Zhiyu Zhang

Besides linear algebra and differential calculus, my deepest interest lies in basic questions around arithmetic geometry, quadratic forms, symmetry and integrals, where diverse branches of mathematics intersect. They can be studied in a well-organized way under the Langlands program, with unexpected applications. You can study a question directly (which is often hard), or relate it to known methods and results.

I enjoy algebraic and analytic aspects of number theory and representation theory, and any reasonable question related to Shimura varieties.

Papers / Preprints

2026

2025

[12]

[11] Complex multiplication and twisted triple product . In preparation.

[10] Higher special cycles and orbital integrals (with X. Griffin Wang and Z. Yun). In preparation.

[9] Central values of Asai L-functions and twisted Gan--Gross--Prasad conjecture: relative trace formulas (with Weixiao Lu and Danielle Wang). Arxiv:2509.16356.

[8] On unitary Shimura varieties at ramified primes (with Yu Luo and Andreas Mihatsch). Arxiv:2504.17484. Submitted.

[7] Weighted cycles on Rapoport--Zink spaces with almost-self dual level structure (with Qiao He and Baiqing Zhu). Submitted.

[6] Non-reductive cycles and twisted arithmetic transfers for Shimura curves . Arxiv:2502.16754. RIMS Kôkyûroku, 2025 .

2024

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

[4] Non-reductive special cycles and twisted arithmetic fudamental lemmas . Arxiv:2406.00986. Submitted.

2023

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

2022

[2] Arithmetic transfers, modularity of arithmetic theta series and geometry of local--global Shimura varieties at parahoric levels. PHD thesis at Massachusetts Institute of Technology, 2022.

2021

[1] Maximal parahoric arithmetic transfers, resolutions and modularity. Arxiv:2112.11994. Duke Mathematical Journal 174 (1), 1-129, 2025.

Some words on past, current and future research

Elliptic curves with complex multiplication

A survey on twisted AFL and arithmetic trace formulas

Formulas about infinite sums and integrals are often interesting. One example is 1+ 1/2^2 + 1/3^2+ ... Another example is the area of a circle or a sphere. After qualitative analysis, we could ask about quantitative formulas. There remain many profound and fascinating phenomena in mathematics that we do not yet fully understand, though we aspire to. Our toolkit of methods—old and new—continues to evolve, shaped both by historical developments and by rethinking foundational ideas and computations, while standing on the shoulders of giants.

In my PHD thesis, I proved the related arithmetic fundamental lemmas and arithmetic transfer for ramified test functions for any prime p > 2. They are used crucially to prove a $p$-adic Gross-Zagier formula and related rank one Bloch-Kato conjecture for the standard L-functions of automorphic forms on $\GL_n \times \GL_{n+1}$ by Disegni-Zhang. See also lectures (C. Li, W. Zhang) of IHES 2022 Langlands summer school. Compared to previous works on the arithmetic fundamental lemma, I improve the global modularity method to a double induction method, overcame the the difficulty of singularity by Atiyah flops of moduli spaces, overcame the difficulty of small primes by relative Cayley maps, and use my new result on ramified modularity of arithmetic theta series. The new result on ramified modularity is proved using modifications in complex and mod p geometry and local duality of special divisors, and has many other applications.

In my recent work, I proposed an arithmetic analog of the twisted GGP conjecture using Hilbert-unitary Shimura varieties. The formulation involves non-reductive cycles (twisted Fourier-Jacobi cycles) and has applications to Bloch-Kato conjectures for twisted symmetric square motives. Moreover, I proved a related twisted arithmetic fundamental lemma (TAFL) for any prime p>2. There are several new ingredients in the formulations and proofs. There is no direct relation between the AFL and TAFL. In the proof, it is necessary to use certain non-reductive cycles to do reductions of arithmetic integrals. Also, we need to globalize via a general non-split quadratic extension $K$, leading to the study of Hilbert-unitary Shimura varieties and new twisted CM cycles. Also, the global modularity method is refined with new relative trace formulas. Moreover, the double induction method would lead to similar results on twisted arithemtic transfers.

There are many nice surveys on automorphic forms and L-functions, see e.g. Arthur and Gelbart. "Lectures on automorphic L-functions." (Durham, 1989), Shahidi, "Automorphic L-functions and functoriality" (2002 ICM report) and Jiang, "On Some Topics in Automorphic Representations" (2007 ICCM report). For the GGP conjecture, see Gross' survey "the road to GGP" (2020). For theta correspondences, see Gan's lectures ''Automorphic Forms and the Theta Correspondence'' and ''Explicit Constructions of Automorphic Forms: Theta Correspondence and Automorphic Descent''.

I am now interested in explicit relative Langlands program, in particular non-reductive period integrals (including theta liftings) and arithmetic applications. For example, I work on Asai L-functions (which occur naturally in many places) and the twisted GGP conjecture, with new arithmetic applications. In the case of a modular form, the twisted period recovers L-function of its Gelbart-Jacquet lift, or its symmetric square L-function. For a beautiful application to analytic number theory, see "Quadratic Hecke sums and mass equidistribution" by Paul D. Nelson.

Friends and Collaborators

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