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In this paper, we establish a one-to-one correspondence between conjugacy classes of any Hecke group and irreducible systems of poles of rational period functions for automorphic integrals on the same group. We use this correspondence to construct irreducible systems of poles and to count poles. We characterize Hecke-conjugation and Hecke-symmetry for poles of rational period functions in terms of the transpose of matrices in conjugacy classes. We construct new rational period functions and families of rational period functions. [ABSTRACT FROM AUTHOR]

Let F (s) = ∑ n = 1 ∞ a n n s be a Dirichlet series in the axiomatically defined class # . The class # is known to contain the extended Selberg class # , as well as all the L -functions of automorphic forms on GL n / K , where K is a number field. Let d be the degree of F (s). We show that ∑ n < X | a n | = Ω (X 1 2 + 1 2 d ) , and hence, that the abscissa of absolute convergence of σ a of F (s) must satisfy σ a ≥ 1 / 2 + 1 / 2 d. [ABSTRACT FROM AUTHOR]

We give explicit expressions for the Fourier coefficients of Eisenstein series twisted by Dirichlet characters and modular symbols on Γ 0 (N) in the case where N is prime and equal to the conductor of the Dirichlet character. We obtain these expressions by computing the spectral decomposition of automorphic functions closely related to these Eisenstein series. As an application, we then evaluate certain sums of modular symbols in a way which parallels past work of Goldfeld, O'Sullivan, Petridis, and Risager. In one case we find less cancelation in this sum than would be predicted by the common phenomenon of "square root cancelation", while in another case we find more cancelation. [ABSTRACT FROM AUTHOR]

In this paper, we establish a very flexible and explicit Voronoï summation formula. This is then used to prove an almost Weyl strength subconvexity result for automorphic L -functions of degree two in the depth aspect. That is, looking at twists by characters of prime power conductor. This is the natural p -adic analogue to the well-studied t -aspect. [ABSTRACT FROM AUTHOR]

We consider the value distribution of the difference between logarithms of two symmetric power L -functions at s = σ > 1 / 2. We prove that certain averages of those values can be written as integrals involving a density function which is constructed explicitly. [ABSTRACT FROM AUTHOR]

In this paper, using methods from geometric analysis and theory of heat kernels, we derive qualitative estimates of automorphic cusp forms defined over quaternion algebras. Using which, we prove an average version of the holomorphic QUE conjecture. We then derive quantitative estimates of classical Hilbert modular cusp forms. This is a generalization of the results from [A. Aryasomayajula, Heat kernel approach for sup-norm bounds for cusp forms of integral and half-integral weight, Arch. Math. 106(2) (2016) 165-173; J. S. Friedman, J. Jorgenson and J. Kramer, Uniform sup-norm bounds on average for cusp forms of higher weights, in Arbeitstagung Bonn 2013, Progress in Mathematics, Vol. 319 (Birkhäuser/Springer, Cham, 2016), pp. 127-154] to higher dimensions. [ABSTRACT FROM AUTHOR]

We establish the Fourier inversion for the smooth vectors in over a number field , using minimal knowledge from automorphic representation theory. We point out a possible way to establish Fourier inversion for larger classes of functions. We also point out the incompleteness of some commonly believed 'proof' of Fourier inversion in the literature. Moreover, the explicit computation of the intertwining operator has independent interests. [ABSTRACT FROM AUTHOR]

In this paper, we define a -factor for generic representations of and prove a local converse theorem for using the -factor we defined. We also give a new proof of the local converse theorem for using a -factor of type which was originally defined by Jacquet in [ Automorphic Forms on , Part 2, Springer Lecture Notes in Mathematics, Vol. 278 (Springer, 1972)]. [ABSTRACT FROM AUTHOR]

Let be a number field of a finite degree and let be the Rankin-Selberg -function associated to unitary cuspidal automorphic representations and of and , respectively. The main result of the paper is an asymptotic formula for evaluation of coefficients appearing in the Laurent (Taylor) series expansion of the logarithmic derivative of the function at . As a corollary, we derive orthogonality and weighted orthogonality relations. [ABSTRACT FROM AUTHOR]

Let be a self-contragredient irreducible unitary cuspidal representation of with , and be the automorphic -function attached to . Assume that is self-contragredient. Under the Generalized Ramanujan Conjecture and Generalized Riemann Hypothesis for , the estimate holds for all real number as . [ABSTRACT FROM AUTHOR]

Let be a normalized Maass cusp form for . For , we define as the supremum of all numbers such that where is the automorphic -function attached to . In this paper, we shall establish the lower bounds of for and obtain asymptotic formulas for the second, fourth and sixth powers of . [ABSTRACT FROM AUTHOR]

Let be a self-contragredient automorphic cuspidal representation of for . Using a refined version of the Selberg orthogonality, we recompute the -level correlation of high non-trivial zeros of the product . In the process, we are able to extract certain low-order terms which suggest the asymptotics of these statistics are not necessarily universal, but depend upon the conductors of the representations and hence the ramification properties of the local components coming from each . The computation of these lower-order terms is unconditional as long as all . [ABSTRACT FROM AUTHOR]

Let f be a holomorphic cusp form of weight k for SL(2, ℤ) with Fourier coefficients λf(n). We study the sum ∑n>0λf(n)ϕ(n/X)e(αn), where . It is proved that the sum is rapidly decaying for α close to a rational number a/q where q2 < X1-ε. The main techniques used in this paper include Dirichlet's rational approximation of real numbers, a Voronoi summation formula for SL(2, ℤ) and the asymptotic expansion for Bessel functions. [ABSTRACT FROM AUTHOR]

We generalize the correspondence between Dirichlet series with finitely many poles that satisfy a functional equation and automorphic integrals with log-polynomial sum period functions. In particular, we extend the correspondence to hold for Dirichlet series with finitely many essential singularities. We also study Dirichlet series with infinitely many poles in a vertical strip. For Hecke groups with λ ≥ 2 and some weights, we prove a similar correspondence for these Dirichlet series. For this case, we provide a way to estimate automorphic integrals with infinite log-polynomial periods by automorphic integrals with finite log-polynomial periods. [ABSTRACT FROM AUTHOR]

Let k ≥ 2 be a fixed integer and let be k infinite sequences of nonnegative integers. Let rk(n) be the number of solutions of . In this paper, we generalize Horváth's result by showing that if for any 1 ≤ u, v ≤ k and any with , there exists an such that , then rk(n) = cn + o(n1/4( n)-βk) cannot hold for any constant c > 0, where νk = 1, βk = 1/2 if 2 | k and if 2 ∤ k. [ABSTRACT FROM AUTHOR]

Dans cet article, on montre que les orbites sous Galois des invariants modulaires associés à des courbes elliptiques complexes sans multiplication complexe variant dans une même classe d'isogénie s'équidistribuent dans la courbe modulaire vers la probabilité hyperbolique. La démonstration repose sur des arguments de théorie ergodique, notamment le théorème de Ratner (cf. [A. Eskin et H. Oh, Ergodic theoretic proof of equidistribution of Hecke points, Ergodic Theory Dynam. Systems 26(1) (2006) 163-167]), ainsi que sur le théorème de l'image ouverte de Serre [J.-P. Serre, Abelian l-Adic Representations and Elliptic Curves (W. A. Benjamin, New York, 1968); Propriétés Galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15(4) (1972) 259-331] dans le cas où les invariants modulaires considérés sont algébriques sur Q, et des résultats de G. Shimura dans le cas transcendant [ Introduction to the Arithmetic Theory of Automorphic Functions, Publications of the Mathematical Society of Japan (Princeton University Press, Princeton, NJ, 1994)]. In this article, it is shown that Galois orbits of invariants associated with non-CM and pairwise isogeneous complex elliptic curves equidistribute in the classical modular curve towards the hyperbolic probability measure. The proof is based on arguments from ergodic theory, especially Ratner's theorem on unipotent flows (cf. [A. Eskin and H. Oh, Ergodic theoretic proof of equidistribution of Hecke points, Ergodic Theory Dynam. Systems 26(1) (2006) 163-167]), as well as on Serre's open image theorem [J.-P. Serre, Abelian l-Adic Representations and Elliptic Curves (W. A. Benjamin, New York, 1968); Propriétés Galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15(4) (1972) 259-331] in case of algebraic invariants, and on G. Shimura's work in the transcendant case [ Introduction to the Arithmetic Theory of Automorphic Functions, Publications of the Mathematical Society of Japan (Princeton University Press, Princeton, NJ, 1994)]. [ABSTRACT FROM AUTHOR]

For a pair (¶, σ) of cuspidal automorphic representations of GLn and GLn-i, both of non-vanishing cohomology, we show algebraicity properties of critical values of the associated Rankin-Selberg //-function twisted by finite-order characters. A certain non-vanishing assumption about the associated archimedean Rankin-Selberg pairings on the cohomology is established for n = 3. [ABSTRACT FROM AUTHOR]

We investigate proportion of nonvanishing central values L(f, 1/2) of L-functions associated to the basis of holomorphic modular forms of fixed weight k with respect to Γ1(q), in the limit when q → ∞ along the primes. Motivation is a contrast between Γ0(q) and Γ1(q) families in the sense of underlying harmonic analysis. [ABSTRACT FROM AUTHOR]

We show starting with relations between Fourier coefficients of weakly parabolic generalized modular forms of negative weight that we can construct automorphic integrals for large integer weights. We finally prove an Eichler isomorphism theorem for weakly parabolic generalized modular forms using the classical approach as in [3]. [ABSTRACT FROM AUTHOR]

Let F ∈ Sk(Sp(2g, ℤ)) be a cuspidal Siegel eigenform of genus g with normalized Hecke eigenvalues μF(n). Suppose that the associated automorphic representation πF is locally tempered everywhere. For each c > 0, we consider the set of primes p for which |μF(p)| ≥ c and we provide an explicit upper bound on the density of this set. In the case g = 2, we also provide an explicit upper bound on the density of the set of primes p for which μF(p) ≥ c. [ABSTRACT FROM AUTHOR]

We define a power series expansion of an holomorphic modular form f in the p-adic neighborhood of a CM point x of type K for a split good prime p. The modularity group can be either a classical conguence group or a group of norm 1 elements in an order of an indefinite quaternion algebra. The expansion coefficients are shown to be closely related to the classical Maass operators and give p-adic information on the ring of definition of f. By letting the CM point x vary in its Galois orbit, the rth coefficients define a p-adic K×-modular form in the sense of Hida. By coupling this form with the p-adic avatars of algebraic Hecke characters belonging to a suitable family and using a Rankin-Selberg type formula due to Harris and Kudla along with some explicit computations of Watson and of Prasanna, we obtain in the even weight case a p-adic measure whose moments are essentially the square roots of a family of twisted special values of the automorphic L-function associated with the base change of f to K. [ABSTRACT FROM AUTHOR]

We re-examine some critical values of symmetric square L-functions for cusp forms of level one. We construct some more of the elements of large prime order in Shafarevich–Tate groups, demanded by the Bloch–Kato conjecture. For this, we use the Galois interpretation of Kurokawa-style congruences between vector-valued Siegel modular forms of genus two (cusp forms and Klingen–Eisenstein series), making further use of a construction due to Urban. We must assume that certain 4-dimensional Galois representations are symplectic. Our calculations with Fourier expansions use the Eholzer–Ibukiyama generalization of the Rankin–Cohen brackets. We also construct some elements of global torsion which should, according to the Bloch–Kato conjecture, contribute a factor to the denominator of the rightmost critical value of the standard L-function of the Siegel cusp form. Then we prove, under certain conditions, that the factor does occur. [ABSTRACT FROM AUTHOR]

A simple construction of Eisenstein series for the congruence subgroup Γ0(p) is given. The construction makes use of the Jacobi triple product identity and Gauss sums, but does not use the modular transformation for the Dedekind eta-function. All positive integral weights are handled in the same way, and the conditionally convergent cases of weights 1 and 2 present no extra difficulty. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a vector-valued nonanalytic Eisenstein series appearing naturally in the Rankin–Selberg convolution of a vector-valued modular cusp form associated to a monomial representation ρ of SL(2,ℤ). This vector-valued Eisenstein series transforms under a representation χρ associated to ρ. We use a method of Selberg to obtain an analytic continuation of this vector-valued nonanalytic Eisenstein series to the whole complex plane. [ABSTRACT FROM AUTHOR]

We conjecture that the only irreducible cuspidal automorphic representations for GL3/ℚ of cohomological type and level 1 are (up to twisting) the symmetric square lifts of classical cuspforms on GL2/ℚ of level 1. We present computational evidence for this conjecture. [ABSTRACT FROM AUTHOR]

Using Ramanujan's differential equations for Eisenstein series and an idea from Ramanujan's unpublished manuscript on the partition function p(n) and the tau function τ(n), we provide simple proofs of Ramanujan's congruences for p(n) modulo 5, 7, and 11. [ABSTRACT FROM AUTHOR]

If one takes the Mellin transform of an automorphic form for GL(n) and then integrates it along the diagonal on GL(n - 1) then one obtains an automorphic form on GL(n - 1). This gives a rank lowering operator. In this paper a more general rank lowering operator is obtained by combining the Mellin transform with a sum of powers of certain fixed differential operators. The analytic continuation of the rank lowering operator is obtained by showing that the spectral expansion consists of sums of Rankin–Selberg L-functions of type GL(n) × GL(n - 1). [ABSTRACT FROM AUTHOR]

Kohnen showed that the zeros of the Eisenstein series Ek in the standard fundamental domain other than i and ρ are transcendental. In this paper, we obtain similar results for a more general class of modular forms, using the earlier works of Kanou, Kohnen and the recent work of Getz. [ABSTRACT FROM AUTHOR]

As automorphic L-functions or Artin L-functions, several classes of L-functions have Euler products and functional equations. In this paper we study the zeros of L-functions which have Euler products and functional equations. We show that there exists a relation between the zeros of the Riemann zeta-function and the zeros of such L-functions. As a special case of our results, we find relations between the zeros of the Riemann zeta-function and the zeros of automorphic L-functions attached to elliptic modular forms or the zeros of Rankin–Selberg L-functions attached to two elliptic modular forms. [ABSTRACT FROM AUTHOR]