## 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**

**References：**

[Bump98] D. Bump, Automorphic forms and representations. No. 55. Cambridge university press, 1998.

[Clozel06] L. Clozel, Motives and automorphic representations. preprint, 2006.

[Coates] J. Coates, Motivic p-adic L-functions. L-functions and arithmetic (Durham, 1989). 1991 Feb 22;153:141-72.

[TYZ21] T. Feng, Z. Yun, W. Zhang, Higher theta series for unitary groups over function fields. arXiv preprint arXiv:2110.07001. 1,2

[GZ86] B. Gross, D. Zagier, Heegner points and derivatives of $L$-series, Invent. Math. 84 (1986), no. 2,225--320. 1

[HLSY22] Q. He, C. Li, Y. Shi, T. Yang. A proof of the Kudla-Rapoport conjecture for Krämer models. arXiv preprint arXiv:2208.07988.

[Kudla97] S. Kudla. Central derivatives of Eisenstein series and height pairings. Ann. of Math.(2), 146(3):545–646, 1997. 1

[Kudla04] S. Kudla, Special cycles and derivatives of Eisenstein series, in Heegner points and Rankin L-series, volume 49 of Math. Sci. Res. Inst. Publ., pages 243–270. Cambridge Univ. Press, Cambridge, 2004. 1

[KR11] S. Kudla, M. Rapoport, Special cycles on unitary Shimura varieties I. Unramified local theory. Invent. math, (2011), 184(3), 629-682.

[Li21] C. Li, From sum of two squares to arithmetic Siegel-Weil formulas, arXiv preprint arXiv:2110.07457, 2021. 1

[LL21] C. Li, Y. Liu, Chow groups and L-derivatives of automorphic motives for unitary groups. Ann. of Math..2021 Nov 1;194(3):817-901. 1

[LZ22] C. Li, W. Zhang, Kudla–Rapoport cycles and derivatives of local densities, Journal of the American Mathematical Society (2022), 35.3, 705-797. 1

[LTXZZ22] Y. Liu, Y. Tian, L. Xiao, W. Zhang and X. Zhu, On the Beilinson–Bloch–Kato conjecture forRankin–Selberg motives, Invent. Math (2022), 228(1), 107-375.[MM92] Y. Manin, Lectures on zeta functions and motives (1992).

[NN16] J. Nekovář, W. Nizioł, Syntomic cohomology and p-adic regulators for varieties over p-adic fields. Algebra Number Theory. 2016 Oct 7;10(8):1695-790.