Qiao He

I am a sixth year graduate student in the Department of Mathematics at University of Wisconsin-Madison. My advisor is Professor Tonghai Yang. I am broadly interested in number theory and arithmetic geometry.

I will be a Ritt Assistant Professor at Columbia University starting in Fall 2023.Email: qhe36 at wisc dot edu Here is my CV.

Activities

I will be a Salgo-Noren Program Associate in the program Algebraic Cycles, L-Values, and Euler Systems during the Spring semester of 2023.

I will be the study group leader for Tsimerman's lecture in Arizona Winter School 2023: Unlikely Intersections.

Research

1. Just-likely intersections on Hilbert modular surfaces, with Asvin G. and Ananth N. Shankar, submitted, 13pp.

We prove an intersection-theoretic result pertaining to curves in certain Hilbert modular surfaces in positive characteristic. Specifically, we show that given two appropriate curves C,D parameterizing abelian surfaces with real multiplication, the set of points (x,y) \in C\times D with surfaces parameterized by x and y isogenous to each other is Zariski dense in C\times D, thereby proving a case of a just-likely intersection conjecture. We also compute the change in Faltings height under appropriate p-power isogenies of abelian surfaces with real multiplication over characteristic p global fields.

2. A proof of the Kudla-Rapoport conjecture for Kramer models, with Chao Li , Yousheng Shi and Tonghai Yang, submitted, 82pp.

We prove the Kudla-Rapoport conjecture for Kramer models proposed in my earlier paper with Yousheng Shi and Tonghai Yang. For the geometric side, we completely avoid explicit calculation and the Tate conjecture for the relevant Deligne-Lusztig varieties. The new observation is that a coarse information about translation invariance and Fourier transform is sufficient for our purpose. As a trade off, the analytic side is substantially more challenging. However, although the formula of a primitive local density polynomial is complicated, as our major technical innovation, we manage to prove a strikingly simple formula for its derivative. After the preparations, we prove the conjecture by applying partial Fourier transform and induction.

3. Kudla-Rapoport conjecture for Kramer models, with Yousheng Shi and Tonghai Yang.

To appear in Compositio Mathematica, 64pp.

Based on the observations we made from our earlier work, we propose a Kudla-Rapoport conjecture for Kramer models, which is a precise identity between arithmetic intersection numbers of special cycles on Kramer models and modified derived local densities of hermitian forms. We also verify the conjecture when n=3. In order to compare the geometric side and analytic side, we introduce the notion of difference cycle, which is an analogue of primitive local density and have surprisingly simple property. By an induction argument, now it suffices to compare the intersection between difference cycles and derivatives of certain primitive local density polynomials, which is significantly simpler than a direct comparison.

4. Kudla program for unitary Shimura varieties. Published version (in Chinese), With Yousheng Shi and Tonghai Yang.

Sci Sin Math, 2021, 51: 1595-1626, doi: 10.1360/SSM-2021-0002 (Special issue in honor of Professor Keqin Feng's 80th birthday).

This is mainly a survey paper, where we review various topics about Kudla program for unitary Shimura varieties.

5. The Kudla-Rapoport conjecture at a ramified prime for U(1,1), with Yousheng Shi and Tonghai Yang.

To appear in Transactions of the AMS.

We establish a Kudla-Rapoport type formula for U(1,1), which is a precise identity between arithmetic intersection numbers of special cycles on 2-dimensional Kramer models and modified derived local densities of hermitian forms. We use Grothendieck-Messing theory and the exceptional isomorphism between the formal completion of Drinfeld upper plane and our Rapport-Zink space to derive an explicit equation for each special divisor. Given the equation, we are able to determine the horizontal part and vertical part of the special divisor, which enables us to calculate the intersection numbers between special divisors explicitly and compare with the analytic side.

6. On conjectures of Samart, with Dongxi Ye.

manuscripta mathematica, 2022, 167: pages 545-588, doi: https://doi.org/10.1007/s00229-021-01279-6

We prove conjectures of Smart on the Mahler measure of Laurent polynomials related with various elliptic curves and K3 surfaces. More precisely, we use properties of spherical theta functions to show that the Mahler measures can be expressed in terms of special values of modular L-functions and Dirichlet L-series. As a byproduct of our method, we derive some new formulas of Samart-type.