian lattices and theta correspondences
Speaker(s): Zhiyu Zhang (MIT)
Time: July 27 - July 31, 2023
Venue: Room 77201, Jingchunyuan 78, BICMR
Abstract：
In this mini-course, we will talk about the Kudla program and some recent breakthroughs [TYZ21] [LL21] [LZ22] , with applications to Beilinson-Bloch--Kato conjectures for tempered automorphic motives appearing in the cohomology of unitary Shimura varieties. The Kudla program is about geometric and arithmetic theta correspondences for certain reductive pairs $(G, H)$ (e.g. $(U(n,n), U(V))$. Building on several landmark works (Gross--Zagier, Hirzebruch--Zagier, Gross--Keating, Kudla--Millson...), Kudla [Kudla97] invented the fundamental program relating special algebraic cycles (on Shimura varieties for $H$) to specific automorphic generating series (Siegel--Eisenstein series for $G$), hence to L-functions via the doubling method. See the excellent surveys [Kudla04] [Li21].
Lecture I: Arithmetic of automorphic motives and hermitian lattices
A central question in arithmetic geometry is to understand rational points of algebraic varieties (or geometric motives) over global fields. The BSD conjecture gives a nice answer for elliptic curves via their L-functions. In general, we have the Beilinson--Bloch--Kato conjectures for motives. We will introduce a local--global approach to these conjectures using special cycles on Shimura varieties, including the Gross--Zagier formula [GZ86] and the arithmetic inner product formula [LL21] in the Kudla program. We will include some backgrounds from Langlands program, automorphic representation theory and Shimura varieties, including L-functions of motives and representation theory of real groups.
Lecture II: Local densities and Kudla--Rapoport conjectures
I will give more introductions to the Kudla program, focusing on the local story[KR11] [LZ22]. For a hermitian lattice $L$ over $p$-adic fields, we will introduce the notion of local density polynomial $\mathrm{Den}(X, L) \in \mathbb Z[X]$. Assume the valuation of $L$ is odd, then the local density $\mathrm{Den}(L)=0$. We will introduce and discuss the proof of the Kudla--Rapoport conjecture for $L$, which predicts a precise identity between the derived local density $\partial \mathrm{Den}(L)$ and the arithmetic intersection number $\mathrm{Int}(L)$ of special cycles on unitary Rapoport--Zink spaces. It is a key local ingredient to establish the arithmetic Siegel--Weil formula and the arithmetic Rallis inner product formula. Time permitted, we will discuss some recent and ongoing works on variants of Kudla--Rapoport conjectures with levels.
Lecture III: Arithmetic modularity, theta liftings and Siegel--Weil formulas
I will give more introductions to the Kudla program, focusing on the global story [Kudla97] [LZ22]. The theta series of a positively definite quadratic lattice is a modular form, and could be described as a theta lifting from definite orthogonal groups and more precisely as certain Eisenstein series (the Siegel--Weil formula) hence related to L-functions ( the Rallis inner product formula). The precise modularity has many applications e.g. the quantitative Lagrange's four--square theorem. We will introduce geometry/arithmetic theta series of non-positively definite hermitian lattices as generating series of special cycles on unitary Shimura varieties. We will discuss recent developments on arithmetic modularity, theta liftings, Siegel--Weil formulas.
Lecture IV: Derived algebraic geometry and higher theta series
I will introduce the higher theta series over moduli of hermitian shtukas following [TYZ21], which is conjectured to be modular and satisfies higher Siegel--Weil formulas. Moreover, we will give a brief introduction to the derived algebraic geometry, and apply it to the construction of virtual fundamental classes for special cycles on the moduli stack of hermitian shtukas.
Time
7.27\28\30\31 : 13:30-15:30
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