Geometry of the Siegel modular threefold with paramodular level structure

Siegel Paramodular Forms of Weight 2 and Squarefree Level

We compute the space $S_2(K(N))$ of weight $2$ Siegel paramodular cusp forms of squarefree level $N<300$. In conformance with the paramodular conjecture of A. Brumer and K. Kramer, the space is only the additive (Gritsenko) lift space of the Jacobi cusp form space $J_{2,N}^{\text{cusp}}$ except for $N=249,295$, when it further contains one nonlift newform. For these two values of $N$, the Hasse-Weil $p$-Euler factors of a relevant abelian surface match the spin $p$-Euler factors of the nonlift newform for the first two primes $p\nmid N$.

到Siegel参模形式的显式Theta提升

An Explicit Theta Lift to Siegel Paramodular Forms

研究数域实二次扩张下的Theta提升问题，构造从含Γ₀级Hilbert模形式的GL(2,E)尖点自守表示到含Siegel参模形式的GSp(4,L)自守表示的显式映射，给出除野分歧外有限位处生成参模不变向量的局部数据。

Appendix to "Twisting of Siegel paramodular forms"

In this appendix we present an expanded version of Section 4 of our paper arXiv:1404.4596, including the proofs of all of the technical lemmas.

Twisting of Siegel paramodular forms

Let $S_k(Γ^{\mathrm{para}}(N))$ be the space of Siegel paramodular forms of level $N$ and weight $k$. Let $p\nmid N$ and let $χ$ be a nontrivial quadratic Dirichlet character mod $p$. Based on our previous work, we define a linear twisting map $\mathcal{T}_χ:S_k(Γ^{\mathrm{para}}(N))\rightarrow S_k(Γ^{\mathrm{para}}(Np^4))$. We calculate an explicit expression for this twist and give the commutation relations of this map with the Hecke operators and Atkin-Lehner involution for primes $\ell\neq p$.

Siegel Paramodular Forms from Exponential Lifts: Slow versus Fast Growth

We investigate the growth of Fourier coefficients of Siegel paramodular forms built by exponentially lifting weak Jacobi forms, focusing on terms with large negative discriminant. To this end we implement a method based on deforming contours that expresses the coefficients of all such terms as residues. We find that there are two types of weak Jacobi forms, leading to two different growth behaviors: the more common type leads to fast, exponential growth, whereas a second type leads to slower growth, akin to the growth seen in ratios of theta functions. We give a simple criterion to distinguish…

Quinary forms and paramodular forms

We work out the exact relationship between algebraic modular forms for a two-by-two general unitary group over a definite quaternion algebra, and those arising from genera of positive-definite quinary lattices, relating stabilisers of local lattices with specific open compact subgroups, paramodular at split places, and with Atkin-Lehner operators. Combining this with the recent work of Rösner and Weissauer, proving conjectures of Ibukiyama on Jacquet-Langlands type correspondences (mildly generalised here), provides an effective tool for computing Hecke eigenvalues for Siegel modular forms of …

The Siegel modular forms of genus 2 with the simplest divisor

We prove that there exist exactly eight Siegel modular forms with respect to the congruence subgroups of Hecke type of the paramodular groups of genus two vanishing precisely along the diagonal of the Siegel upper half-plane. This is a solution of a question formulated during the conference "Black holes, Black Rings and Modular Forms" (ENS, Paris, August 2007). These modular forms generalize the classical Igusa form and the forms constructed by Gritsenko and Nikulin in 1998.

Some Siegel threefolds with a Calabi-Yau model

In the following we describe some examples of Calabi-Yau manifolds, which arise as desingularizations of certain Siegel threefolds. The first Siegel modular variety with a Calabi-Yau model and the essentially only one up to now has been discovered by Barth and Nieto.

算术子群的形式西格尔模形式

Formal Siegel modular forms for arithmetic subgroups

本文研究算术子群的形式西格尔模形式，证明在仿射覆盖数有上界时其均可由经典全纯西格尔模形式的傅里叶-雅可比展开给出，并验证属2至4时该条件恒成立，进而应用于属2无平方水平参数模群。

Some Siegel threefolds with a Calabi-Yau model II

In the paper [FSM] we described some Siegel modular threefolds which admit a Calabi-Yau model. Using a different method we give in this paper an enlarged list of such varieties that admits a Calabi-Yau model in the following weak sense: there exists a desingularization in the category of complex spaces of the Satake compactification which admits a holomorphic three-form without zeros and whose first Betti number vanishes Basic for our method is the paper [GN] of van Geemen and Nygaard.

某个 Calabi-Yau 三维流的剩余参数模性

Residual paramodularity of a certain Calabi-Yau threefold

本文证明了 Hilbert 新形式之间的 Hecke 特征值同余，从而得出某个 Calabi-Yau 三维流第 3 阶上同调的模 5 表示源自 Siegel 模形式 $F_{79}$，验证了 Golyshev 和 van Straten 猜想的弱形式。

Quaternionic loci in Siegel's modular threefold

Let $\mathcal Q_D$ be the set of moduli points on Siegel's modular threefold whose corresponding principally polarized abelian surfaces have quaternionic multiplication by a maximal order $\mathcal O$ in an indefinite quaternion algebra of discriminant $D$ over $\mathbb Q$ such that the Rosati involution coincides with a positive involution of the form $α\mapstoμ^{-1}\overlineαμ$ on $\mathcal O$ for some $μ\in\mathcal O$ with $μ^2+D=0$. In this paper, we first give a formula for the number of irreducible components in $\mathcal Q_D$, strengthening an earlier result of Rotger. Then for each irr…

Endoscopic lifts to the Siegel modular threefold related to Klein's cubic threefold

Let $A^{lev}_{11}$ be the moduli space of (1,11)-polarized abelian surfaces with level structure of canonical type. Let $χ$ be a finite character of order 5 with conductor 11. In this paper we construct five endoscopic lifts $Π_i,0\le i\le 4$ from two elliptic modular forms $f\otimesχ^i$ of weight 2 and $g\otimesχ^i$ of weight 4 with complex multiplication by $Q(\sqrt{-11})$ such that ${Π_i}_\infty$ gives a non-holomorphic differential form on $A^{lev}_{11}$ for each $i$. Then the spinor L-function is of form $L(f\otimesχ^i,s-1)L(g\otimesχ^i,s)$ such that $L(g\otimesχ^i,s)$ does not appear in …

On some Siegel threefold related to the tangent cone of the Fermat quartic surface

Let $Z$ be the quotient of the Siegel modular threefold $\mathcal{A}^{\rm sa}(2,4,8)$ which has been studied by van Geemen and Nygaard. They gave an implication that some 6-tuple $F_Z$ of theta constants which is in turn known to be a Klingen type Eisenstein series of weight 3 should be related to a holomorphic differential $(2,0)$-form on $Z$. The variety $Z$ is birationally equivalent to the tangent cone of Fermat quartic surface in the title. In this paper we first compute the L-function of two smooth resolutions of $Z$. One of these, denoted by $W$, is a kind of Igusa compactification such…

On higher regulators of Siegel threefolds II: the connection to the special value

We establish a connection between motivic cohomology classes over the Siegel threefold and special values of the degree four $L$-function of some cuspidal automorphic representations of $\mathrm{GSp}(4)$. Our computation relies on our previous work [Le15] and on an integral representation of the $L$-function due to Piatetski-Shapiro.

On Siegel eigenvarieties at Saito-Kurokawa points

We study the geometry of the $p$-adic Siegel eigenvariety $\mathcal{E}$ of paramodular tame level at certain Saito-Kurokawa points having a critical slope. For $k \geq 2$ let $f$ be a cuspidal new eigenform of $\mathrm{S}_{2k-2}(Γ_0(N))$ ordinary at a prime $p\nmid N$ with sign $ε_f=-1$ and write $α$ for the $p$-adic unit root of the Hecke polynomial of $f$ at $p$. Let $π_α$ be the semi-ordinary $p$-stabilization of the Saito-Kurokawa lift of the cusp form $f$ to $\mathrm{GSp}(4)$ of weight $(k,k)$ and paramodular tame level. Under the assumption that the dimension of the Selmer group $H^1_{f,…

Siegel modular forms of degree two attached to Hilbert modular forms

Let E/Q be a real quadratic field and pi_0 a cuspidal, irreducible, automorphic representation of GL(2,A_E) with trivial central character and infinity type (2,2n+2) for some non-negative integer n. We show that there exists a non-zero Siegel paramodular newform F with weight, level, Hecke eigenvalues, epsilon factor and L-function determined explicitly by pi_0. We tabulate these invariants in terms of those of pi_0 for every rational prime p.

亏格二拟Siegel模形式与环面Calabi-Yau三流形的Gromov-Witten理论

Genus Two Quasi-Siegel Modular Forms and Gromov-Witten Theory of Toric Calabi-Yau Threefolds

本文发展了亏格二拟Siegel模形式与拟Siegel Jacobi形式的微分环理论，并将其应用于带膜的局部Calabi-Yau三流形的拓扑递归，证明了开闭Gromov-Witten势本质上是亏格二拟Siegel模形式。

On the strict endoscopic part of modular Siegel threefolds

In this paper we study the non-holomorphic strict endoscopic parts of inner cohomology spaces of a modular Siegel threefold respect to local systems. First we show that there is a non-zero subspace of the strict endoscopic part such that it is constructed by global theta lift of automorphic froms of $(GL(2)\times GL(2))/G_{m}$. Secondly, we present an explicit analytic calculation of levels of lifted forms into GSp(4), based on the paramodular representations theory for $GSp(4; F)$. Finally, we prove the conjecture, by C. Faber and G. van der Geer, that gives a description of the strict endosc…

Siegel modular forms of degree three and the cohomology of local systems

We give an explicit conjectural formula for the motivic Euler characteristic of an arbitrary symplectic local system on the moduli space A_3 of principally polarized abelian threefolds. The main term of the formula is a conjectural motive of Siegel modular forms of a certain type; the remaining terms admit a surprisingly simple description in terms of the motivic Euler characteristics for lower genera. The conjecture is based on extensive counts of curves of genus three and abelian threefolds over finite fields. It provides a lot of new information about vector-valued Siegel modular forms of d…

Equidistribution theorems for holomorphic Siegel modular forms for $GSp_4$; Hecke fields and $n$-level density

This paper is a continuation of the author's previous wotk. We supplement four results on a family of holomorphic Siegel cusp forms for $GSp_4/\mathbb{Q}$. First, we improve the result on Hecke fields. Namely, we prove that the degree of Hecke fields is unbounded on the subspace of genuine forms which do not come from functorial lift of smaller subgroups of $GSp_4$ under a conjecture in local-global compatibility and Arthur's classification for $GSp_4$. Second, we prove simultaneous vertical Sato-Tate theorem. Namely, we prove simultaneous equidistribution of Hecke eigenvalues at finitely many…

区分希格拉模形式

Distinguishing Siegel modular forms

本文通过研究海格尔特征值的多项式关系，证明了希格拉模形式之间的扭关系，扩展了强多重性一理论，并提供了一种统一区分椭圆和希格拉模形式的方法。

Siegel modular forms of genus 2 attached to elliptic curves

The object of this article is to construct certain classes of arithmetically significant, holomorphic Siegel cusp forms F of genus 2, which are neither of Saito-Kurokawa type, in which case the degree 4 spinor L-function L(s, F) is divisible by an abelian L-function, nor of Yoshida type, in which case L(s,F) is a product of L-series of a pair of elliptic cusp forms. Two key examples are the forms F defined by the symmetric cube of a non-CM elliptic curve E over the rationals, and those defined by anticyclotomic twists (of even non-zero weight) of the base change of such an E to an imaginary qu…

Siegel 模形式权重 $(3,3)$ 的 Kato 显式互反律

Kato explicit reciprocity law for Siegel modular forms of weight $(3, 3)$

本文将 Kato 显式互反律从模曲线推广到两条模曲线的乘积，并利用嵌入 Siegel 三维簇，得到权重 $(3,3)$ 的 Siegel 模形式所附 $p$-进表示的唯一次临界扭转的显式互反律。

Jacobians among Abelian threefolds: a formula of Klein and a question of Serre

Let k be a field and f be a Siegel modular form of weight h \geq 0 and genus g>1 over k. Using f, we define an invariant of the k-isomorphism class of a principally polarized abelian variety (A,a)/k of dimension g. Moreover when (A,a) is the Jacobian of a smooth plane curve, we show how to associate to f a classical plane invariant. As straightforward consequences of these constructions, when g=3 and k is a subfield of the complex field, we obtain (i) a new proof of a formula of Klein linking the modular form χ_{18} to the square of the discriminant of plane quartics ; (ii) a proof that one ca…

L-functions of $S_3(\G_2(2,4,8))$

The space of Siegel cuspforms of degree $2$ of weight $3$ with respect to the congruence subgroup $\G_2(2,4,8)$ was studied by van Geemen and van Straten in Math. computation. {\bf 61} (1993). They showed the space is generated by six-tuple products of Igusa $þ$-constants, and all of them are Hecke eigenforms. They gave conjecture on the explicit description of the Andrianov $L$-functions. In J. Number Theory. {\bf 125} (2007), we proved some conjectures by showing that some products are obtained by the Yoshida lift, a construction of Siegel eigenforms. But, other products are not obtained by …