Katz-Sarnak哲学指出，当导子趋于无穷时，L函数族零点在中心点附近的统计特性与随机矩阵系综的特征值在矩阵规模趋于无穷时的统计特性一致。尽管大量结果支持这一猜想，但S. J. Miller观察到，对于有限导子，椭圆曲线族中中心点附近的零点会呈现出截然不同的行为。这催生了Dueñez、Huynh、Keating、Miller和Snaith提出的切除模型（excised model），该模型对给定椭圆曲线的二次扭转的预测与数据完美吻合。该模型的核心要素是将L函数中心值的离散化与基于特征多项式在1处的值对矩阵进行切除相关联，并利用（单能级密度、对关联等统计量中的）低阶项来调整矩阵规模。我们讨论了作者们近期在将该模型推广到一类具有奇素数级的给定全纯尖点新形式的有限导子二次扭转族方面取得的进展。特别地，我们预测权重大于2的形式几乎不存在排斥效应。
英文摘要：
The Katz-Sarnak philosophy states that statistics of zeros of $L$-function families near the central point as the conductors tend to infinity agree with those of eigenvalues of random matrix ensembles as the matrix size tends to infinity. While numerous results support this conjecture, S. J. Miller observed that for finite conductors, very different behavior can occur for zeros near the central point in elliptic curve families. This led to the excised model of Dueñez, Huynh, Keating, Miller, and Snaith, whose predictions for quadratic twists of a given elliptic curve are beautifully fit by the data. The key ingredients are relating the discretization of central values of the $L$-functions to excising matrices based on the value of the characteristic polynomials at 1 and using lower order terms (in statistics such as the one-level density and pair-correlation) to adjust the matrix size. We discuss recent successes by the authors in extending this model to a family of quadratic twists of finite conductor of a given holomorphic cuspidal newform of level an odd prime level. In particular, we predict very little repulsion for forms with weight greater than 2.
In this article, we discuss Iwasawa Main Conjecture for $p$-adic families of elliptic modular cuspforms. After the overview on the situation of the ordinary case of Hida family, we will introduce a Coleman map for Coleman family for the non-ordinary case (Coleman family) which was obtained as a joint work with Filippo Nuccio [NO16] and we give some results on Iwasawa Theory for Coleman families as applications of . First, we give a construction of a two-variable $p$-adic $L$-function for a Coleman family thanks to the ingredients given in [NO16] (Theorem 5.1). Combining this result and Coleman…
This is a survey of results on the construction of holomorphic cusp forms on tube domains originally initiated by Ikeda. Besides a survey it includes conjectures and possible applications of our work.
We study the second moment of the L-function associated to a holomorphic primitive cusp form of even weight perturbed by a new family of mollifiers. This family is a natural extension of the mollifers considered by Conrey and by Bui, Conrey and Young. As an application, we improve the current lower bound on critical zeros of holomorphic primitive cusp forms.
In this article, we compute the Petersson norm of a family of weight one cusp forms constructed by Hecke and express it in terms of the Rademacher symbol and the regulator of real quadratic field.
The goal of this paper is to construct the p-adic analytic family of overconvergent half-integral weight modular forms using Hecke-equivariant overconvergent Shintani lifting. The classical Shintani map is the Hecke-equivariant map from the space of cusp forms of integral weight to the space of cusp forms of half-integral weight. Glenn Stevens proved in [St1] that there is Lamda-adic lifting of this map to the Hida family of ordinary cusp forms of integral weight and consequently constructed Lamda-adic modular eigen form of half-integral weight. The natural thing to do is generalizing his resu…
Let $ Γ$ be a congruence subgroup of $ \mathrm{Sp}_{2n}(\mathbb Z) $. Using Poincaré series of $ K $-finite matrix coefficients of integrable discrete series representations of $ \mathrm{Sp}_{2n}(\mathbb R) $, we construct a spanning set for the space $ S_m(Γ) $ of Siegel cusp forms of weight $ m\in\mathbb Z_{>2n} $. We prove the non-vanishing of certain elements of this spanning set using Muić's integral non-vanishing criterion for Poincaré series on locally compact Hausdorff groups. Moreover, using the representation theory of $ \mathrm{Sp}_{2n}(\mathbb R) $, we study the Petersson inner pro…
Let $k\geq 2$ be an integer and $j$ an integer satisfying $1\leq j \leq 4k-5$. We define a family $\{ C_{j,k}(z) \}_{1\leq j \leq 4k-5} $ of eta quotients, and prove that this family constitute a basis for the space $S_{2k} (Γ_0 (12))$ of cusp forms of weight $2k$ and level $12$. We then use this basis together with certain properties of modular forms at their cusps to prove an extension of the Ramanujan-Mordell formula.
The Katz-Sarnak Density Conjecture states that the behavior of zeros of a family of $L$-functions near the central point (as the conductors tend to zero) agree with the behavior of eigenvalues near 1 of a classical compact group (as the matrix size tends to infinity). Using the Petersson formula, Iwaniec, Luo and Sarnak \cite{ILS} proved that the behavior of zeros near the central point of holomorphic cusp forms agree with the behavior of eigenvalues of orthogonal matrices for suitably restricted test functions. We prove a similar result for level 1 cuspidal Maass forms, the other natural fami…
We construct an explicit family of modular iterated integrals which involves cusp forms. This leads to a new method of producing "invariant versions" of iterated integrals of modular forms. The construction will be based on an extension of higher-order modular forms which, in contrast to the standard higher-order forms, applies to general Fuchsian groups of the first kind and, as such, is of independent interest.
We prove a strong form of the trivial zero conjecture at the central point for the $p$-adic $L$-function of a non-critically refined self-dual cohomological cuspidal automorphic representation of $\mathrm{GL}_2$ over a totally real field, which is Iwahori spherical at places above $p$. In the case of a simple zero we adapt the approach of Greenberg and Stevens, based on the functional equation for the $p$-adic $L$-function of a nearly finite slope family and on improved $p$-adic $L$-functions that we construct using automorphic symbols and overconvergent cohomology. For higher order zeros we d…
We specify sufficient conditions for the square modulus of the local parameters of a family of GL(n) cusp forms to be bounded on average. These conditions are global in nature and are at present satisfied for n less than or equal to 4. As an application, we show that Rankin-Selberg L-functions on GL(m) x GL(n), when m and n are less than or equal to 4, satisfy the standard convexity bound.
Suppose that p > 5 is a rational prime. Starting from a well-known p-adic analytic family of ordinary elliptic cusp forms of level p due to Hida, we construct a certain p-adic analytic family of holomorphic Siegel cusp forms of arbitrary even genus and of level p associated with Hida's p-adic analytic family via the Duke-Imamoglu lifting provided by Ikeda. Moreover, we also give a similar results for the Siegel Eisenstein series of even genus with trivial Nebentypus.
Let F be a totally real field and G=GSp(4)_{/F}. In this paper, we show under a weak assumption that, given a Hecke eigensystem lambda which is (p,P)-ordinary for a fixed parabolic P in G, there exists a several variable p-adic family underline{lambda} of Hecke eigensystems (all of them (p,P)-nearly ordinary) which contains lambda. The assumption is that lambda is cohomological for a regular coefficient system. If F=Q, the number of variables is three. Moreover, in this case, we construct the three variable p-adic family rho_{underline{lambda}} of Galois representations associated to underline…
Suppose that p > 5 is a rational prime. Starting from a well-known p-adic analytic family of ordinary elliptic cusp forms of level p due to Hida, we construct a certain p-adic analytic family of holomorphic Siegel cusp forms of arbitrary even genus and of level p associated with Hida's p-adic analytic family via the Duke-Imamoglu lifting which has been established by Ikeda. Moreover, we also give a similar results for the Siegel Eisenstein series of even genus with trivial Nebentypus.
For a fixed SL(3, Z) Maass form g, we consider the family of L-functions L(g \times u_j, s) where u_j runs over the family of Hecke-Maass cusp forms on SL(2,Z). We obtain an estimate for the second moment of this family of L-functions at the special points \half + it_j consistent with the Lindelöf Hypothesis. We also obtain a similar upper bound on the sixth moment of the family of Hecke-Maass cusp forms at these special points; this is apparently the first occurrence of a Lindelöf-consistent estimate for a sixth power moment of a family of GL(2) L-functions.
We prove equidistribution theorems for a family of holomorphic Siegel cusp forms of general degree in the level aspect. Our main contribution is to estimate unipotent contributions for general degree in the geometric side of Arthur's invariant trace formula in terms of Shintani zeta functions. Several applications including the vertical Sato-Tate theorem and low-lying zeros for standard $L$-functions of holomorphic Siegel cusp forms are discussed. We also show that the "non-genuine forms" which come from non-trivial endoscopic contributions by Langlands functoriality classified by Arthur are n…
We construct p-adic families of Klingen Eisenstein series and L-functions for cuspforms (not necessarily ordinary) unramified at an odd prime p on definite unitary groups of signature (r, 0) (for any positive integer r) for a quadratic imaginary field split at p. When r=2, we show that the constant term of the Klingen Eisenstein family is divisible by a certain p-adic L-function.
We consider sign changes of Fourier coefficients of Hecke-Maass cusp forms for the group $\mathrm{SL}_3(\mathbb Z)$. When the underlying form is self-dual, we show that there are $\gg_\varepsilon X^{5/6-\varepsilon}$ sign changes among the coefficients $\{A(m,1)\}_{m\leq X}$ and that there is a positive proportion of sign changes for many self-dual forms. Similar result concerning the positive proportion of sign changes also hold for the real-valued coefficients $A(m,m)$ for generic $\mathrm{GL}_3$ cusp forms, a result which is based on a new effective Sato-Tate type theorem for a family of $\…
In [6], Kohnen proves that if $Γ=Γ_0(N)$ where $N$ is a square-free integer, then any modular function of weight $0$ for $Γ$ having a divisor supported at the cusps is an $η$-product. Under the condition of having rational Fourier coefficients, we are able to extend Kohnen's result to the case where $N$ is the square of a prime. If the rationality condition does not hold, we show that the statement is no longer true by providing a family of counter-examples that generalizes naturally the Dedekind $η$-function. This paper fits within the framework of generalized modular forms in the sense of Kn…
Let p>2 be a prime and let X be a compactified PEL Shimura variety of type (A) or (C) such that p is an unramified prime for the PEL datum. Using the geometric approach of Andreatta, Iovita, Pilloni, and Stevens we define the notion of families of overconvergent locally analytic p-adic modular forms of Iwahoric level for X. We show that the system of eigenvalues of any finite slope cuspidal eigenform of Iwahoric level can be deformed to a family of systems of eigenvalues living over an open subset of the weight space. To prove these results, we actually construct eigenvarieties of the expected…
We derive a formula for Greenberg's $L$-invariant of Tate twists of the symmetric sixth power of an ordinary non-CM cuspidal newform of weight $\geq4$, under some technical assumptions. This requires a "sufficiently rich" Galois deformation of the symmetric cube which we obtain from the symmetric cube lift to $\GSp(4)_{/\QQ}$ of Ramakrishnan--Shahidi and the Hida theory of this group developed by Tilouine--Urban. The $L$-invariant is expressed in terms of derivatives of Frobenius eigenvalues varying in the Hida family. Our result suggests that one could compute Greenberg's $L$-invariant of all…
We obtain a Weyl law with power savings for the universal families of cuspidal automorphic representations, ordered by analytic conductor, of $\mathrm{GL}_2$ over $\mathbb{Q}$, as well as for Hecke characters over any number field. The method proceeds by establishing the requisite analytic properties of the underlying conductor zeta function.
The purpose of this paper is to investigate the number of classical weight one specializations of a non-CM ordinary Hida family of parallel weight Hilbert cusp forms. It is known that a specialization of a primitive ordinary Hida family at any arithmetic points of weight at least two is a classical holomorphic Hilbert cusp form. However, this is not always the case for weight one specializations. Balasubramanyam, Ghate and Vatsal proved that such a Hida family admits infinitely many classical weight one specializations if and only if it is of CM type. We give an explicit upper bound on the num…
We prove a version of Bagchi's Theorem and of Voronin's Universality Theorem for family of primitive cusp forms of weight $2$ and prime level, and discuss under which conditions the argument will apply to general reasonable family of automorphic $L$-functions.
Breaking Universality in the Lower Order Terms in the 1-level and 2-level Density of Holomorphic Cusp Newforms
University of Maryland（马里兰大学）；University of Chicago（芝加哥大学）；University of Washington（华盛顿大学）；Missouri State University（密苏里州立大学）；University of Michigan（密歇根大学）；University of Rochester（罗切斯特大学）；Princeton University（普林斯顿大学）；Yale University（耶鲁大学）；Williams College（威廉姆斯学院）；Columbia University（哥伦比亚大学）；Carleton College（卡尔顿学院）
We study low-lying zeros of $L$-functions attached to holomorphic cusp forms of level $1$ and large weight. In this family, the Katz--Sarnak heuristic with orthogonal symmetry type was established in the work of Iwaniec, Luo and Sarnak for test functions $ϕ$ satisfying the condition supp$(\widehat ϕ) \subset(-2,2)$. We refine their density result by uncovering lower-order terms that exhibit a sharp transition when the support of $\widehat ϕ$ reaches the point $1$. In particular the first of these terms involves the quantity $\widehat ϕ(1)$ which appeared in previous work of Fouvry--Iwaniec and…
Let $j(z)$ be the modular $j$-invariant function. Let $τ$ be an algebraic number in the complex upper half plane $\mathbb{H}$. It was proved by Schneider and Siegel that if $τ$ is not a CM point, i.e., $[\mathbb{Q}(τ):\mathbb{Q}]\neq2$, then $j(τ)$ is transcendental. Let $f$ be a harmonic weak Maass form of weight $0$ on $Γ_0(N)$. In this paper, we consider an extension of the results of Schneider and Siegel to a family of values of $f$ on Hecke orbits of $τ$. For a positive integer $m$, let $T_m$ denote the $m$-th Hecke operator. Suppose that the coefficients of the principal part of $f$ at t…
In this paper we prove one side divisibility of the Iwasawa-Greenberg main conjecture for Rankin-Selberg product of a weight two cusp form and an ordinary CM form of higher weight, using congruences between Klingen Eisenstein series and cusp forms on $\mathrm{GU}(3,1)$, generalizing earlier result of the third-named author to allow non-ordinary cusp forms. The main result is a key input in the third author's proof for Kobayashi's $\pm$-main conjecture for supersingular elliptic curves. The new ingredient here is developing a semi-ordinary Hida theory along an appropriate smaller weight space, …
The correspondence principle in physics between quantum mechanics and classical mechanics suggests deep relations between spectral and geometric entities of Riemannian manifolds. We survey---in a way intended to be accessible to a wide audience of mathematicians---a mathematically rigorous instance of such a relation that emerged in recent years, showing a dynamical interpretation of certain Laplace eigenfunctions of hyperbolic surfaces.
It is known due to Baker and Montgomery that almost all Fekete polynomials under certain ordering have at least one zero on the interval (0, 1). In terms of the positive-definiteness, Fekete polynomial has no zero on the interval (0, 1) if and only if the corresponding automorphic form is positive-definite. On generalizing their result, we formulate an axiomatic result about sets of automorphic forms satisfying certain averages when suitably ordered, which ensures that almost all forms are not positive-definite within such sets. We then apply the result to various families, including the famil…
We consider the family of Rankin-Selberg convolution L-functions of a fixed SL(3, Z) Maass form with the family of Hecke-Maass cusp forms on SL(2, Z). We estimate the second moment of this family of L-functions with a "long" integration in t-aspect. These L-functions are distinguished by their high degree (12) and large conductors (of size T^{12}).
We consider the Rankin-Selberg L-functions associated with a fixed modular form of full level and holomorphic cuspidal newforms of large even weight, fixed level and fixed primitive nebentypus. We compute the second moment of this family in fairly general ranges, and obtain an asymptotic formula with a power saving error term. A special case treats the fourth moment of L-functions associated with holomorphic cusp forms.
The Katz-Sarnak density conjecture states that, in the limit as the conductors tend to infinity, the behavior of normalized zeros near the central point of families of L-functions agree with the N -> oo scaling limits of eigenvalues near 1 of subgroups of U(N). Evidence for this has been found for many families by studying the n-level densities; for suitably restricted test functions the main terms agree with random matrix theory. In particular, all one-parameter families of elliptic curves with rank r over Q(T) and the same distribution of signs of functional equations have the same limiting …
In this paper, we aimed at constructing a two-variable Coleman map for a given $p$-adic family of eigen cuspforms with a fixed non-zero slope (Coleman family). A Coleman map is a machinary which transforms a hypothetical $p$-adic family of zeta elements to a $p$-adic $L$-function. The result would be a non-ordinary generalization of a two-variable Coleman map for a given Hida deformation obtained by the second-named author.
Let $η>0$ be a fixed positive number, let $N$ be a sufficiently large number. In this paper, we study the second moment of the sum of Hecke eigenvalues over primes in short intervals (whose length is $η\log N$) on average (with some weights) over the family of weight $k$ holomorphic Hecke cusp forms. We also generalize the above result to Hecke-Maass cusp forms for $SL(2,\mathbb{Z})$ and $SL(3,\mathbb{Z}).$ By applying the Hardy-Littlewood prime 2-tuples conjecture, we calculate the exact values of the mean values.
We prove the following uniformity principle: if one of the Galois representations in the family attached to a genus two Siegel cusp form of weight $k>3$, "semistable" and with multiplicity one, is reducible (for an odd prime $p$),then all the representations in the family are reducible. This, combined with Serre's conjecture (which is now a theorem) gives a proof of the Endoscopy Conjecture.
We present an explicit approach to the GL(3) Kuznetsov formula. As an application, for a restricted class of test functions, we obtain the low-lying zero densities for the following three families: cuspidal GL(3) Maass forms phi, the symmetric square family sym^2 phi on GL(6), and the adjoint family Ad phi on GL(8). Hence we can identify their symmetry types; they are: unitary, unitary, and symplectic, respectively.
Cusp restrictions and Bunke--Naumann invariants with level structure
C. N. Yang Institute for Theoretical Physics, Stony Brook University（纽约州立大学石溪分校杨振宁理论物理研究所）；Weinberg Institute, The University of Texas at Austin（德克萨斯大学奥斯汀分校温伯格研究所）
In this paper, we will first investigate the linear relations of a one parameter family of Siegel Poincaré series. Then we give the applications to the non-vanishing of Fourier coefficients of Siegel cusp eigenforms and the central values.
An automorphic self dual L-function has the super-positivity property if all derivatives of the completed L-function at the central point $s=1/2$ are non-negative and all derivatives at a real point $s > 1/2$ are positive. In this paper we prove that at least 12% of L-functions associated to Hecke basis cusp forms of weight $2$ and large prime level $q$ have the super-positivity property. It is also shown that at least 49% of such L-functions have no real zeros on $ \Re(s) > 0$ except possibly at $s = 1/2.$
In this paper we study the compatible family of degree-4 Scholl representations $ρ_{\ell}$ associated with a space $S$ of weight $κ> 2$ noncongruence cusp forms satisfying Quaternion Multiplications over a biquadratic field $K$. It is shown that when either $K$ is totally real or $κ$ is odd, $ρ_\ell$ is automorphic, that is, its associated L-function has the same Euler factors as the L-function of an automorphic form for $GL_4(\mathbb Q)$. Further, it yields a relation between the Fourier coefficients of noncongruence cusp forms in $S$ and those of certain automorphic forms via the three-term …
We consider the logarithm of the central value $\log L(1/2)$ in the orthogonal family ${L(s,f)}_{f \in H_k}$ where $H_k$ is the set of weight $k$ Hecke-eigen cusp form for $SL_2(\mathbb{Z})$, and in the symplectic family ${L(s,χ_{8d})}_{d \asymp D}$ where $χ_{8d}$ is the real character associated to fundamental discriminant $8d$. Unconditionally, we prove that the two distributions are asymptotically bounded above by Gaussian distributions, in the first case of mean $-\frac{1}{2} \log \log k$ and variance $\log \log k$, and in the second case of mean $\frac{1}{2}\log \log D$ and variance $\log…
We study the distribution, in the space of Satake parameters, of local components of Siegel cusp forms of genus 2 and growing weight, subject to a specific weighting which allows us to apply results concerning Bessel models and a variant of Petersson's formula. We obtain for this family a quantitative local equidistribution result, and derive a number of consequences. In particular, we show that the computation of the density of low-lying zeros of the spinor L-functions (for restricted test functions) gives global evidence for a well-known conjecture of Böcherer concerning the arithmetic natur…
We characterize Maass cusp forms for any cofinite Hecke triangle group as 1-eigenfunctions of appropriate regularity of a transfer operator family. This transfer operator family is associated to a certain symbolic dynamics for the geodesic flow on the orbifold arising as the orbit space of the action of the Hecke triangle group on the hyperbolic plane. Moreover we show that the Selberg zeta function is the Fredholm determinant of the transfer operator family associated to an acceleration of this symbolic dynamics.
A family of global integrals representing a product of tensor product (partial) $L$-functions: $ L^S(s,π\timesτ_1)L^S(s,π\timesτ_2)... L^S(s,π\timesτ_r) $ are established in this paper, where $π$ is an irreducible cuspidal automorphic representation of a quasi-split classical group of Hermitian type and $τ_1,...,τ_r$ are irreducible unitary cuspidal automorphic representations of $\GL_{a_1},...,\GL_{a_r}$, respectively. When $r=1$ and the classical group is an orthogonal group, this was studied by Ginzburg, Piatetski-Shapiro and Rallis in 1997 and when $π$ is generic and $τ_1,...,τ_r$ are not …
We prove an equidistribution theorem for a family of holomorphic Siegel cusp forms for $GSp_4/\mathbb{Q}$ in various aspects. A main tool is Arthur's invariant trace formula. While Shin and Shin-Templier used Euler-Poincaré functions at infinity in the formula, we use a pseudo-coefficient of a holomorphic discrete series to extract holomorphic Siegel cusp forms. Then the non-semisimple contributions arise from the geometric side, and this provides new second main terms $A, B_1$ in the main theorem which have not been studied and a mysterious second term $B_2$ also appears in the second main te…
