Supplement to a Shimura's theorem on Eisenstein series

Pullbacks of Eisenstein series from GU(3,3) and critical L-values for GSp(4) X GL(2)

Let F be a genus two Siegel newform and g a classical newform, both of squarefree levels and of equal weight l. We prove a pullback formula for certain Eisenstein series -- thus generalizing a construction of Shimura -- and use this to derive an explicit integral representation for the degree eight L-function L(s, F X g). This integral representation involves the pullback of a simple Siegel-type Eisenstein series on the unitary group GU(3,3). As an application, we prove a reciprocity law -- predicted by Deligne's conjecture -- for the critical special values L(m, F X g) where m is an integer, …

Vanishing theorems for the mod $p$ cohomology of some simple Shimura varieties

We show that the mod $p$ cohomology of a simple Shimura variety treated in Harris-Taylor's book vanishes outside a certain nontrivial range after localizing at any non-Eisenstein ideal of the Hecke algebra. In cases of low dimensions, we show the vanishing outside the middle degree.

关于一般同余子群上艾森斯坦级数空间的注记

A note on the spaces of Eisenstein series on general congruence subgroups

本文提出利用主同余子群的Hecke艾森斯坦级数理论研究一般同余子群上谱艾森斯坦级数的新方法，得到相关空间的基与系数性质，并简化证明了Eichler-Shimura同构定理。

酉 Shimura 簇的模高 II: 除子的算术生成级数

Modular Heights of Unitary Shimura Varieties II: Arithmetic Generating Series of Divisors

本文通过定义并计算酉 Shimura 簇上除子的算术生成级数及其算术相交数，推导了酉 Shimura 曲线的模高公式及其上 CM 点的高度公式。

酉 Shimura 簇的模高度 III：主定理的证明

Modular Heights of Unitary Shimura Varieties III: Proof of the Main Theorem

通过比较混合 Eisenstein-theta 级数的全纯投影与除子生成级数的算术度，证明了 CM 数域上酉 Shimura 簇的模高度公式。

Mock modular Eisenstein series with Nebentypus

By the theory of Eisenstein series, generating functions of various divisor functions arise as modular forms. It is natural to ask whether further divisor functions arise systematically in the theory of mock modular forms. We establish, using the method of Zagier and Zwegers on holomorphic projection, that this is indeed the case for certain (twisted) "small divisors" summatory functions $σ_ψ^{\mathrm{sm}}(n)$. More precisely, in terms of the weight 2 quasimodular Eisenstein series $E_2(τ)$ and a generic Shimura theta function $θ_ψ(τ)$, we show that there is a constant $α_ψ$ for which $$ \math…

Special cycles and derivatives of Eisenstein series

This article sketches relations among algebraic cycles for the Shimura varieties defined by arithmetic quotients of symmetric domains for O(n,2), theta functions, values and derivatives of Eisenstein series and values and derivatives of certain L-functions. In the geometric case, results of joint work with John Millson imply that the generating functions for the classes in cohomology of certain algebraic cycles of codimension r are Siegel modular forms of genus r and weight n/2+1. A result of Borcherds shows that, for r=1, the same is true for the generating function for the classes of such di…

Generating series of intersection volumes of special cycles on unitary Shimura varieties

We form a generating series of regularized volumes of intersections of special cycles on a non-compact unitary Shimura variety with a fixed base change cycle. We show that it is a Hilbert modular form by identifying it with a theta integral, which we show converges even though the parameters lie outside the classical convergence range of Weil. By applying the regularized Siegel-Weil formulas of Ichino and Gan-Qiu-Takeda, we show the modular form is the restriction of a hermitian modular form of degree n related to Siegel Eisenstein series on U(n,n). An essential fact used is a computation show…

Arithmetic degrees of special cycles and derivatives of Siegel Eisenstein series

Let V be a rational quadratic space of signature (m,2). A conjecture of Kudla relates the arithmetic degrees of top degree special cycles on an integral model of a Shimura variety associated with SO(V) to the coefficients of the central derivative of an incoherent Siegel Eisenstein series of genus m+1. We prove this conjecture for the coefficients of non-singular index T when T is not positive definite. We also prove it when T is positive definite and the corresponding special cycle has dimension 0. To obtain these results, we establish new local arithmetic Siegel-Weil formulas at the archimed…

Improper Intersections of Kudla-Rapoport divisors and Eisenstein series

We consider a certain family of Kudla-Rapoport cycles on an integral model of a Shimura variety attached to a unitary group of signature (1,1), and prove that the arithmetic degrees of these cycles can be identified with the Fourier coefficients of the central derivative of an Eisenstein series of genus 2. The integral model in question parametrizes abelian surfaces equipped with a non-principal polarization and an action of an imaginary quadratic number ring, and in this setting the cycles are degenerate: they may contain components of positive dimension. This result can be viewed as confirma…

关于非紧Unitary Shimura簇上同调的一般部分

On the generic part of the cohomology of non-compact unitary Shimura varieties

本文证明了偶数维quasi-split unitary群相关的非紧Shimura簇的模l上同调一般部分集中在中度以上，并通过计算紧化Shimura簇上Hodge-Tate周期映射的纤维，推导出无自对偶假设的potential automorphy定理。

Special cycles on unitary Shimura varieties II: global theory

We introduce moduli spaces of abelian varieties which are arithmetic models of Shimura varieties attached to unitary groups of signature (n-1, 1). We define arithmetic cycles on these models and study their intersection behaviour. In particular, in the non-degenerate case, we prove a relation between their intersection numbers and Fourier coefficients of the derivative at s=0 of a certain incoherent Eisenstein series for the group U(n, n). This is done by relating the arithmetic cycles to their formal counterpart from Part I via non-archimedean uniformization, and by relating the Fourier coeff…

Derivatives of Eisenstein series and Faltings heights

We prove a relation between a generating series for the heights of Heegner cycles on the arithmetic surface associated to a Shimura curve and the second term in the Laurent expansion at s=1/2 of an Eisenstein series of weight 3/2 for SL(2). On the geometric side, a typical coefficient of the generating series involves the Faltings heights of abelian surfaces isogenous to a product of CM elliptic curves, an archimedean contribution, and contributions from vertical components in the fibers of bad reduction. On the analytic side, these terms arise via the derivatives of local Whittaker functions.…

On the analytic properties of a cubic Dirichlet series associated to a cubic metaplectic form

In this paper we study the analytic properties of a certain cubic Dirichlet series associated to a metaplectic form $f$ over the cubic cover of $GL_2.$ Such a sum generalizes the work of Shimura in studying a similar quadratic Dirichlet series for a half-weight modular form $f.$ Shimura connects the analytic properties of his Dirichlet series to the L-function of a holomorphic modular form via a converse theorem. This connection, and its higher cover generalizations, has been given the name: Shimura's correspondence. Even assuming Shimura's correspondence for the cubic cover of $GL_2,$ the ana…

Representations of SL_2(R) and nearly holomorphic modular forms

In this semi-expository note, we give a new proof of a structure theorem due to Shimura for nearly holomorphic modular forms on the complex upper half plane. Roughly speaking, the theorem says that the space of all nearly holomorphic modular forms is the direct sum of the subspaces obtained by applying appropriate weight-raising operators on the spaces of holomorphic modular forms and on the one-dimensional space spanned by the weight 2 nearly holomorphic Eisenstein series. While Shimura's proof was classical, ours is representation-theoretic. We deduce the structure theorem from a decompositi…

高秩退化Eisenstein级数的均值与量子方差

Mean Values and Quantum Variance for Degenerate Eisenstein Series of Higher Rank

本研究通过Watson-Ichino型公式和Jutila渐近公式，证明高秩退化Eisenstein级数平方内积的均值突破广义Lindelöf阈值，并计算量子方差，扩展了Zhang和Huang的工作。

几何埃西森系列、 intertwining 运算符与 Shin 的平均公式

Geometric Eisenstein Series, Intertwining Operators, and Shin's Averaging Formula

本文构造了几何埃西森函子，并基于其与 Verdier 对偶性的良好交互，提供了局部 Shimura 变形上同调的显式平均公式，扩展了 Shin 的公式到非 minuscule 情况。