Hirzebruch-Zagier类与五次数域上的有理椭圆曲线

Borcherds products and arithmetic intersection theory on Hilbert modular surfaces

We prove an arithmetic version of a theorem of Hirzebruch and Zagier saying that Hirzebruch-Zagier divisors on a Hilbert modular surface are the coefficients of an elliptic modular form of weight two. Moreover, we determine the arithmetic self-intersection number of the line bundle of modular forms equipped with its Petersson metric on a regular model of a Hilbert modular surface, and study Faltings heights of arithmetic Hirzebruch-Zagier divisors.

The Geometric Theta Correspondence for Hilbert Modular Surfaces

In a series of papers we have been studying the geometric theta correspondence for non-compact arithmetic quotients of symmetric spaces associated to orthogonal groups. It is our overall goal to develop a general theory of geometric theta liftings in the context of the real differential geometry/topology of non-compact locally symmetric spaces of orthogonal and unitary groups which generalizes the theory of Kudla-Millson in the compact case. In this paper we study in detail the geometric theta lift for Hilbert modular surfaces. In particular, we will give a new proof and an extension (to all f…

Twisted Teichmüller curves

Let $X_D$ denote the Hilbert modular surface $\HH \times \HH^- / \SL_2(\OD)$. In \cite{HZ76}, F. Hirzebruch and D. Zagier introduced Hirzebruch-Zagier cycles, that could also be called twisted diagonals. These are maps $\HH \to \HH \times \HH^-$ given by $z \mapsto (Mz,-M^σz)$ where $M \in \GL_2^+(K)$ and $σ$ denotes the Galois conjugate. The projection of a twisted diagonal to $X_D$ yields a Kobayashi curve, i.e. an algebraic curve which is a geodesic for the Kobayashi metric on $X_D$. Properties of Hirzebruch-Zagier cycles have been abundantly studied in the literature.\\Teichmüller curves a…

The rings of Hilbert modular forms for $\mathbb{Q}(\sqrt{29})$ and $\mathbb{Q}(\sqrt{37})$

We use Borcherds products and their restrictions to Hirzebruch-Zagier curves to determine generators and relations for the graded rings of Hilbert modular forms for the fields $\mathbb{Q}(\sqrt{29})$ and $\mathbb{Q}(\sqrt{37})$. These seem to be the first cases where the graded ring can be computed despite obstructions to the existence of Borcherds products with arbitrary divisors.

Arithmetic Intersection on a Hilbert Modular Surface and the Faltings Height

In this paper, we prove an explicit arithmetic intersection formula between arithmetic Hirzebruch-Zagier divisors and arithmetic CM cycles in a Hilbert modular surface over $\mathbb Z$. As applications, we obtain the first `non-abelian' Chowla-Selberg formula, which is a special case of Colmez's conjecture; an explicit arithmetic intersection formula between arithmetic Humbert surfaces and CM cycles in the arithmetic Siegel modular variety of genus two; Lauter's conjecture about the denominators of CM values of Igusa invariants; and a result about bad reductions of CM genus two curves.

与实二次数域中理想相关的椭圆Eisenstein级数

Elliptic Eisenstein series associated to ideals in real quadratic number fields

本文研究了实二次数域中的椭圆Eisenstein级数，计算了满足plus空间条件的非全纯Eisenstein级数的Fourier展开，并揭示了权重2时Fourier系数与Hirzebruch--Zagier除子几何的关联。

On class number relations, intersections, and GL(2)-tale over the function field side

The aim of this paper is to study class number relations over function fields and the intersections of Hirzebruch-Zagier type divisors on the Drinfeld-Stuhler modular surfaces. The main bridge is a particular "harmonic" theta series with nebentypus. Using the strong approximation theorem, the Fourier coefficients of this series are expressed in two ways; one comes from modified Hurwitz class numbers and another gives the intersection numbers in question. An elaboration of this approach enables us to interpret these class numbers as a "mass sum" over the CM points on the Drinfeld-Stuhler modula…

Hilbert 模曲面上的自守 Green 函数

Automorphic Green functions on Hilbert modular surfaces

本文将 Hilbert 模曲面上的自守 Green 函数理论推广至任意理想，给出 Fourier 展开、正则化、尖点增长、局部 Borcherds 乘积及生成级数模性等结果。

通过正则化theta提升的实二次域环类扩张中椭圆曲线的$L$-函数

$L$-functions of elliptic curves in ring class extensions of real quadratic fields via regularized theta liftings

利用正则化theta提升，将椭圆曲线$E/\mathbf{Q}$的$L$-函数在环类特征扭曲下的中心导数表示为模曲线$X_0(N) \times X_0(N)$上实测地线循环的算术除子的自守格林函数和，并关联到BSD常数和周期。

Modular Curves, Modular Surfaces, and Modular Fourfolds

This article surveys some recent work of the author on Hilbert modular fourfolds X. After some preliminaries on the cohomology and special, codimension 2 cycles Z on X of Hirzebruch-Zagier type, a proof of the Tate conjecture for X over abelian fields is exposed. It is followed by a description of a method to construct classes in the Bloch's Chow group CH^3(X,1) by using Hecke translates of the cycles Z as above with suitable intersections in translates of modular curves. The article ends with the introduction of a modular Gersten complex for a general Shimura variety X and the corresponding g…

Igusa class polynomials, embeddings of quartic CM fields, and arithmetic intersection theory

Bruinier and Yang conjectured a formula for an intersection number on the arithmetic Hilbert modular surface, CM(K).T_m, where CM(K) is the zero-cycle of points corresponding to abelian surfaces with CM by a primitive quartic CM field K, and T_m is the Hirzebruch-Zagier divisors parameterizing products of elliptic curves with an m-isogeny between them. In this paper, we examine fields not covered by Yang's proof of the conjecture. We give numerical evidence to support the conjecture and point to some interesting anomalies. We compare the conjecture to both the denominators of Igusa class polyn…

关于平方判别式Humbert曲面的几何性质

On the geometry of the Humbert surface of square discriminant

该研究确定了平方判别式Humbert曲面的Enriques--Kodaira型，通过构造对角Hirzebruch--Zagier除子分析Fricke型对合不动点，得到含扩张Cartan子群模曲线情形的亏格公式。