This paper surveys the connection between the elliptic curve
E_D: x^3 + y^3 = D
and a certain metaplectic form on the cubic cover of GL(3) which has the property that its m,n^{th} Whittaker--Fourier coefficient is essentially the L--series of the curve E_{m^2n}. One may obtain information about the collective behavior the curves E_D by exploiting this connection; for example, one can prove:
Theorem: Fix any prime p \ne 3, and any congruence class c mod p. Then there are infinitely many D congruent to c mod p such that the curve E_D has no rational solutions.
This paper is fairly self-contained; no prior knowledge of algebraic number theory, analytic number theory or metaplectic forms is assumed. On the other hand, this paper is a survey, no proofs are included.
In this paper, we extend Ginzburg-Rallis' integral representation for the exterior cube automorphic $L$-function of ${\rm GL}_6\times {\rm GL}_1$ to that of the quasi-split unitary similitude group ${\rm GU}_6$ and establish its analytic properties to determine the poles of this $L$-function. Furthermore, we introduce the automorphic induction for automorphic representations of ${\rm GU}_n$ and then show that the weak Langlands functorial lift for the automorphic induction exists for generic cuspidal automorphic representations. By using this automorphic induction, we give a conjectural criter…
In this paper, we give an asymptotic formula for the second moment of Dirichlet twists of an automorphic $L$-function $L(s, π)$ on the critical line averaged over characters and conductors, where $π$ denotes an irreducible tempered cuspidal automorphic representation of $\textrm{GL}_{4}$ with unitary central character. We give some hybrid bound for the error term with respect to the size of conductors of Dirichlet characters and that of the automorphic representation.
We calculate mean values of $\operatorname{GL}_n$-automorphic $L$-functions twisted by primitive even Dirichlet characters of prime-power conductor, at arbitrary points within the critical strip, by derivation of special Voronoi summation formulae. Our calculation is novel in that the twisted sum can be expressed in terms of the average itself, and also that it sees the derivation of various new summation formulae in the setting of prime-power modulus. One consequence, as we explain, is to show the analytic continuation and additive summation formulae for hyper-Kloosterman Dirichlet series ass…
We prove that sums of length about $q^{3/2}$ of Hecke eigenvalues of automorphic forms on $SL_3(\Zz)$ do not correlate with $q$-periodic functions with bounded Fourier transform. This generalizes the earlier results of Munshi and Holowinsky--Nelson, corresponding to multiplicative Dirichlet characters, and applies in particular to trace functions of small conductor modulo primes.
We study the nonvanishing of twists of automorphic L-functions at the centre of the critical strip. Given a primitive character χmodulo D satisfying some technical conditions, we prove that the twisted L-functions L(f.χ,s) do not vanish at s=1/2 for a positive proportion of primitive forms of weight 2 and level q, for large prime q. We also investigate the central values of high derivatives of L(f.χ,s), and from that derive an upper bound for the average analytic rank of the studied L-functions.
We provide Fourier expansions of vector-valued eigenfunctions of the hyperbolic Laplacian that are twist-periodic in a horocycle direction. The twist may be given by any endomorphism of a finite-dimensional vector space; no assumptions on invertibility or unitarity are made. Examples of such eigenfunctions include vector-valued twisted automorphic forms of Fuchsian groups. We further provide a detailed description of the Fourier coefficients and explicitly identify each of their constituents, which intimately depend on the eigenvalues of the twisting endomorphism and the size of its Jordan blo…
Let $d\in\mathbb{N}$, and let $π$ be a fixed cuspidal automorphic representation of $\mathrm{GL}_{d}(\mathbb{A}_{\mathbb{Q}})$ with unitary central character. We determine the limiting distribution of the family of values $-\frac{L'}{L}(1+it,π\otimesχ_D)$ as $D$ varies over fundamental discriminants. Here, $t$ is a fixed real number and $χ_D$ is the real character associated with $D$. We establish an upper bound on the discrepancy in the convergence of this family to its limiting distribution. As an application of this result, we obtain an upper bound on the small values of $\left|\frac{L'}{L}…
Let pi be an automorphic representation on GL(r, A_Q) for r=1, 2, or 3. Let d be a fundamental discriminant and chi_d the corresponding quadratic Dirichlet character. We consider the question of the least d, relative to the data (level, weight or eigenvalue) of pi, such that the central value of the twisted L-series is nonzero, i.e. L(1/2, pi x chi_d) != 0. For example, let N be the level of pi. Using multiple Dirichlet series, we prove the nonvanishing of a central twisted $L$-value with d << N^{1/2} for GL(1), d << N^1 for GL(2), and d << N^2 for GL(3), the last case assuming that a certain …
The zeros and poles of standard automorphic $L$-functions attached to representations of classical groups are linked to the nonvanishing of lifts in the theory of the theta correspondence. The results of this paper show that when a cuspidal representation $σ$ of a symplectic group $G$ lifts to a cuspidal representation $π= θ_ψ(σ)$ of an orthogonal group $H$ attached to a quadratic space $V$ of dimension $m$, the Fourier coefficients of automorphic forms in $σ$ are linked to periods of automorphic forms in $π$. Consequently, when our results are combined with the Rallis inner product formula in…
We prove upper bounds for geodesic periods of automorphic forms over general rank one locally symmetric spaces. Such periods are integrals of automorphic forms restricted to special totally geodesic cycles of the ambient manifold and twisted with automorphic forms on the cycles. The upper bounds are in terms of the Laplace eigenvalues of the two automorphic forms, and they generalize previous results for real hyperbolic manifolds to the context of all rank one locally symmetric spaces.
Let $f$ and $g$ be holomorphic or Maass cusp forms for $\rm SL_2(\mathbb{Z})$ with normalized Fourier coefficients $λ_f(n)$ and $λ_g(n)$, respectively. In this paper, we prove nontrivial estimates for the sum $$ \sum_{n=1}^{\infty}λ_f(n) λ_g(n)e\left(t φ\left(\frac{n}{X}\right)\right)V\left(\frac{n}{X}\right), $$ where $e(x)=e^{2πix}$, $V(x)\in \mathcal{C}_c^{\infty}(1,2)$, $t\geq 1$ is a large parameter and $φ(x)$ is some nonlinear real valued smooth function. Applications of these estimates include a subconvex bound for the Rankin-Selberg $L$-function $L(s,f\otimes g)$ in the $t$-aspect, an …
Additive twists are important invariants associated to holomorphic cusp forms; they encode the Eichler--Shimura isomorphism and contain information about automorphic $L$-functions. In this paper we prove that central values of additive twists of the $L$-function associated to a holomorphic cusp form $f$ of even weight $k$ are asymptotically normally distributed. This generalizes (to $k\geq 4$) a recent breakthrough of Petridis and Risager concerning the arithmetic distribution of modular symbols. Furthermore we give as an application an asymptotic formula for the averages of certain 'wide' fam…
We introduce a class of multiplicative functions in which each function satisfies some statistic conditions, and then prove that above functions are not correlated with finite degree polynomial nilsequences. Besides, we give two applications of this result. One is that the twisting of coefficients of automorphic $L$-function on $GL_m (m \geq 2)$ and polynomial nilsequences has logarithmic decay; the other is that the mean value of the Möbius function, coefficients of automorphic $L$-function and polynomial nilsequences also has logarithmic decay.
Let $π$ be a Hecke--Maass cusp form for $\rm SL_3(\mathbb{Z})$ with normalized Hecke eigenvalues $λ_π(n,r)$. Let $f$ be a holomorphic or Maass cusp form for $\rm SL_2(\mathbb{Z})$ with normalized Hecke eigenvalues $λ_f(n)$. In this paper, we are concerned with obtaining nontrivial estimates for the sum \begin{equation*} \sum_{r,n\geq 1}λ_π(n,r)λ_f(n)e\left(t\,φ(r^2n/N)\right)V\left(r^2n/N\right), \end{equation*} where $e(x)=e^{2πix}$, $V(x)\in \mathcal{C}_c^{\infty}(0,\infty)$, $t\geq 1$ is a large parameter and $φ(x)$ is some real-valued smooth function. As applications, we give an improved s…
The present paper is devoted to the relations between Deligne's conjecture on critical values of motivic $L$-functions and the multiplicative relations between periods of arithmetically normalized automorphic forms on unitary groups. In the first place, we combine the Ichino--Ikeda--Neal-Harris (IINH) formula -- which is now a theorem -- with an analysis of cup products of coherent cohomological automorphic forms on Shimura varieties to establish relations between certain automorphic periods and critical values of Rankin-Selberg and Asai $L$-functions of ${\rm GL}(n)\times{\rm GL}(m)$ over CM …
We show that for an elliptic divisibility sequence on a twist of the Fermat cubic, u^3+v^3=m, with m cube-free, all the terms beyond the first have a primitive divisor.
In 1987, Zagier and Kramarz published a paper in which they presented evidence that a positive proportion of the even-signed cubic twists of the elliptic curve $x^3+y^3=1$ should have positive rank. We extend their data, showing that it is more likely that the proportion goes to zero.
We extend the strong multiplicity one theorem of Jacquet, Piatetski-Shapiro and Shalika. Let $π$ be a unitary, cuspidal, automorphic representation of $GL_n(\A_K)$. Let $S$ be a set of finite places of $K$, such that the sum $\sum_{v\in S}Nv^{-2/(n^2+1)}$ is convergent. Then $π$ is uniquely determined by the collection of the local components $\{π_v\mid v\not\in S, ~v \~\text{finite}\}$ of $π$. Combining this theorem with base change, it is possible to consider sets $S$ of positive density, having appropriate splitting behavior with respect to solvable extensions of $K$, and where $π$ is deter…
We discover new Voronoi formulae for automorphic forms on GL($n$) for $n\geq 4$. There are $[n/2]$ different Voronoi formulae on GL($n$), which are Poisson summation formulae weighted by Fourier coefficients of the automorphic form with twists by some hyper-Kloosterman sums.
Firstly we prove that the Voronoi formula of Miller-Schmid type applies to automorphic forms on GL(3) for the congruence subgroup $Γ_0(N)$, when the conductor of the additive character in the formula is a multiple of $N$. As an application, we produce a result about the functional equation of $L$-function of the automorphic form on GL(3) twisted by Dirichlet characters. Secondly we prove that a similar formula applies to automorphic forms on GL(3) for the congruence subgroup $Γ_0(N)$, when the conductor of the additive character in the formula is coprime with $N$.
This paper continues the authors previous investigations concerning orders of Tate-Shafarevich groups in quadratic twists of a given elliptic curve, and for the family of the Neumann-Setzer type elliptic curves. Here we present the results of our search for the (analytic) orders of Tate-Shafarevich groups for the cubic twists of $X_0(27)$. Our calculations extend those given by Zagier and Kramarz \cite{ZK} and by Watkins \cite{Wat}. Our main observations concern the asymptotic formula for the frequency of orders of Tate-Shafarevich groups. In the last section we propose a similar asymptotic fo…
We prove the non-vanishing of special $L$-values of cuspidal automorphic forms on GL(2) twisted by Hecke characters of prime power orders and totally split prime power conductors. Main ingredients of the proof are estimating the Galois averages of fast convergent series expressions of special $L$-values and considering Shintani cone decomposition.
We introduce the zeta function of the prehomogenous vector space of binary cubic forms, twisted by the real analytic Eisenstein series. We prove the meromorphic continuation of this zeta function and identify its poles and their residues. We also identify the poles and residues of the zeta function when restricted to irreducible binary cubic forms. This zeta function can be used to prove the equidistribution of the lattice shape of cubic rings.
The space of toroidal automorphic forms was introduced by Zagier in the 1970s: a GL_2-automorphic form is toroidal if it has vanishing constant Fourier coefficients along all embedded non-split tori. The interest in this space stems (amongst others) from the fact that an Eisenstein series of weight s is toroidal for a given torus precisely if s is a non-trivial zero of the zeta function of the quadratic field corresponding to the torus. In this paper, we study the structure of the space of toroidal automorphic forms for an arbitrary number field F. We prove that it decomposes into a space span…
Let $π$ be a $SL(3,\mathbb Z)$ automorphic form. Let $χ=χ_1χ_2$ be a Dirichlet character with $χ_i$ primitive modulo $M_i$. Suppose $M_1$, $M_2$ are primes such that $\sqrt{M_2}M^{4δ}<M_1<M_2M^{-3δ}$, where $M=M_1M_2$ and $0<δ<1/28$. In this paper we will prove the following subconvex bound $$ L(\t1/2,π\otimesχ)\ll_{π,\varepsilon} M^{3/4}-δ+\varepsilon}. $$
We obtain a sharp refinement of the strong multiplicity one theorem for the case of unitary non-dihedral cuspidal automorphic representations for GL(2). Given two unitary cuspidal automorphic representations for GL(2) that are not twist-equivalent, we also find sharp lower bounds for the number of places where the Hecke eigenvalues are not equal, for both the general and non-dihedral cases. We then construct examples to demonstrate that these results are sharp.
In this note we study the symmetric powers of strongly modular icosahedral representations $ρ$ of ${\rm Gal} (\bar{F}/F)$, $F$ a number field, and their twisted $L$--functions. We prove that for such $ρ$, there exists a cuspidal automorphic representation $Π= Π_{\infty} \otimes Π_{f}$ of $GL_{6} (\mathbb{A}_{F})$ such that $L (s, {\rm sym}^{5} (ρ)) = L (s, Π_{f})$. One sees that ${\rm sym}^{5} (ρ)$ is twist equivalent to $ρ' \otimes {\rm sym}^{2} (ρ)$ for another modular icosahedral representation $ρ'$, and our theorem is a special case of a cuspidality criterion formulated and proved in this …
In the present paper, we will show that three apparently disjoint objects: Galois representations arising from twenty-seven lines on a cubic surface (number theory and arithmetic algebraic geometry), Picard modular forms (automorphic forms), rigid Calabi-Yau threefolds and their arithmetic (Diophantine geometry) are intimately related to Hessian polyhedra and their invariants. We construct a Galois representation whose image is a proper subgroup of $W(E_6)$, the Weyl group of the exceptional Lie algebra $E_6$. We give a conjecture about the identification of two different kinds of $L$-function…
We determine the number of level 1, polarized, algebraic regular, cuspidal automorphic representations of GL_n over Q of any given infinitesimal character, for essentially all n <= 8. For this, we compute the dimensions of spaces of level 1 automorphic forms for certain semisimple Z-forms of the compact groups SO_7, SO_8, SO_9 (and G_2) and determine Arthur's endoscopic partition of these spaces in all cases. We also give applications to the 121 even lattices of rank 25 and determinant 2 found by Borcherds, to level one self-dual automorphic representations of GL_n with trivial infinitesimal c…
Waldspurger's formula gives an identity between the norm of a torus period and an L-function of the twist of an automorphic representation on GL(2). For any two Hecke characters of a fixed quadratic extension, one can consider the two torus periods coming from integrating one character against the automorphic induction of the other. Because the corresponding L-functions agree, (the norms of) these periods---which occur on different quaternion algebras---are closely related. In this paper, we give a direct proof of an explicit identity between the torus periods themselves.
This thesis contributes to the analytic theory of automorphic L-functions. We prove an approximate functional equation for the central value of the L-series attached to an irreducible cuspidal automorphic representation of GL(m) over a number field. We investigate the decay rate of the cutoff function and its derivatives in terms of the analytic conductor introduced by Iwaniec and Sarnak. We also see that the truncation can be made uniformly explicit at the cost of an error term. The results extend to products of central values. We establish, via the Hardy-Littlewood circle method, a nontrivia…
This paper generalises previous work of the author to the setting of overconvergent $p$-adic automorphic forms for a definite quaternion algebra over a totally real field. We prove results which are analogues of classical `level raising' results in the theory of mod $p$ modular forms. Roughly speaking, we show that an overconvergent eigenform whose associated local Galois representation at some auxiliary prime $ł$ is (a twist of) a direct sum of trivial and cyclotomic characters lies in a family of eigenforms whose local Galois representation at $ł$ is generically (a twist of) a ramified exten…
In this article, we study the co-period integral attached to an automorphic form on $\GL(2)$ and two exceptional theta series on the cubic Kazhdan-Patterson cover of $\GL(2)$. In the local aspect, we show the $\Hom$-space is always of one dimension and conduct the unramified calculations. In the global aspect, we give the Euler decomposition for the co-period integrals of Eisenstein series and propose an Ichino-Ikeda type conjecture relating the co-period integrals of cuspidal forms to the central critical value of symmetric cube $L$-functions. We also deduce from the local multiplicity one re…
This article describes cubic function fields $L/K$ with prescribed ramification, where $K$ is a rational function field. We give general equations for such extensions, an explicit procedure to obtain a defining equation when the purely cubic closure $K'/K$ of $L/K$ is of genus zero, and a description of the twists of $L/K$ up to isomorphism over $K$. For cubic function fields of genus at most one, we also describe the twists and isomorphism classes obtained when one allows Möbius transformations on $K$. The article concludes by studying the more general case of covers of elliptic and hyperelli…
We are interested in finding for which positive integers $D$ we have rational solutions for the equation $x^3+y^3=D.$ The aim of this paper is to compute the value of the $L$-function $L(E_D, 1)$ for the elliptic curves $E_D: x^3+y^3=D$. For the case of $p$ prime $p\equiv 1\mod 9$, two formulas have been computed by Rodriguez-Villegas and Zagier. We have computed formulas that relate $L(E_D, 1)$ to the square of a trace of a modular function at a CM point. This offers a criterion for when the integer $D$ is the sum of two rational cubes. Furthermore, when $L(E_D, 1)$ is nonzero we get a formul…
We investigate a question of Cooper adjacent to the Virtual Haken Conjecture. Assuming certain conjectures in number theory, we show that there exist hyperbolic rational homology 3-spheres with arbitrarily large injectivity radius. These examples come from a tower of abelian covers of an explicit arithmetic 3-manifold. The conjectures we must assume are the Generalized Riemann Hypothesis and a mild strengthening of results of Taylor et al on part of the Langlands Program for GL_2 of an imaginary quadratic field. The proof of this theorem involves ruling out the existence of an irreducible two …
