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 archimedean and non-archimedian places.
A generalization of Serre's $p$-adic Eisenstein series in the case of Siegel modular forms is studied and a coincidence between a $p$-adic Siegel Eisenstein series and a genus theta series associated with a quaternary quadratic form is proved.
We find nice representatives for the 0-dimensional cusps of the degree $n$ Siegel upper half-space under the action of $Γ_0(\stufe)$. To each of these we attach a Siegel Eisenstein series, and then we make explicit a result of Siegel, realizing any integral weight average Siegel theta series of arbitrary level $\stufe$ and Dirichlet character $χ_L$ modulo $\stufe$ as a linear combination of Siegel Eisenstein series.
We construct a basis for the space of half-integral weight Siegel Eisenstein series of level 4N where N is odd and square-free. Then we restrict our attention to those Eisenstein series generated from elements of $Γ_0(4)$, commenting on why this restriction is necessary for our methods. We directly apply to these forms all Hecke operators attached to odd primes, and we realize the images explicitly as linear combinations of Siegel Eisenstein series. Using this information, we diagonalize the subspace of Eisenstein series generated from elements of $Γ_0(4)$, obtaining a multiplicity-one result.
We study nearly holomorphic Siegel Eisenstein series of general levels and characters on $\mathbb{H}_{2n}$, the Siegel upper half space of degree $2n$. We prove that the Fourier coefficients of these Eisenstein series (once suitably normalized) lie in the ring of integers of $\mathbb{Q}_p$ for all sufficiently large primes $p$. We also prove that the pullbacks of these Eisenstein series to $\mathbb{H}_n \times \mathbb{H}_n$ are cuspidal under certain assumptions.
The aim of this article is to study the derivative of "incoherent" Siegel-Eisenstein series on symplectic groups over function fields. By the Siegel-Weil formula for "coherent" Siegel-Eisenstein series, we can relate the non-singular Fourier coefficients of the derivative in question to the arithmetic of quadratic forms. Restricting to the special case when the incoherent quadratic space has dimension 2, we explicitly compute all the Fourier coefficients, and connect the derivative with the special cycles on the coarse moduli schemes of rank 2 Drinfeld modules with "complex multiplication."
We compute the Fourier coefficients of all degree 2 Siegel-Eisenstein series of square-free level $N$ transforming with the trivial character. We then apply use these formulae to present some explicit examples of higher representation numbers attached to non-unimodular quadratic forms.
In this paper we obtain a weighted average formula for special values of $L$-functions attached to normalized elliptic modular forms of weight $k$ and full level. These results are obtained by studying the pullback of a Siegel Eisenstein series and working out an explicit spectral decomposition.
Eisenstein classes of Siegel varieties are motivic cohomology classes defined as pull-backs by torsion sections of the polylogarithm prosheaf on the universal abelian scheme. By reduction to the Hilbert-Blumenthal case, we prove that the Betti realization of these classes on Siegel varieties of arbitrary genus have non-trivial residue on zero dimensional strata of the Baily-Borel compactification. A direct corollary is the non-vanishing of a higher regulator map.
We evaluate the action of Hecke operators on Siegel Eisenstein series of degree 2, square-free level and arbitrary character, without using knowledge of their Fourier coefficients. From this we construct a basis of simultaneous eigenforms for the full Hecke algebra, and we compute their eigenvalues. As well, we obtain Hecke relations among the Eisenstein series. Using these Hecke relations in the case that $\stufe$ is square-free and the character is trivial, we generate a basis for the space of Eisenstein series.
If E(z,s) is the nonholomorphic Eisenstein series on the upper half plane, then for all y sufficiently large, E(z,s) has a "Siegel zero." That is E(z,β)=0 for a real number βjust to the left of one. We give a generalization of this result to Eisenstein series formed with real valued automorphic forms on a finite central covering of the adele points of a connected reductive algebraic group over a global field.
We compute Hecke eigenform bases of spaces of level one, degree~three Siegel modular forms and 2-Euler factors of the eigenforms through weight 22. Our method uses the Fourier coefficients of Siegel Eisenstein series, which are fully known and computationally tractable by the work of H. Katsurada; we also use P. Garrett's decomposition of the pullback of the Eisenstein series through the Witt map. Our results support I. Miyawaki's conjectural lift, and they give examples of eigenforms that are congruence neighbors.
For any rational prime $p$, we define a certain $p$-stabilization of holomorphic Siegel Eisenstein series for the symplectic group ${\rm Sp}(2n)_{/\mathbb{Q}}$ of an arbitrary genus $n \ge 1$. In addition, we derive an explicit formula for the Fourier coefficients and conclude their $p$-adic interpolation problems. Consequently, for any odd prime $p$, we deduce the existence of a $Λ$-adic form (in the sense of A. Wiles, H. Hida and R.L. Taylor) such that after taking a suitable constant multiple, it interpolates $p$-adic analytic families of the above-mentioned $p$-stabilized Siegel Eisenstein…
We evaluate the action of Hecke operators on Siegel Eisenstein series of arbitrary degree, level and character. For square-free level, we simultaneously diagonalise the space with respect to all the Hecke operators, computing the eigenvalues explicitly, and obtain a multiplicity-one result. For arbitrary level, we simultaneously diagonalise the space with respect to the Hecke operators attached to primes not dividing the level, again computing the eigenvalues explicitly.
For any rational prime $p$, we define a certain $p$-stabilization of holomorphic Siegel Eisenstein series for the symplectic group $\text{Sp}(2n)_{/\mathbb{Q}}$ of an arbitrary genus $n \ge 1$. In addition, we derive an explicit formula for the Fourier coefficients and conclude their $p$-adic interpolation problems. Consequently, for any odd prime $p$, we deduce the existence of a $Λ$-adic form (in the sense of A. Wiles, H. Hida and R.L. Taylor) such that after taking a suitable constant multiple, it interpolates $p$-adic analytic families of the above-mentioned $p$-stabilized Siegel Eisenstei…
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…
We give a formula for certain values and derivatives of Siegel series and use them to compute Fourier coefficients of derivatives of the Siegel Eisenstein series of weight g/2 and genus g. When g=4, the Fourier coefficient is approximated by a certain Fourier coefficient of the central derivative of the Siegel Eisenstein series of weight 2 and genus 3, which is related to the intersection of 3 arithmetic modular correspondences. Applications include a relation between weighted averages of representation numbers of symmetric matrices.
We prove the higher Siegel--Weil formula for \emph{corank one} terms, relating (1) the $r^{\rm th}$ central derivatives of corank one Fourier coefficients of Siegel--Eisenstein series, and (2) the degrees of special cycles of virtual dimension 0 on the moduli stack of Hermitian shtukas with $r$ legs. Notably, the formula holds for all $r$, regardless of the order of vanishing of the Eisenstein series. This extends earlier work of Feng--Yun--Zhang, who proved the higher Siegel--Weil formula for the non-singular (corank zero) terms.
Let $k > 3$ be an integer and $p$ a prime with $p > 2k-2$. Let $f$ be a newform of weight $2k-2$ and level 1 so that $f$ is ordinary at $p$ and $\barρ_{f}$ is irreducible. Under some additional hypotheses we prove that $ord_{p}(L_{alg}(k,f)) \leq ord_{p}(# S)$ where $S$ is the Pontryagin dual of the Selmer group associated to $ρ_{f} \otimes ε^{1-k}$ with $ε$ the $p$-adic cyclotomic character. We accomplish this by first constructing a congruence between the Saito-Kurokawa lift of $f$ and a non-CAP Siegel cusp form. Once this congruence is established, we use Galois representations to obtain th…
Let $k \geq 2$ and $N$ be positive integers and let $χ$ be a Dirichlet character modulo $N$. Let $f(z)$ be a modular form in $M_k(Γ_0(N),χ)$. Then we have a unique decomposition $f(z)=E_f(z)+S_f(z)$, where $E_f(z) \in E_k(Γ_0(N),χ)$ and $S_f(z) \in S_k(Γ_0(N),χ)$. In this paper we give an explicit formula for $E_f(z)$ in terms of Eisenstein series. Then we apply our result to certain families of eta quotients and to representations of positive integers by $2k$-ary positive definite quadratic forms in order to give an alternative version of Siegel's formula for the weighted average number of re…
Let $G$ be a linear split algebraic group. The degenerate Eisenstein series associated to a maximal parabolic subgroup $E_{P}(f^{0},s,g)$ with the spherical section $f^{0}$ is studied in the first part of the thesis. In this part, we study the poles of $E_{P}(f^{0},s,g)$ in the region $\operatorname{Re} s >0$. We determine when the leading term in the Laurent expansion of $E_{P}(f^{0},s,g)$ around $s=s_0$ is square integrable. The second part is devoted to finding identities between the leading terms of various Eisenstein series at different points. We present an algorithm to find this data an…
We define a family of arithmetic zero cycles in the arithmetic Chow group of a modular curve X_0(N), for N>3 odd and squarefree, and identify the arithmetic degrees of these cycles as q-coefficients of the central derivative of a Siegel Eisenstein series of genus two. This parallels work of Kudla-Rapoport-Yang for Shimura curves.
It is known that average Siegel theta series lie in the space of Siegel Eisenstein series. Also, every lattice equipped with an even integral quadratic form lies in a maximal lattice. Here we consider average Siegel theta series of degree 2 attached to maximal lattices; we construct maps for which the average theta series is an eigenform. We evaluate the action of these maps on Siegel Eisenstein series of degree 2 (without knowing their Fourier coefficients), and then realise the average theta series as an explicit linear combination of the Eisenstein series.
We derive the Fourier expansion of scalar-valued Eisenstein series for O(2, n+2) using classical methods of Siegel, Braun, Zagier, Bruinier and others. We assume that the underlying lattice splits two hyperbolic planes. Finally we prove for the Eisenstein series at the standard cusp that they belong to the Maass space for O(2, n + 2), an analogue of the "Spezialschar" for Siegel modular forms introduced by Maass, at least at all local places p, where the localization of the underlying lattice is maximal.
Let $E_n$ be the Siegel Eisenstein series of degree $n$ and weight $k$ with a complex parameter $s$. In this paper, using a differential operator $D$ by Ibukiyama which sends a scalar valued Siegel modular form to the tensor product of two vector valued Siegel modular forms, under a certain condition, we give a formula of $DE_{p+q}$ on $H_p\times H_q$, where $H_n$ is the Siegel upper half space of degree $n$. Furthermore, we give some applications of this formula, i.e., analytic properies of standard $L$-functions and the Klingen Eisenstein series and algebraicity results for Siegel modular fo…
Shimura studied the analytic properties of the non-holomorphic Siegel Eisenstein series and derived a residue formula. Herein, we provide a refinement of his result for several types of Eisenstein series.
We compute the intertwining relation between the Hecke operators and the Siegel lowering operators on Siegel modular forms of arbitrary level $N$ and character $χ$ by using formulas for the action of the Hecke operators on Fourier expansions. Using an explicit description of the Satake compactification of $Γ_0^{(n)}(N) \backslash \mathfrak{H}_n$ when $N$ is squarefree we extend this to give intertwining relations for each cusp. As an application we give formulas for the action of Hecke operators on the space of Klingen--Eisenstein series of squarefree level $N$, for primes $p \nmid N$.
We formulate and prove a version of the arithmetic Siegel-Weil formula for (zero dimensional) Shimura varieties attached to tori, equipped with some additional data. More precisely, we define a family of ``special" divisors in terms of Green functions at archimedean and non-archimedean places, and prove that their degrees coincide with the Fourier coefficients of the central derivative of an Eisenstein series. The proof relies on the usual Siegel-Weil formula to provide a direct link between both sides of the identity, and in some sense, offers a more conceptual point of view on prior results …
We formulate and prove a local arithmetic Siegel--Weil formula for GSpin Rapoport--Zink spaces, which is a precise identity between the arithmetic intersection numbers of special cycles on GSpin Rapoport--Zink spaces and the derivatives of local representation densities of quadratic forms. As a first application, we prove a semi-global arithmetic Siegel--Weil formula as conjectured by Kudla, which relates the arithmetic intersection numbers of special cycles on GSpin Shimura varieties at a place of good reduction and the central derivatives of nonsingular Fourier coefficients of incoherent Sie…
The Fourier coefficients of the Siegel-Eisenstein series are p-adically continued for all primes p, as meromorphic functions, using the reciprocal of a product of L-functions. A construction of p-adic meromorphic families of such series is given in relation to the geometry of homogeneous spaces. Applications are given to p-adic L-functions, to Siegel's Mass Formula, to p-adic analytic families of automorphic representations. Based on author's talk for the Conference Automorphic Forms and Related Geometry, Assessing the Legacy of I.I. Piatetski-Shapiro
Siegel modular forms in the space of the mod $p$ kernel of the theta operator are constructed by the Eisenstein series in some odd-degree cases. Additionally, a similar result in the case of Hermitian modular forms is given.
We describe connections between the Fourier coefficients of derivatives of Eisenstein series and invariants from the arithmetic geometry of the Shimura varieties $M$ associated to rational quadratic forms $(V,Q)$ of signature $(n,2)$. In the case $n=1$, we define generating series $\hatϕ_1(τ)$ for 1-cycles (resp. $\hatϕ_2(τ)$ for 0-cycles) on the arithmetic surface $\Cal M$ associated to a Shimura curve over $\Bbb Q$. These series are related to the second term in the Laurent expansion of an Eisenstein series of weight $\frac32$ and genus 1 (resp. genus 2) at the Siegel--Weil point, and these …
We consider the Hermitian Eisenstein series $E^{(\mathbb{K})}_k$ of degree $2$ and weight $k$ associated with an imaginary-quadratic number field $\mathbb{K}$ and determine the influence of $\mathbb{K}$ on the arithmetic and the growth of its Fourier coefficients. We find that they satisfy the identity $E^{{(\mathbb{K})}^2}_4 = E^{{(\mathbb{K})}}_8$, which is well-known for Siegel modular forms of degree $2$, if and only if $\mathbb{K} = \mathbb{Q} (\sqrt{-3})$. As an application, we show that the Eisenstein series $E^{(\mathbb{K})}_k$, $k=4,6,8,10,12$ are algebraically independent whenever $\…
The aim of this paper is to generalize the Maass relation for generalized Cohen-Eisenstein series of degree two and of degree three. Here the generalized Cohen-Eisenstein series are certain Siegel modular forms of half-integral weight, and generalized Maass relations are certain relations among Fourier-Jacobi coefficients of them.
Let $\ell$ be a prime, $k$ a finitely generated field of characteristic different from $\ell$, and $X$ a smooth geometrically connected curve over $k$. Say a semisimple representation of $π_1^{\mathrm{et}}(X_{\bar k})$ is arithmetic if it extends to a finite index subgroup of $π_1^{\mathrm{et}}(X)$. We show that there exists an effective constant $N=N(X,\ell)$ such that any semisimple arithmetic representation of $π_1^{\mathrm{et}}(X_{\bar k})$ into $\mathrm{GL}_n(\bar{\mathbb{Z}_\ell})$, which is trivial mod $\ell^N$, is in fact trivial. This extends a previous result of the second author fro…
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, …
The real analytic Eisenstein series is a special function that has been studied classically. Its generalization to the case of many variables has been studied extensively. Moreover, the analytic properties of certain Eisenstein series on the Siegel modular groups have also been investigated. The purpose of this study is to provide concrete forms of the residue of E_0^{(m)}(z,s) at s=m/2.
We investigate the relation between Klingen's decomposition of the space of Siegel modular forms and Dulinski's analogous decomposition of the space of Jacobi forms.
It is well-known that the Fourier coefficients of Siegel-Eisenstein series can be expressed in terms of the Siegel series. The functional equation of the Siegel series of a quadratic form over $\mathbb{Q}_p$ was first proved by Katsurada. In this paper, we prove the functional equation of the Siegel series over a non-archimedean local field by using the representation theoretic argument by Kudla and Sweet.
We show that certain $p$-adic Eisenstein series for quaternionic modular groups of degree 2 become "real" modular forms of level $p$ in almost all cases. To prove this, we introduce a $U(p)$ type operator. We also show that there exists a $p$-adic Eisenstein series of the above type that has transcendental coefficients. Former examples of $p$-adic Eisenstein series for Siegel and Hermitian modular groups are both rational (i.e., algebraic).
We construct natural Green forms for special cycles in orthogonal and unitary Shimura varieties, in all codimensions, and, for compact Shimura varieties of type O(p,2) and U(p,1), we show that the resulting local archimedean height pairings are related to special values of derivatives of Siegel Eisentein series. A conjecture put forward by Kudla relates these derivatives to arithmetic intersections of special cycles, and our results settle the part of his conjecture involving local archimedean heights.
In this paper, we proved an arithmetic Siegel-Weil formula and the modularity of some arithmetic theta function on the modular curve $X_0(N)$ when $N$ is square free. In the process, we also construct some generalized Delta function for $Γ_0(N)$ and proved some explicit Kronecker limit formula for Eisenstein series on $X_0(N)$.
In this paper, we will study the arithmetic of the Eisenstein part of the modular Jacobians. In the first section, we introduce some general preliminaries of the arithmetic theory of modular curves that we will need later. In the second section, we give an example of modular abelian varieties due to Gross and study its properties in some details. In the third section, we define Eisenstein quotients of the modular Jacobians in general and give a criterion of the non-triviality of Heegner points on such Eisenstein quotients. The last two sections return to the concrete examples when the level of…
