Zhiyu Zhang

Papers / Preprints

Some words on past, current and future research

Elliptic curves with complex multiplication

A survey on twisted AFL and arithmetic trace formulas

I enjoy asking and studying reasonable questions (and simpler analogs), via new approaches and tools (proposing first non-trivial steps, efforts needed to complete), or reducions to combinations of known methods and results. Besides linear algebra and calculus, my deepest interest lies in symmetry and geometry, where diverse branches of mathematics intersect. Conceptually, I enjoy generalizations / analogs of old ideas in new useful languages and categories, such as linear algebra over a base (still evolving). Concretely, I enjoy numbers, polynomial equations, power series, integrals, metrics and related harmonic analysis. I study them in a well-organized way generalizing classical Langlands program, with unexpected applications and exotic examples. I enjoy all questions from practice, utility, curiosity and creativity.

• basic questions: Gross-Zagier formula, Langlands program, Kudla program, Tate / Hodge conjecture and BSD conjecture (Millennium Prize).
• basic examples: power series, algebraic cycles and rational points, families / moduli of curves, surfaces and motives.
• locally symmetric spaces, moduli spaces, period maps and arithmetic models (Shimura varieties), completing ramified and relative stories.
• functions on these spaces (automorphic forms), period integrals and L-functions. The Waldspurger period (twisted Hecke period) has deep applications, to elliptic curves and sum of squares. I am working on more general twisted period integrals and applications.
• Lie / profinite / Iwahori-Weyl groups, monoids, cylces on affine flag varieties, components, geometric representation theory and applications.

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 arithmetic fundamental lemmas and arithmetic transfers for some 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 my new result on ramified modularity of arithmetic theta series.

Modularity of arithmetic theta series is a nice topic. In my thesis, I prove a new result on ramified modularity, based on modifications in complex and mod p geometry and local duality of special divisors. Now, it has found 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 in other contexts.

I really enjoy twisted versions of things, and generalizations from complex numbers to ramified cases. Deligne–Lusztig theory may be a generalization of parabolic induction, even when there is no stable parabolic; Weil representation for unitary groups may be a generalization of theta series and parabolic inductions, even when there is no parabolic; Hodge theory may be generalized to combinatorical objects, even when there are no algebraic varieties.

I am 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 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", Paul D. Nelson.

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'', ''Explicit Constructions of Automorphic Forms: Theta Correspondence and Automorphic Descent", Theta book (Gan-Kudla-Takeda), Chen-Bo Zhu's "Lectures on local theta correspondence".