Romyar T. Sharifi
In this note, we describe a conjecture, that, for an odd prime p, relates special values of a cup product pairing on cyclotomic p-units in the pth cyclotomic field to the L-values of newforms satisfying modulo p congruences with Eisenstein series.
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We extend to positive real weights Haberland's formula giving a cohomological description of the Petersson scalar product of modular cusp forms of positive even weight. This relation is based on the cup product of an Eichler cocycle and a Knopp cocycle. We also consider the cup product of two Eichler cocycles attached to modular forms. In the classical context of integral weights at least $2$ this cup product is uninteresting. We show evidence that for real weights this cup product may very well be non-trivial. We approach the question whether the cup product is a non-trivial coinvariant by du…
We present an algorithm to compute all Borcherds product paramodular cusp forms of a specified weight and level, describing its implementation in some detail and giving examples of its use.
The purpose of this paper is to study products of Fourier coefficients of an elliptic cusp form, $a(n)a(n + r)$ $(n \geq 1)$ for a fixed positive integer $r$, concerning both non-vanishing and non-negativity.
We investigate the inner product between meromorphic modular forms from certain distinguished subspaces. In particular, we compute the degenerate parts of these subspaces and further compute the inner product on some of the remaining quotient spaces.
We begin by explaining how to compute Fourier expansions at all cusps of any modular form of integral or half-integral weight thanks to a theorem of Borisov-Gunnells and explicit expansions of Eisenstein series at all cusps. Using this, we then give a number of methods for computing arbitrary Petersson products. All this is available in the current release of the Pari/GP package.
We prove a bound for the Fourier coefficients of a cusp form of integral weight which is not a newform by computing an explicit orthogonal basis for the space of cusp forms of given integral weight and level. In contrast to previous work on special cases of this problem we use elementary methods for the computation of Petersson products and avoid using the Rankin $L$-function.
In this paper, we present Tate's theory of augmented cup products in profinite cohomology in a modern constructive style. As an application, we interpret pairings between groups associated to curves constructed by Lichtenbaum, in terms of augmented cup products.
We show, for levels of the form $N = p^a q^b N'$ with $N'$ squarefree, that in weights $k \geq 4$ every cusp form $f \in \mathcal{S}_k(N)$ is a linear combination of products of certain Eisenstein series of lower weight. In weight $k=2$ we show that the forms $f$ which can be obtained in this way are precisely those in the subspace generated by eigenforms $g$ with $L(g, 1) \neq 0$. As an application of such representations of modular forms we can calculate Fourier expansions of modular forms at arbitrary cusps and we give several examples of such expansions in the last section.
In 2006, in a paper published in Compositio, titled "Bounds on canonical Green's functions", J. Jorgenson and J. Kramer proved a certain key identity which relates the two natural metrics, namely the hyperbolic metric and the canonical metric defined on a compact hyperbolic Riemann surface. In this article, we extend this identity to product of noncompact hyperbolic Riemann orbisurfaces of finite volume, which can be realized as a quotient space of the action of a Fuchsian subgroup of first kind on the hyperbolic upper half plane.
Let $Λ= \{λ_{k}\}$ denote a sequence of complex numbers and assume that that the counting function $#\{λ_{k} \in Λ: | λ_{k}| < T\} =O(T^{n})$ for some integer $n$. From Hadamard's theorem, we can construct an entire function $f$ of order at most $n$ such that $Λ$ is the divisor $f$. In this article we prove, under reasonably general conditions, that the superzeta function $\Z_{f}(s,z)$ associated to $Λ$ admits a meromorphic continuation. Furthermore, we describe the relation between the regularized product of the sequence $z-Λ$ and the function $f$ as constructed as a Weierstrass product. In t…
We compute the special values for the spinor L-function L(s,F12) in the critical strip s={12,...,19}, where F12 is the unique (up to a scalar) Siegel cusp form of degree 3 and weight 12, which was constructed by Miyawaki. These values are proportional to the product of Petersson inner products of Ramanujan's Delta by itself and the cusp form of weight 20 for SL(2,Z) by itself by a rational number and some power of Pi. We also verify this result numerically using Dokchitser's ComputeL PARI package. To our knowledge this is the first example of a spinor L-function of Siegel cusp forms of degree …
We give a new expression for the inner product of two kernel functions associated to a cusp form. Among other applications, it yields an extension of a formula of Kohnen and Zagier, and another proof of Manin's Periods Theorem. Cohen's representation of these kernels as series is also generalized.
We extend the usual notion of Petersson inner product on the space of cuspidal Jacobi forms to include non-cuspidal forms as well. This is done by examining carefully the relation between certain "growth-killing" invariant differential operators on $\mathbf H_2$ and those on $\mathbf{H}_1 \times \mathbf{C}$ (here $\mathbf H_n$ denotes the Siegel upper half space of degree $n$). As applications, we can understand better the growth of Petersson norms of Fourier Jacobi coefficients of Klingen Eisenstein series, which in turn has applications to finer issues about representation numbers of quadrat…
We connect Veronese embeddings to splitting varieties of cup products. We then give an algorithm for constructing splitting varieties for cup products with $\mathbb Z/n$ coefficients, with an explicit calculation for $n=3$. An application to the automatic realization of Galois groups is given.
We give a general formula relating self cup products in cohomology to connecting maps in nonabelian cohomology, and apply it to obtain a formula for the self cup product associated to the Weil pairing.
Cusp forms of weight 1/2 and pairs of quadratic forms
本文证明了 Kohnen 子空间中半整权尖点形式四个傅里叶系数乘积的谱求和公式，并将其与整数系数二次型对的广义类数联系起来。
In this paper, we study the average behaviour of the coefficients of triple product L-functions and some related L-functions corresponding to normalized primitive holomorphic cusp form $f(z)$ of weight $k$ for the full modular group $SL(2,\ \mathbb{Z}).$ Here we call $f(z)$ a primitive cusp form if it is an eighenfunction of all Hecke operators simultaneously.
This paper concerns cup product pairings in étale cohomology related to work of M. Kim and of W. McCallum and R. Sharifi. We will show that by considering Ext groups rather than cohomology groups, one arrives at a pairing which combines invariants defined by Kim with a pairing defined by McCallum and Sharifi. We also prove a formula for Kim's invariant in terms of Artin maps in the case of cyclic unramified Kummer extensions. One consequence is that for all $n > 1$, there are infinitely many number fields $F$ over which there are both trivial and non-trivial Kim invariants associated to cyclic…
We present an explicit description, in terms of central simple algebras, of a cup-product map which occurs in the statement of local Tate duality for Galois modules of prime order p. Given cocycles f and g, we construct a central simple algebra of dimension p^2 whose class in the Brauer group gives the cup-product of f with g. This algebra is as small as possible.
We explicitly construct cusp forms on the orthogonal group of signature $(1,8n+1)$ for an arbitrary natural number $n$ as liftings from Maass cusp forms of level one. In our previous works, the fundamental tool to show the automorphy of the lifting was the converse theorem by Maass. In this paper, we use the Fourier expansion of the theta lifts by Borcherds instead. We also study cuspidal representations generated by such cusp forms and show that they are irreducible and that all of their non-archimedean local components are non-tempered while the archimedean component is tempered, if the Maas…
We give a short proof of a formula of de Shalit, expressing the cup product of two vector valued one forms of the second kind on a Mumford curve in terms of Coleman integrals and residues. The proof uses the notion of double indices on curves and their reciprocity laws.
Let K be a number field containing the group of n-th roots of unity and S a set of primes of K including all those dividing n and all real archimedean places. We consider the cup product on the first Galois cohomology group of the maximal S-ramified extension of K with coefficients in n-th roots of unity, which yields a pairing on a subgroup of the multiplicative group of K containing the S-units. In this general situation, we determine a formula for the cup product of two elements which pair trivially at all local places. Our primary focus is the case that K is the cyclotomic field of p-th ro…
We present a method for certifying the existence of an arbitrary Maass cusp form for any level and character. This is accomplished by producing a bound on the difference between the $Δ$-eigenvalue of an authentic Maass cusp form and a purported approximation of a $Δ$-eigenvalue, arrived at by any means. We apply this method to a proposed non-CM level 5 form with quadratic character, to present the first certified $Δ$-eigenvalue of such a form. This work generalises the method for certifying level 1 forms presented by Booker, Strömbergsson and Venkatesh, and is motivated by the production of pu…
We use Galois cohomology to study the $p$-rank of the class group of $\mathbf{Q}(N^{1/p})$, where $N \equiv 1 \bmod{p}$ is prime. We prove a partial converse to a theorem of Calegari--Emerton, and provide a new explanation of the known counterexamples to the full converse of their result. In the case $p = 5$, we prove a complete characterization of the $5$-rank of the class group of $\mathbf{Q}(N^{1/5})$ in terms of whether or not $\prod_{k=1}^{(N-1)/2} k^{k}$ and $\frac{\sqrt{5} - 1}{2}$ are $5$th powers mod $N$.
We study the meromorphy of various Euler products of degree two attached to cusp forms including Ramanujan's $Δ$-function.
We obtain a generalisation of the Quantum Unique Ergodicity for holomorphic cusp forms on $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ in the weight aspect. We show that correlations of masses coming from off-diagonal terms dissipate as the weight tends to infinity. This corresponds to classifying the possible quantum limits along any sequence of Hecke eigenforms of increasing weight. Our new ingredient is to incorporate the spectral theory of weight $k$ automorphic functions to the method of Holowinsky-Soundararajan. For Holowinsky's shifted convolution sums approach, we need to develop …
Subconvexity Implies Effective Quantum Unique Ergodicity for Hecke-Maaß Cusp Forms on $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathrm{SL}_2(\mathbb{R})$
本文证明 Hecke-Maaß 尖点形式的相空间量子唯一遍历性可由三重积 $L$-函数的次凸界推出，填补了相空间层面此前仅有 Eisenstein 级数结果的空白，证明基于 Watson-Ichino 公式及 Whittaker 函数积分的精细分析。
By studying an explicit form of the Eichler--Shimura map for ${\rm GL}_3$, we describe a precise relation between critical values of the complete $L$-function for the Rankin--Selberg convolution ${\rm GL}_3 \times {\rm GL}_2$ and the cohomological cup product of certain rational cohomology classes which are uniquely determined up to rational scalar multiples from the cuspidal automorphic representations under consideration. This refines rationality results on critical values due to Raghuram et al.
We explicitly compute the special values of the standard $L$-function $L(s, F_{12}, \mathrm{St})$ at the critical points $s\in\{-8, -6, -4, -2, 0, 1, 3, 5, 7, 9\}$, where $F_{12}$ is the unique (up to a scalar) Siegel cusp form of degree $3$ and weight $12$, which was constructed by Miyawaki. These values are proportional to the product of the Petersson norms of symmetric square of Ramanujan's $Δ$ and the cusp form of weight $20$ for ${\rm SL}_2(\mathbb{Z})$ by a rational number and some power of $π$. We use the Rankin-Selberg method and apply the Holomorphic projection to compute these values…
We classify the simple even lattices of square free level and signature (2,n) for n > 3. A lattice is called simple if the space of cusp forms of weight 1+n/2 for the dual Weil representation of the lattice is trivial. For a simple lattice every formal principal part obeying obvious conditions is the principal part of a vector valued modular form. Using this, we determine all holomorphic Borcherds products of singular weight (arising from vector valued modular forms with non-negative principal part) for the simple lattices. We construct the corresponding vector valued modular forms by eta prod…
Let $Γ\subset\mathrm{PSL}_{2}(\mathbb{R})$ be a Fuchsian subgroup of the first kind acting on the upper half-plane $\mathbb{H}$. Consider the $d_{2k}$-dimensional space of cusp forms $\mathcal{S}_{2k}^Γ$ of weight $2k$ for $Γ$, and let $\{f_{1},\ldots,f_{d_{2k}}\}$ be an orthonormal basis of $\mathcal{S}_{2k}^Γ$ with respect to the Petersson inner product. In this paper we will give effective upper and lower bounds for the supremum of the quantity $S_{2k}^Γ(z):=\sum_{j=1}^{d_{2k}}\vert f_{j}(z)\vert^{2}\,\mathrm{Im}(z)^{2k}$ as $z$ ranges through $\mathbb{H}$.
A generalized Riemann hypothesis states that all zeros of the completed Hecke $L$-function $L^*(f,s)$ of a normalized Hecke eigenform $f$ on the full modular group should lie on the vertical line $Re(s)=\frac{k}{2}.$ It was shown by Kohnen that there exists a Hecke eigenform $f$ of weight $k$ such that $L^*(f,s) \neq 0$ for sufficiently large $k$ and any point on the line segments $Im(s)=t_0, \frac{k-1}{2} < Re(s) < \frac{k}{2}-ε, \frac{k }{2}+ε< Re(s) < \frac{k+1}{2},$ for any given real number $t_0$ and a positive real number $ε.$ This paper concerns the non-vanishing of the product $L^*(f,s…
We work out instances of a general conjecture on congruences between Hecke eigenvalues of induced and cuspidal automorphic representations of a reductive group, modulo divisors of certain critical L-values, in the case that the group is a split orthogonal group. We provide some numerical evidence in the case that the group is SO(4,3) and the L-function is the spinor L-function of a genus 2, vector-valued, Siegel cusp form. We also consider the case that the group is SO(4,4) and the L-function is a triple product L-function.
An additive problem in the Fourier coefficients of cusp forms
本文研究了任意级和特征的尖点形式傅里叶系数平移乘积之和的估计，并作为应用推广了前人工作，导出了其特征扭曲关联L函数在临界线上的次凸估计。
We show that every elliptic modular form of integral weight greater than $1$ can be expressed as linear combinations of products of at most two cusp expansions of Eisenstein series. This removes the obstruction of nonvanishing central $\mathrm{L}$-values present in all previous work. For weights greater than $2$, we refine our result further, showing that linear combinations of products of exactly two cusp expansions of Eisenstein series suffice.
We study kernel functions of L-functions and products of L-functions of Hilbert cusp forms over real quadratic fields. This extends the results on elliptic modualr forms by Diamantis and C. O'Sullivan. .
Reflective modular varieties and their cusps
本文利用障碍理论分类奇异权反射自守乘积，证明恰有11个，并将其尖点与Schellekens列表中的根系及Co_0共轭类联系起来，从而给出该列表的复几何证明。
We define graded hyper-algebras of vector-valued Siegel modular forms, which allow us to study tensor products of the latter. We also define vector-valued Hecke operators for Siegel modular forms at all places of ${\mathbb Q}$, acting on these hyper-algebras. These definitions bridge the classical and representation theoretic approach to Siegel modular forms. Combining both the product structure and the action of Hecke operators, we prove in the case of elliptic modular forms that all cusp forms of sufficiently large weight can be obtained from products involving only two fixed Eisenstein seri…
We compute the p-adic Abel-Jacobi map of the product of a Hilbert modular surface and a modular curve at a null-homologous (modified) embedding of the modular curve in this product, evaluated on differentials associated to a Hilbert cuspidal form f of weight (2,2) and a cuspidal form of weight 2. We generalize this computation to suitable null-homological cycles in the fibre products of the universal families on the surface and the curve, evaluated at differentials associated to f and g of higher weights. We express the values of the p-adic Abel-Jacobi map at these weights in terms of a p-adic…
Let $\G\subset \mathrm{SL}_{2}(\R)$ be a cofinite Fuchsian subgroup, and let $i\infty$ be a cusp of $\G$. For $k\in\Z_{\geq 0}$, let $\Sk$ denote the complex vector space of cusp forms of weight-$k$, with respect to the Fuchsian subgroup $\G$. Let $f\in\Sk$ be a cusp form of weight-$k$, which is normalized, with respect to the Petersson inner-product on $\Sk$. For any $n\in\Z_{\geq 1}$, let $a_{n}$ denote the $n$-th Fourier coefficient of $f$ at $i\infty$. Then, for any $k\in\Z_{\geq 5}$, we show that \begin{align*} \big|a_{n}\big|=O_{f,\G}\big(n^{\frac{k-1}{2}+\frac{2}{k}}\big), \end{align*} …
Cup product of inhomogeneous Tate cochains, and Galois cohomology of tori over local fields that split over cyclic extensions
本文给出Tate上同调中杯积的非齐次上链公式，并用于计算分裂于循环扩张的局部域上环面的一阶上同调类的显式上循环。
The nondegeneracy of the Petersson inner product on cusp forms, and the fact that Hecke operators are self-adjoint with respect to the Petersson product, together imply that the cusp forms have a basis consisting of Hecke eigenforms. In the literature on topological modular forms, no topological analogue of the Petersson product is to be found, and it is not known which topological spaces have the property that their topological cusp forms admit a basis consisting of eigenforms for the action of Baker's topological Hecke operators. In this note we define and study a natural topological Peterss…
Explicit Construction of Maass Wave Forms and Their Petersson Inner Products
本文通过显式构造实二次域上的Maass波形并计算其Petersson内积，推广了Maass的原始工作，并给出了与二面体Artin表示相关的例子。
In this paper we prove the following conditional result: Let F be a number field, and pi a cusp form on GL(2)/F which is not solvable polyhedral. Assume that all the symmetric powers sym^m(pi) are modular, i.e., define automorphic forms on GL(m+1)/F. If sym^6(pi) is cuspidal, then all the symmetric powers are cuspidal, for all m. Moreover, sym^6(pi) is Eisenteinian iff sym^5(pi) is an abelian twist of the functorial product of pi with the symmetric square of a cusp form pi' on GL(2)/F.
In this note, an analogous statement to the Nagell-Lutz theorem does not hold for the Jacobian of a certain curve of genus 2 over $\mathbb{C}(t)$. As a by-product, we give a (2, 3, 6) quasi-torus decomposition for the dual curve of a smooth cubic.
The rational cuspidal subgroup of J_0(N)
本文在整除N的最大完全平方数为奇素数幂或两个奇素数幂乘积时，确定J₀(N)的有理尖点子群结构，证明其等于X₀(N)的有理尖点除子类群，而后者结构已由第一作者在[14]中确定。
By a transfer operator approach to Maass cusp forms and the Selberg zeta function for cofinite Hecke triangle groups, M. Möller and the author found a factorization of the Selberg zeta function into a product of Fredholm determinants of transfer-operator-like families: $Z(s) = \det(1-\mc L_s^+)\det(1-\mc L_s^-)$. In this article we show that the operator families $\mc L_s^\pm$ arise as families of transfer operators for the triangle groups underlying the Hecke triangle groups, and that for $s\in\C$, $\Rea s=\tfrac12$, the operator $\mc L_s^+$ (resp. $\mc L_s^-$) has a 1-eigenfunction if and on…
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