[Paper Review] Periods, cycles, and $L$-functions: a relative trace formula approach

Wei Zhang|arXiv (Cornell University)|Dec 23, 2017

Advanced Algebra and Geometry6 references3 citations

TL;DR

This paper develops a relative trace formula approach to connect automorphic periods, special cycles on Shimura varieties, and central values/derivatives of L-functions, extending classical formulas of Gross–Zagier and Waldspurger. It establishes the arithmetic fundamental lemma conjecture for unitary Rapoport–Zink spaces and proves higher-order Gross–Zagier formulas in the function field setting via Drinfeld Shtukas.

ABSTRACT

This is a report for the author's talk in ICM-2018. Motivated by the formulas of Gross--Zagier and Waldspurger, we review conjectures and theorems on automorphic period integrals, special cycles on Shimura varieties, and their connection to central values of L-functions and their derivatives. We focus on the global Gan--Gross--Prasad conjectures, their arithmetic versions and some variants in the author's joint work with Rapoport and Smithling. We discuss the approach of relative trace formulas and the arithmetic fundamental lemma conjecture. In the function field setting, Z. Yun and the author obtain a formula for higher order derivatives of L-functions in terms of special cycles on the moduli space of Drinfeld Shtukas.

Motivation & Objective

• To generalize classical formulas of Dirichlet, Gross–Zagier, and Waldspurger to higher-rank automorphic L-functions using relative trace formulas.
• To establish a connection between arithmetic intersection numbers on Rapoport–Zink spaces and derivatives of orbital integrals.
• To formulate and prove the arithmetic fundamental lemma (AFL) conjecture for unitary groups over p-adic fields.
• To extend the Gross–Zagier formula to higher-order derivatives of L-functions in the function field setting using Drinfeld Shtukas.
• To provide a geometric interpretation of special cycles and their intersection theory in relation to L-values and automorphic periods.

Proposed method

• Utilizes the relative trace formula to relate automorphic period integrals over spherical subgroups to L-values.
• Applies the arithmetic fundamental lemma (AFL) conjecture to relate orbital integrals of characteristic functions to arithmetic intersection numbers on Rapoport–Zink spaces.
• Employs derived intersection theory on formal moduli spaces of Drinfeld Shtukas to compute intersection products.
• Uses the derived tensor product of structure sheaves to define the intersection number Int(g) for translates of diagonal cycles.
• Applies the theory of spherical harmonic analysis and orbital integrals in the p-adic setting to relate trace formulas to L-function derivatives.
• Leverages Yun’s function field analog of the AFL conjecture to derive higher-order Gross–Zagier formulas in the function field setting.

Experimental results

Research questions

• RQ1 How can the Gross–Zagier and Waldspurger formulas be generalized to higher-rank reductive groups and automorphic L-functions?
• RQ2 What is the precise relationship between arithmetic intersection numbers on Rapoport–Zink spaces and derivatives of L-functions?
• RQ3 To what extent does the arithmetic fundamental lemma (AFL) conjecture hold for unitary groups over p-adic fields?
• RQ4 Can higher-order derivatives of L-functions be expressed in terms of special cycles on moduli spaces of Drinfeld Shtukas?
• RQ5 What is the role of relative trace formulas in linking automorphic periods, special cycles, and L-values in arithmetic geometry?

Key findings

• The arithmetic fundamental lemma (AFL) conjecture is proven for n=2,3 and for minuscule elements g∈G♭(F) when p≥n/2+1.
• The AFL conjecture is verified for minuscule elements in G♭(F) under the condition p≥n/2+1, extending earlier results.
• A higher-order Gross–Zagier formula is established in the function field setting via the intersection theory of special cycles on Drinfeld Shtuka moduli spaces.
• The derivative of the orbital integral ∂Orb(γ,1_G′(O_F)) is shown to be proportional to the arithmetic intersection number Int(g) via the identity ω(γ)∂Orb(γ,1_G′(O_F)) = −2·Int(g)·log q.
• The AFL conjecture is proven for n=2,3 and extended to minuscule elements in G♭(F) with p≥n/2+1, using derived intersection theory on Rapoport–Zink spaces.
• Yun’s function field analog of the AFL conjecture provides a geometric formula for higher-order derivatives of L-functions in terms of special cycles on moduli spaces of Drinfeld Shtukas.