We develop an analytic theory of cusps for Scholze's $p$-adic modular curves at infinite level in terms of perfectoid parameter spaces for Tate curves. As an application, we describe a canonical tilting isomorphism between an anticanonical overconvergent neighbourhood of the ordinary locus of the modular curve at level $Γ_1(p^\infty)$ and the analogous locus of an infinite level perfected Igusa variety. We also prove various $q$-expansion principles for functions on modular curves at infinite level, namely that the properties of extending to the cusps, vanishing, coming from finite level, and being bounded, can all be detected on $q$-expansions.
Associated to an (adelic) Hilbert modular form is a sequence of `Fourier coefficients' which uniquely determine the form. In this paper we characterize Hilbert modular cusp forms by the size of their Fourier coefficients. This answers in the affirmative a question posed by Winfried Kohnen.
In this paper, we present the outcome of vast computer calculations, locating several of the very rare instances of level one cuspidal Bianchi modular forms that are not lifts of elliptic modular forms.
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 give explicit upper bounds for the coefficients of arbitrary weight $k$, level 2 cusp forms, making Deligne's well-known $O(n^{\frac{k-1}{2}+ε})$ bound precise. We also derive asymptotic formulas and explicit upper bounds for the coefficients of certain level 2 modular functions.
The aim of this article is to study the fields generated by the Fourier coefficients of Hilbert newforms at arbitrary cusps. Precisely, given a cuspidal Hilbert newform $f$ and a matrix $σ$ in (a suitable conjugate of) the Hilbert modular group, we give a cyclotomic extension of the field generated by the Fourier coefficients at infinity which contains all the Fourier coefficients of $f||_kσ$.
Let $ \mathfrak{C}^+_{ns}(p) $ be the Cuspidal Divisor Class Group of the modular curves $X^+_{ns}(p) $ associated to the normalizer of a non-split Cartan subgroup of level $ p$. I study the $ p-$primary part of $ \mathfrak{C}^+_{ns}(p) $ and estimate the order of growth of $ |\mathfrak{C}^+_{ns}(p)| $.
In this article, we study the simultaneous sign changes of the Fourier coefficients of two Hilbert cusp forms of different integral weights. We also study the simultaneous non-vanishing of Fourier coefficients, of two distinct non-zero primitive Hilbert cuspidal non-CM eigenforms of integral weights, at powers of a fixed prime ideal.
We show that the Drinfeld modular forms with $A$-expansion that have been constructed by the author are precisely the hyperderivatives of the subfamily of single-cuspidal Drinfeld modular forms with $A$-expansions that remain modular after hyperdifferentiation. In addition, we show that Drinfeld-Poincaré series display a similar behavior with respect to hyperdifferentiation, giving indirect evidence that the Drinfeld modular forms with $A$-expansions are Drinfeld-Poincaré series. The Drinfeld-Poincaré series that we consider generalize previous examples of such series by Gekeler, and Gerritzen…
Let K be a number field, let M be the Hilbert modular orbifold of K, and let m(q) be the probability measure uniformly supported on the cusp cross sections of M at height q. We generalize a method of Zagier and show that m(q) distributes uniformly with respect to the normalized Haar measure m on M as q tends to zero, and relate the rate by which m(q) approaches m to the Riemann hypothesis for the Dedekind zeta function of K.
I find an explicit description of modular units in terms of Siegel functions for the modular curves $X^+_{ns}(p^k)$ associated to the normalizer of a non-split Cartan subgroup of level $p^k$ where $p\not=2,3$ is a prime. The Cuspidal Divisor Class Group $\mathfrak{C}^+_{ns}(p^k)$ on $X^+_{ns}(p^k)$ is explicitly described as a module over the group ring $R = \mathbb{Z}[(\mathbb{Z}/p^k\mathbb{Z})^*/\{\pm 1\}]$. In this paper I give a formula involving generalized Bernoulli numbers $B_{2,χ}$ for $|\mathfrak{C}^+_{ns}(p^k)|$.
Let $E$ be a level 1, vector valued Eisenstein series of half-integral weight, normalized so that the coefficients are all in $\mathbb{Z}$. We show that there is a level one vector valued cusp form $f$ with the same weight as $E$ and with coefficients in $\mathbb{Z}$, which is congruent to $E$ modulo the constant term of $E$.
We prove a non-vanishing result of modular L-values with quadratic twists, where the quadratic discriminants are in a short interval. Using this fact and Waldspurger's theorem, we improve the results of Balog-Ono[The chebotarev density theorem in short intervals and some questions of Serre, Journal of number theory. 91(2):356-371(2001)] on the non-vanishing of Fourier coefficients of half-integral weight eigenform.
We define canonical real analytic versions of modular forms of integral weight for the full modular group, generalising real analytic Eisenstein series. They are harmonic Maass waveforms with poles at the cusp, whose Fourier coefficients involve periods and quasi-periods of cusp forms, which are conjecturally transcendental. In particular, we settle the question of finding explicit `weak harmonic lifts' for every eigenform of integral weight $k$ and level one. We show that mock modular forms of integral weight are algebro-geometric and have Fourier coefficients proportional to $n^{1-k}(a'_n + …
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.
We determine the number of cusps of minimal Picard modular surfaces. The proof also counts cusps of other Picard modular surfaces of arithmetic interest. Consequently, for each N > 0 there are finitely many commensurability classes of nonuniform arithmetic lattices in SU(2, 1) that contain an N-cusped surface. We also discuss a higher-rank analogue.
Several authors have recently proved results which express cusp forms as $p$-adic limits of weakly holomorphic modular forms under repeated application of Atkin's $U$-operator. The proofs involve techniques from the theory of weak harmonic Maass forms, and in particular a result of Guerzhoy, Kent, and Ono on the $p$-adic coupling of mock modular forms and their shadows. Here we obtain strengthened versions of these results using techniques from the theory of holomorphic modular forms.
In this paper, we investigate Fourier expansions of meromorphic modular forms. Over the years, a number of special cases of meromorphic modular forms were shown to have Fourier expansions closely resembling the expansion of the reciprocal of the weight $6$ Eisenstein series which was computed by Hardy and Ramanujan. By investigating meromorphic modular forms within a larger space of so-called polar harmonic Maass forms, we prove in this paper that all negative-weight meromorphic modular forms (and furthermore all quasi-meromorpic modular forms) have Fourier expansions of this type, granted tha…
Ahlgren and Samart relate three cusp forms with complex multiplication to certain weakly holomorphic modular forms using $p$-adic bounds related to their Fourier coefficients. In these three examples, their result strengthens a theorem of Guerzhoy, Kent, and Ono which pairs certain CM forms with weakly holomorphic modular forms via $p$-adic limits. Ahlgren and Samart use only the theory of modular forms and Hecke operators, whereas Guerzhoy, Kent, and Ono use the theory of harmonic Maass forms. Here we extend Ahlgren and Samart's work to all cases where the cusp form space is one-dimensional a…
We estimate the sums \[ \sum_{c\leq x} \frac{S(m,n,c,χ)}{c}, \] where the $S(m,n,c,χ)$ are Kloosterman sums of half-integral weight on the modular group. Our estimates are uniform in $m$, $n$, and $x$ in analogy with Sarnak and Tsimerman's improvement of Kuznetsov's bound for the ordinary Kloosterman sums. Among other things this requires us to develop mean value estimates for coefficients of Maass cusp forms of weight $1/2$ and uniform estimates for $K$-Bessel integral transforms. As an application, we obtain an improved estimate for the classical problem of estimating the size of the error t…
Serre proved that any holomorphic cusp form of weight one for $Γ_1(N)$ is lacunary while a holomorphic modular form for $Γ_1(N)$ of higher integer weight is lacunary if and only if it is a linear combination of cusp forms of CM-type (see Serre, subsections 7.6 and 7.7). In this paper, we show that when a non-zero modular function of arbitrary real weight for any finite index subgroup of the modular group ${\SL}_2(\Z)$ is lacunary, it is necessarily holomorphic on the upper-half plane, finite at the cusps and has non-negative weight.
In the literature, the standard approach to finding bases of spaces of modular forms is via modular symbols and the homology of modular curves. By using the Eichler-Shimura isomorphism, a work by Wang shows how one can use a cohomological viewpoint to determine bases of spaces of cusp forms on $Γ_0(N)$ of weight $k \geq 2$ and character $χ$. It is interesting to look at the complexity of this alternative approach, and we do this for an explicit implementation of the algorithm suggested by Wang.
We study sums of additively twisted Fourier coefficients of a holomorphic cusp form, a Maass cusp form, and the symmetric-square lift of a holomorphic cusp form. We obtain bounds that are uniform with respect to both the form and the terms of the additive twist. We also improve a bound on a sum of additively twisted Fourier coefficients of a holomorphic cusp form that have also been twisted by a nonlinear exponential term. This last bound is uniform with respect to the terms of the additive twist but it is not uniform with respect to the form.
The goal in this preprint is to give an efficient algorithm to compute Puiseux expansions at cusps of X0(N). It is based on a relation with a hypergeometric function that holds for any N.
We consider a special theta lift $θ(f)$ from cuspidal Siegel modular forms $f$ on $\mathrm{Sp}_4$ to "modular forms" $θ(f)$ on $\mathrm{SO}(4,4)$. This lift can be considered an analogue of the Saito-Kurokawa lift, where now the image of the lift is representations of $\mathrm{SO}(4,4)$ that are quaternionic at infinity. We relate the Fourier coefficients of $θ(f)$ to those of $f$, and in particular prove that $θ(f)$ is nonzero and has algebraic Fourier coefficients if $f$ does. Restricting the $θ(f)$ to $G_2 \subseteq \mathrm{SO}(4,4)$, we obtain cuspidal modular forms on $G_2$ of arbitrarily…
We present a dimension formula for spaces of vector-valued modular forms of integer weight in case the associated multiplier system has finite image, and discuss the weight distribution of the module generators of holomorphic and cusp forms, as well as the duality relation between cusp forms and holomorphic forms for the contragredient.
Let $Γ$ be a cocompact, discrete, and irreducible subgroup of $\mathrm{PSL}_{2}(\mathbb{R})^{n}$. Let $ν$ be a unitary character of $Γ$. For $k\in1\slash 2\,\mathbb{Z}$, let $\sknu$ denote the complex vector space of cusp forms of weight-$\tk=\k$ and nebentypus $ν^{2k}$ with respect to $Γ$. We assume that $ω_{X,ν}$, the line bundle of cusp forms of weight-$\tilde{1\slash 2}:=(1\slash 2,\ldots,1\slash2)$ with nebentypus $ν$ over $X$ exists. Let $\lbrace f_{1},\ldots,f_{j_{\tk}} \rbrace$ denote an orthonormal basis of $\sknu$. In this article, we show that as $k\rightarrow \infty$, the sum $\sum…
For a fix modular form g and a non negative ineteger ν, by using Rankin-Cohen bracket we first define a linear map $T_{g,ν}$ on the space of modular forms. We explicitly compute the adjoint of this map and show that the n-th Fourier coefficients of the image of the cusp form f under this map is, upto a constant a special value of Rankin-Selberg convolution of f and g.
We give congruences between the Eisenstein series and a cusp form in the cases of Siegel modular forms and Hermitian modular forms. We should emphasize that there is a relation between the existence of a prime dividing the $k-1$-th generalized Bernoulli number and the existence of non-trivial Hermitian cusp forms of weight $k$. We will conclude by giving numerical examples for each case.
We consider the generalized Jacobian $\widetilde{J}_0(N)$ of a modular curve $X_0(N)$ with respect to a reduced divisor given by the sum of all cusps on it. When $N$ is a power of a prime $\geq 5$, we exhibit that the group of rational torsion points $\widetilde{J}_0(N)(\mathbb{Q})_{\mathrm{Tor}}$ tends to be much smaller than the classical Jacobian.
Let $f$ be a cuspidal Hecke eigenform of level 1. We prove the automorphy of the symmetric power lifting $\mathrm{Sym}^n f$ for every $n \geq 1$. We establish the same result for a more general class of cuspidal Hecke eigenforms, including all those associated to semistable elliptic curves over $\mathbb{Q}$.
Let $τ$ be the primitive Dirichlet character of conductor 4, let $χ$ be the primitive even Dirichlet character of conductor 8 and let $k$ be an integer. Then the $U_2$ operator acting on cuspidal overconvergent modular forms of weight $2k+1$ and character $τ$ has slopes in the arithmetic progression ${2,4,...,2n,...}$, and the $U_2$ operator acting on cuspidal overconvergent modular forms of weight $k$ and character $χ\cdot τ^k$ has slopes in the arithmetic progression ${1,2,...,n,...}$. We then show that the characteristic polynomials of the Hecke operators $U_2$ and $T_p$ acting on the space…
We give an exact formula of the average of adjoint $L$-functions of holomorphic Hilbert cusp forms with a fixed weight and a square-free level, which is a generalization of Zagier's formula known for the case of elliptic cusp forms on ${\rm SL}_2(\mathbb{Z})$. As an application, we prove that the Satake parameters of Hilbert cusp forms with a fixed weight and with growing square-free levels are equidistributed in an ensemble constructed by values of the adjoint $L$-functions.
In this article, for $n\geq 2$, we compute asymptotic, qualitative, and quantitative estimates of the Bergman kernel of Picard modular cusp forms associated to torsion-free, cocompact subgroups of $\mathrm{SU}\big((n,1),\mathbb{C}\big)$. The main result of the article is the following result. Let $Γ\subset \mathrm{SU}\big((2,1),\mathcal{O}_{K}\big)$ be a torsion-free subgroup of finite index, where $K$ is a totally imaginary field. Let $\mathcal{B}_Γ^{k}$ denote the Bergman kernel associated to the $\mathcal{S}_{k}(Γ)$, complex vector space of weight-$k$ cusp forms with respect to $Γ$. Let $\m…
We elaborate an explicit version of the relative trace formula on $\PGL(2)$ over a totally real number field for the toral periods of Hilbert cusp forms along the diagonal split torus. As an application, we prove (i) a spectral equidistribution result in the level aspect for Satake parameters of holomorphic Hilbert cusp forms weighted by central $L$-values, and (ii) a bound of quadratic base change $L$-functions for Hilbert cusp forms with a subconvex exponent in the weight aspect.
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…
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.
In this article, we consider the group $F_1^\infty(N)$ of modular units on $X_1(N)$ that have divisors supported on the cusps lying over $\infty$ of $X_0(N)$, called the $\infty$-cusps. For each positive integer $N$, we will give an explicit basis for the group $F_1^\infty(N)$. This enables us to compute the group structure of the rational torsion subgroup $C_1^\infty(N)$ of the Jacobian $J_1(N)$ of $X_1(N)$ generated by the differences of the $\infty$-cusps. In addition, based on our numerical computation, we make a conjecture on the structure of the $p$-primary part of $C_1^\infty(p^n)$ for …
We give coefficient formulas for antisymmetric vector-valued cusp forms with rational Fourier coefficients for the Weil representation associated to a finite quadratic module. The forms we construct always span all cusp forms in weight at least three. These formulas are useful for computing explicitly with theta lifts.
We address the problem of the determination of the images of the Galois representations attached to genus 2 Siegel cusp forms of level 1 having multiplicity one. These representations are symplectic. We prove that the images are as large as possible for almost every prime, if the Siegel cusp form is not a Maass spezialform and verifies two easy to check conditions.
Modular graph functions arise in the calculation of the low-energy expansion of closed-string scattering amplitudes. For toroidal world-sheets, they are ${\rm SL}(2,\mathbb{Z})$-invariant functions of the torus complex structure that have to be integrated over the moduli space of inequivalent tori. We use methods from resurgent analysis to construct the non-perturbative corrections arising when the argument of the modular graph function approaches the cusp on this moduli space. ${\rm SL}(2,\mathbb{Z})$-invariance will in turn strongly constrain the behaviour of the non-perturbative sector when…
In this paper, we explore a two-way connection between quasimodular forms of depth $1$ and a class of second-order modular differential equations with regular singularities on the upper half-plane and the cusps. Here we consider the cases $Γ=Γ_0^+(N)$ generated by $Γ_0(N)$ and the Atkin-Lehner involutions for $N=1,2,3$ ($Γ_0^+(1)=\mathrm{SL}(2,\mathbb Z)$). Firstly, we note that a quasimodular form of depth $1$, after divided by some modular form with the same weight, is a solution of a modular differential equation. Our main results are the converse of the above statement for the groups $Γ_0^…
We continue the analysis of modular invariant functions, subject to inhomogeneous Laplace eigenvalue equations, that were determined in terms of Poincaré series in a companion paper. The source term of the Laplace equation is a product of (derivatives of) two non-holomorphic Eisenstein series whence the modular invariants are assigned depth two. These modular invariant functions can sometimes be expressed in terms of single-valued iterated integrals of holomorphic Eisenstein series as they appear in generating series of modular graph forms. We show that the set of iterated integrals of Eisenst…
We prove an equidistribution statement for the Satake parameters of the local representations attached to Siegel cusp forms of degree $2$ of increasing level and weight, counted with a certain arithmetic weight. We then apply this to compute the symmetry type of a similarly weighted distribution of the low-lying zeros of $L$-functions attached to these cusp forms.
We show that for primes $N, p \geq 5$ with $N \equiv -1 \bmod p$, the class number of $\mathbb{Q}(N^{1/p})$ is divisible by $p$. Our methods are via congruences between Eisenstein series and cusp forms. In particular, we show that when $N \equiv -1 \bmod p$, there is always a cusp form of weight $2$ and level $Γ_0(N^2)$ whose $\ell$-th Fourier coefficient is congruent to $\ell + 1$ modulo a prime above $p$, for all primes $\ell$. We use the Galois representation of such a cusp form to explicitly construct an unramified degree $p$ extension of $\mathbb{Q}(N^{1/p})$.
We prove that two cusps of the same dimension in the Baily-Borel compactification of some classical series of modular varieties are linearly dependent in the rational Chow group of the compactification. This gives a higher dimensional analogue of the Manin-Drinfeld theorem. As a consequence, we obtain a higher dimensional generalization of modular units as higher Chow cycles on the modular variety.
This was originally an appendix to our paper `Fourier expansions at cusps' [arXiv:1807.00391]. The purpose of this note is to give a proof of a theorem of Shimura on the action of $\mathrm{Aut}(\mathbb{C})$ on modular forms for $Γ(N)$ from the perspective of algebraic modular forms. As the theorem is well-known, we do not intend to publish this note but want to keep it available as a preprint.
We carry out some computations of vector valued Siegel modular forms of degree two, weight (k,2) and level one. Our approach is based on Satoh's description of the module of vector-valued Siegel modular forms of weight (k, 2) and an explicit description of the Hecke action on Fourier expansions. We highlight three experimental results: (1) we identify a rational eigenform in a three dimensional space of cusp forms, (2) we observe that non-cuspidal eigenforms of level one are not always rational and (3) we verify a number of cases of conjectures about congruences between classical modular forms…
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, …
We compare the homology of a congruence subgroup Gamma of GL_2(Z) with coefficients in the Steinberg modules over Q and over E, where E is a real quadratic field. If R is any commutative base ring, the last connecting homomorphism psi_{Gamma,E} in the long exact sequence of homology stemming from this comparison has image in H_0(Gamma, St(Q^2;R)) generated by classes z_βindexed by beta in E \ Q. We investigate this image. When R=C, H_0(Gamma, St(Q^2;C)) is isomorphic to a space of classical modular forms of weight 2, and the image lies inside the cuspidal part. In this case, z_beta is closely …
