Zhiyu Zhang

Papers by topics

Automorphic forms, L-functions, periods and relative Langlands duality

1. On general twisted Gan-Gross-Prasad conjectures and twisted Asai L-functions (with Weixiao Lu and Danielle Wang). In preparation.

Cycles on moduli spaces and arithmetic Langlands duality

2. Mixed cycles on Rapoport--Zink space with special maximal parahoric level structure. (with Qiao He and Baiqing Zhu). In preparation.

3. Non-reductive special cycles and twisted arithmetic fudamental lemmas . Arxiv:2406.00986. Submitted.

4. Kudla-Rapoport conjecture at unramified primes with maximal parahoric level (with Sungyoon Cho and Qiao He). Arxiv:2312.16906. Submitted.

5. Arithmetic transfers, modularity of arithmetic theta series and geometry of local--global Shimura varieties at parahoric levels. PHD thesis at MIT, 2022.

6. Maximal parahoric arithmetic transfers, resolutions and modularity. Arxiv:2112.11994 . Duke Mathematical Journal , to appear.

Taylor expansion, Shtukas, Hitchin moduli and higher Langlands duality

7. Relative Fundamental Lemmas for Spherical Hecke Algebras and Multiplicative Hitchin Fibrations: the Jacquet--Rallis Case (with X. Griffin Wang). Arxiv:2408.15155. Submitted.

Geometry, decomposition and invariants of moduli spaces and cycles

8. On unitary Shimura varieties at ramified primes. (With Yu Luo and Andreas Mihatsch). In preparation.

Research Statement

Examples: solving Diophantine equations, volumes, eigenvalues, L-functions, counting points, generating series, orbital integrals, intersection numbers, and shapes of geometric spaces with symmetry....

I study moduli spaces (e.g. moduli of motives and G-bundles), tautological classes / correspondences and special functions / cycles / sheaves on them, which give better understandings of moduli spaces.

I study modular forms, automorphic forms and relative duality in the Langlands program, where fundamental lemmas are keys to establish Langlands functoriality, e.g. endoscopic classifications, character relations, base change, local Jacquet-Langlands transfer and Langlands for classical groups.

In my PHD thesis, I extended a global method to prove (almost) modularity of arithmetic theta series and establish arithmetic fundamental lemmas with ramifications (necessary for applications: no level 1 weight 2 modular form exists), and further in my work on non-reductive cycles and twisted AFL, in an framework for arithmetic relative Langlands program (including certain non-algebraic cycles). I have been studying related applications to Kudla program, intersection numbers, Gross-Zagier formulas, explicit mod p geometry and function field analogs.

Currently I enjoy thinking about

I hope to formulate new conjectures and geometric objects (with examples) and prove new theorems. There are still many exciting things to be further explored.

Conjectures on L-functions (generalizing zeta functions): Hasse-Weil conjecture, Riemann hypothesis, Deligne conjecture, relative Langlands duality, Braverman-Kazhdan-Ngo program, Zagier's conjecture, construction of p-adic L-functions, relations to Mahler measures, Stark's conjecture (and Kronecker limit formula), Beilinson conjecture, and Bloch-Kato conjecture.

Related Topics: arithmetic of quadratic lattices, enumerative geometry, invariant theory, spectral theory, sheaf theory, Lie groups and Lie algebras, group actions on manifolds, moduli spaces, equidistribution and dynamics, analytic geometry, analytic methods, algebraic topology, Hodge theory, combinatorics, Hamiltonian geometry, mirror symmetry and mathematical physics.

Friends and Collaborators

I learnt a lot from other people on math and other things.