Let $V_1,V_2,V_3$ be a triple of even dimensional vector spaces over a number field $F$ equipped with nondegenerate quadratic forms $\mathcal{Q}_1,\mathcal{Q}_2,\mathcal{Q}_3$, respectively. Let $Y \subset \prod_{i=1}^3 V_i$ be the closed subscheme consisting of $(v_1,v_2,v_3)$ such that $\mathcal{Q}_1(v_1)=\mathcal{Q}_2(v_2)=\mathcal{Q}_3(v_3)$. One has a Poisson summation formula for this scheme under suitable assumptions on the functions involved, but the relevant Fourier transform was previously only defined as a correspondence. In the current paper we employ a novel global-to-local argument to prove that this Fourier transform is well-defined on the Schwartz space of $Y(\mathbb{A}_F).$ To execute the global-to-local argument, we introduce boundary terms and thereby extend the Poisson summation formula to a broader class of test functions. This is the first time a summation formula with boundary terms has been proven for a spherical variety that is not a Braverman-Kazhdan space.
In this short note, we treat an unbalanced shifted convolution sum of Fourier coefficients of cusp forms by a rather simple argument. Our result improves previous results established by more advanced approaches.
Let $V_1,V_2,V_3$ be a triple of even dimensional vector spaces over a number field $F$ equipped with nondegenerate quadratic forms $\mathcal{Q}_1,\mathcal{Q}_2,\mathcal{Q}_3$, respectively. Let \begin{align*} Y \subset \prod_{i=1}V_i \end{align*} be the closed subscheme consisting of $(v_1,v_2,v_3)$ on which $\mathcal{Q}_1(v_1)=\mathcal{Q}_2(v_2)=\mathcal{Q}_3(v_3)$. Motivated by conjectures of Braverman and Kazhdan and related work of Lafforgue, Ngô, and Sakellaridis we prove an analogue of the Poisson summation formula for certain functions on this space.
We treat an unbalanced shifted convolution sum of Fourier coefficients of cusp forms. As a consequence, we obtain an upper bound for correlation of three Hecke eigenvalues of holomorphic cusp forms $\sum_{H\leq h\leq 2H}W\big(\frac{h}{H}\big)\sum_{X\leq n\leq 2X}λ_{1}(n-h)λ_{2}(n)λ_{3}(n+h)$, which is nontrivial provided that $H\geq X^{2/3+\varepsilon}$. The result can be viewed as a cuspidal analogue of a recent result of Blomer on triple correlations of divisor functions.
The possible torsion groups of elliptic curves induced by Diophantine triples over quadratic fields, which do not appear over Q, are Z/2Z x Z/10Z, Z/2Z x Z/12Z and Z/4Z x Z/4Z. In this paper, we show that all these torsion groups indeed appear over some quadratic field. Moreover, we prove that there are infinitely many Diophantine triples over quadratic fields which induce elliptic curves with these torsion groups.
We state and prove an identity which represents the most general eta-products of weight 1 by binary quadratic forms. We discuss the utility of binary quadratic forms in finding a multiplicative completion for certain eta-quotients. We then derive explicit formulas for the Fourier coefficients of certain eta-quotients of weight 1 and level 47, 71, 135,648 1024, and 1872.
Let $-d$ be a a negative discriminant and let $T$ vary over a set of representatives of the integral equivalence classes of integral binary quadratic forms of discriminant $-d$. We prove an asymptotic formula for $d \to \infty$ for the average over $T$ of the number of representations of $T$ by an integral positive definite quaternary quadratic form and obtain results on averages of Fourier coefficients of linear combinations of Siegel theta series. We also find an asymptotic bound from below on the number of binary forms of fixed discriminant $-d$ which are represented by a given quaternary f…
Let $k \geq 1$ be a cube-free integer with $k \equiv 1 \pmod {9}$ and $\gcd(k, 7\cdot 571)=1$. In this paper, we prove the existence of infinitely many triples of imaginary quadratic fields $\mathbb{Q}(\sqrt{d})$, $\mathbb{Q}(\sqrt{d+1})$ and $\mathbb{Q}(\sqrt{d+k^2})$ with $d \in \mathbb{Z}$ such that the class number of each of them is divisible by $3$. This affirmatively answers a weaker version of a conjecture of Iizuka \cite{iizuka-jnt}.
In this article, we establish an average behaviour of the normalised Fourier coefficients of the Hecke eigenforms supported at the integers represented by any primitive integral positive definite binary quadratic form of fixed discriminant $D < 0$ when the class number $h(D) = 1$. We also obtain a quantitative result for the number of sign changes of the sequence of the normalised Fourier coefficients $λ_{f}(n)$ of the Hecke eigenforms $f$ where $n$ is represented by any primitive integral positive definite binary quadratic form of fixed discriminant $D < 0$ when the class number $h(D) = 1$ in…
In this work we prove a prime number type theorem involving the normalised Fourier coefficients of holomorphic and Maass cusp forms, using the classical circle method. A key point is in a recent paper of Fouvry and Ganguly, based on Hoffstein-Ramakrishnan's result about the non-existence of the Siegel zeros for $GL(2)$ $L$-functions, which allows us to improve preceding estimates.
Let $τ_3(n)$ be the triple divisor function which is the number of solutions of the equation $d_1d_2d_3=n$ in natural numbers. It is shown that $$ \sum_{1\leq n_1,n_2,n_3\leq \sqrt{x}}τ_3(n_1^2+n_2^2+n_3^2)=c_1x^{\frac{3}{2}}(\log x)^2+ c_2x^{\frac{3}{2}}\log x +c_3x^{\frac{3}{2}} +O_{\varepsilon}(x^{\frac{11}{8}+\varepsilon}) $$ for some constants $c_1$, $c_2$ and $c_3$.
We prove several results about integers represented by positive definite quadratic forms, using a Fourier analysis approach. In particular, for an integer $\ell\geq 1$, we improve the error term in the partial sums of the number of representations of integers that are a multiple of $\ell$. This allows us to obtain unconditional Brun-Titchmarsh-type results in short intervals, and a conditional Cramér-type result on the maximum gap between primes represented by a given positive definite quadratic form.
In this paper, we investigate sign changes of Fourier coefficients of half-integral weight cusp forms. In a fixed square class $t\mathbb{Z}^2$, we investigate the sign changes in the $tp^2$-th coefficient as $p$ runs through the split or inert primes over the ring of integers in a quadratic extension of the rationals. We show that sign changes occur in both sets of primes when there exists a prime dividing the discriminant of the field which does not divide the level of the cusp form and find an explicit condition that determines whether sign changes occur when every prime dividing the discrim…
We give an explicit construct of a harmonic weak Maass form $F_Θ$ that is a "lift" of $Θ^3$, where $Θ$ is the classical Jacobi theta function. Just as the Fourier coefficients of $Θ^3$ are related to class numbers of imaginary quadratic fields, the Fourier coefficients of the "holomorphic part" of $F_Θ$ are associated to class numbers of real quadratic fields.
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 show that the eighth power of the Jacobi triple product is a Jacobi--Eisenstein series of weight $4$ and index $4$ and we calculate its Fourier coefficients. As applications we obtain explicit formulas for the eighth powers of theta-constants of arbitrary order and the Fourier coefficients of the Ramanujan Delta-function $Δ(τ)=η^{24}(τ)$, $η^{12}(τ)$ and $η^{8}(τ)$ in terms of Cohen's numbers $H(3,N)$ and $H(5,N)$. We give new formulas for the number of representations of integers as sums of eight higher figurate numbers. We also calculate the sixteenth and the twenty-fourth powers of the J…
We study the relation of the middle convolution to the $\ell$-adic Fourier transformation in the étale context. Using Katz' work and Laumon's theory of local Fourier transformations we obtain a detailed description of the local monodromy and the determinant of Katz' middle convolution functor $\MC_χ$ in the tame case. The theory of local $ε$-constants then implies that the property of an étale sheaf of having an at most quadratic determinant is often preserved under $\MC_χ$ if $χ$ is quadratic.
Eigenvectors of the discrete Fourier transform can be expressed using Ramanujan theta functions. New theta function identities, Ramanujan theta function identities, and generating functions for the quadratic numbers are a consequence.
We define "values" of the elliptic modular j-function at real quadratic irrationalities by using Hecke's hyperbolic Fourier expansions, and present some observations based on numerical experiments.
Let $L/F$ be a quadratic extension of totally real number fields. For any prime $p$ unramified in $L$, we construct a $p$-adic $L$-function interpolating the central values of the twisted triple product $L$-functions attached to a $p$-nearly ordinary family of unitary cuspidal automorphic representations of $\text{Res}_{L\times F/F}(\text{GL}_{2})$. Furthermore, when $L/\mathbb{Q}$ is a real quadratic number field and $p$ is a split prime, we prove a $p$-adic Gross-Zagier formula relating the value of the $p$-adic $L$-function outside the range of interpolation to the syntomic Abel-Jacobi imag…
Let $p>5$ be a fixed prime. We obtain an asymptotic formula related to small solutions of quadratic congruences of the form $x_1^2+x_2^2\equiv x_3^2\bmod{p^n}$ where $\max\{|x_1|,|x_2|,|x_3|\}\le p^{νn}$ with $ν>1/2$.
In this paper, we study lower order terms of the $1$-level density of low-lying zeros of quadratic Hecke L-functions in the Gaussian field. Assuming the Generalized Riemann Hypothesis, our result is valid for even test functions whose Fourier transforms are supported in $(-2, 2)$. Moreover, we apply the ratios conjecture of $L$-functions to derive these lower order terms as well. Up to the first lower order term, we show that our results is consistent with each other, when the Fourier transforms of the test functions are supported in $(-2, 2)$.
In this work we provide a meromorphic continuation in three complex variables of two types of triple shifted convolution sums of Fourier coefficients of holomorphic cusp forms. The foundations of this construction are based in the continuation of the spectral expansion of a special truncated Poincaré series recently developed by Jeffrey Hoffstein. As a result we are able to produce previously unstudied and nontrivial asymptotics of truncated shifted sums which we expect to correspond to off-diagonal terms in the third moment of automorphic L-functions.
In this article we give an analogue of Hecke and Sturm bounds for Hilbert modular forms over real quadratic fields. Let $K$ be a real quadratic field and $\Om_K$ its ring of integers. Let $Γ$ be a congruence subgroup of $\SL_2(\Om_K)$ and $M_{(k_1,k_2)}(Γ)$ the space of Hilbert modular forms of weight $(k_1,k_2)$ for $Γ$. The first main result is an algorithm to construct a finite set $S$, depending on $K$, $Γ$ and $(k_1,k_2)$, such that if the Fourier expansion coefficients of a form $G \in M_{(k_1,k_2)}(Γ)$ vanish on the set $S$, then $G$ is the zero form. The second result corresponds to th…
The elliptic modular j function enjoys many beautiful properties. Its Fourier coefficients are related to the Monster group, and its CM values generate abelian extensions over imaginary quadratic fields. Kaneko gave an arithmetic formula for the Fourier coefficients expressed in terms of the traces of the CM values. In this article, we are concerned with analogues of Kaneko's result for the McKay-Thompson series of square-free level.
Previous work by Rubinstein and Gao computed the n-level densities for families of quadratic Dirichlet L-functions for test functions where the sum of the supports of the Fourier transforms is at most 2, and showed agreement with random matrix theory predictions in this range for n < 4 but only in a restricted range for larger n. We extend these results and show agreement for n < 8, and reduce higher n to a Fourier transform identity. The proof involves adopting a new combinatorial perspective to convert all terms to a canonical form, which facilitates the comparison of the two sides.
We study the $1$-level density and the pair correlation of zeros of quadratic Dirichlet $L$-functions in function fields, as we average over the ensemble $\mathcal{H}_{2g+1}$ of monic, square-free polynomials with coefficients in $\mathbb{F}_q[x]$. In the case of the $1$-level density, when the Fourier transform of the test function is supported in the restricted interval $(\frac{1}{3},1)$, we compute a secondary term of size $q^{-\frac{4g}{3}}/g$, which is not predicted by the Ratios Conjecture. Moreover, when the support is even more restricted, we obtain several lower order terms. For examp…
We establish explicit Ichino's formulae for the central values of the triple product $L$-functions with emphasis on the calculations for the real place. The key ingredient for our computations is Proposition 6.8 which generalizes a result of Michel-Venkatesh. As an application we prove the optimal upper bound of a sum of restricted $L^2$-norms of the $L^2$-normalized newforms on certain quadratic extensions with prime level and bounded spectral parameter following the methods of Blomer.
We investigate integrality and divisibility properties of Fourier coefficients of meromorphic modular forms of weight $2k$ associated to positive definite integral binary quadratic forms. For example, we show that if there are no non-trivial cusp forms of weight $2k$, then the $n$-th coefficients of these meromorphic modular forms are divisible by $n^{k-1}$ for every natural number $n$. Moreover, we prove that their coefficients are non-vanishing and have either constant or alternating signs. Finally, we obtain a relation between the Fourier coefficients of meromorphic modular forms, the coeff…
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."
For the ternary quadratic form Q(x) = x^2 + y^2 - z^2 and a non-zero Pythagorean triple x_0 in Z^3 lying on the cone Q(x) = 0, we consider an orbit O = x_0 Gamma of a finitely generated subgroup Gamma < SO_Q(Z) with critical exponent exceeding 1/2. We find infinitely many Pythagorean triples in O whose hypotenuse, area, and product of side lengths have few prime factors, where "few" is explicitly quantified. We also compute the asymptotic of the number of such Pythagorean triples of norm at most T, up to bounded constants.
We prove the logarithmic Sarnak conjecture for sequences of subquadratic word growth. In particular, we show that the Liouville function has at least quadratically many sign patterns. We deduce the main theorem from a variant which bounds the correlations between multiplicative functions and sequences with subquadratically many sign patterns which occur with positive logarithmic density. This allows us to actually prove that our multiplicative functions do not locally correlate with sequences of subquadratic word growth. We also prove a conditional result which shows that if the $κ-1$-Fourier …
