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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., Comment: 2 Figures. Differences to version 1. There are several improvments in the presentation, including a better fitting title and abract. We also have corrected some minor mathematical errors

We prove analogues of the Bombieri–Vinogradov Theorem and the Barban–Davenport–Halberstam Theorem on primes in arithmetic progressions for Fourier coefficients of cusp forms.

Fix an integer \(\kappa \geqslant 2\). Let P be prime and let \(k> \kappa \) be an even integer. For f a holomorphic cusp form of weight k and full level and g a primitive holomorphic cusp form of weight \(2 \kappa \) and level P, we prove hybrid subconvexity bounds for \(L \left( \frac{1}{2}, \text {Sym}^2 f \otimes g\right) \) in the k and P aspects when \(P^{\frac{13}{64} + \delta } < k < P^{\frac{3}{8} - \delta }\) for any \(0 < \delta < \frac{11}{128}\). These bounds are achieved through a first moment method (with amplification when \(P^{\frac{13}{64}} < k \leqslant P^{\frac{4}{13}}\)).

The Galois representations associated to weight 1 eigenforms over $$\bar{\mathbb {F}}_{p}$$ are remarkable in that they are unramified at $$p$$ , but the effective computation of these modular forms presents challenges. One complication in this setting is that a weight 1 cusp form over $$\bar{\mathbb {F}}_{p}$$ need not arise from reducing a weight 1 cusp form over $$\bar{\mathbb {Q}}$$ . In this article we propose a unified Hecke stability method for computing spaces of weight 1 modular forms of a given level in all characteristics simultaneously. Our main theorems outline conditions under which a finite-dimensional Hecke module of ratios of modular forms must consist of genuine modular forms. We conclude with some applications of the Hecke stability method that are motivated by the refined inverse Galois problem.

Let $F$ be the function field of an elliptic curve $X$ over $\F_q$. In this paper, we calculate explicit formulas for unramified Hecke operators acting on automorphic forms over $F$. We determine these formulas in the language of the graph of an Hecke operator, for which we use its interpretation in terms of $\P^1$-bundles on $X$. This allows a purely geometric approach, which involves, amongst others, a classification of the $\P^1$-bundles on $X$. We apply the computed formulas to calculate the dimension of the space of unramified cusp forms and the support of a cusp form. We show that a cuspidal Hecke eigenform does not vanish in the trivial $\P^1$-bundle. Further, we determine the space of unramified $F'$-toroidal automorphic forms where $F'$ is the quadratic constant field extension of $F$. It does not contain non-trivial cusp forms. An investigation of zeros of certain Hecke $L$-series leads to the conclusion that the space of unramified toroidal automorphic forms is spanned by the Eisenstein series $E(\blanc,s)$ where $s+1/2$ is a zero of the zeta function of $X$---with one possible exception in the case that $q$ is even and the class number $h$ equals $q+1$., Comment: 26 pages

In general, it is difficult to determine the dimension of the space of Siegel modular forms of low weights. In particular, the dimensions of the spaces of cusp forms are known in only a few cases. In this paper, we calculate the dimension of the space of Siegel–Eisenstein series of weight 1, which is a certain subspace of a complement of the space of cusp forms.

Tsuneo Arakawa formulated a theta lifting from elliptic cusp forms to automorphic forms on Sp(1,q) in his unpublished note, which was inspired by “Kudla lifting”, i.e. a theta lifting from elliptic modular forms to holomorphic automorphic forms on SU(1,q). We prove that the images of Arakawa’s theta lifting belong to the space of bounded automorphic forms generating quaternionic discrete series, which are non-holomorphic forms. In the appendix we provide the construction of Eisenstein series and Poincare series generating such discrete series.

This paper deals with Deligne's conjecture on the critical values of L-functions. Let Z G ⊗ h (s) denote the tensor product L-function attached to a Siegel modular form G of weight k and an elliptic cusp form h of weight l. We assume that the first Fourier-Jacobi coefficient of G is not identically zero. Then Deligne's conjecture is fully proven for Z G ⊗ h (s), when l≤2k−2 and partly for the remaining case.

Let f be a non-zero cusp form of even integral weight k ≥ 2 and level N with real Fourier coefficients a(n) (n ∈ N). Using a classical theorem of Landau together with the facts that the Hecke L-function of f is entire and the Rankin-Selberg zeta function of f has a pole at its abscissa of convergence with residue essentially equal to the square of the Petersson norm of f , one easily shows that there are infinitely many n ∈ N such that a(n) > 0 as well as infinitely many n with a(n) 1, Siegel’s argument does not apply and the result above is not known. Now suppose that N is squarefree and f is a newform which is a normalized Hecke eigenform (so a(1) = 1 and a(n) is the n-th Hecke eigenvalue of f ). We shall prove

In this paper we construct a lifting map from a vector space of generalized Jacobi cusp forms to a certain subspace of elliptic cusp forms and vice versa such that both mappings are adjoint with respect to the Petersson scalar products.

Let M be a geometrically finite pinched negatively curved Riemannian manifold with at least one cusp. Inspired by the theory of Diophantine approximation of a real (or complex) number by rational ones, we develop a theory of approximation of geodesic lines starting from a given cusp by ones returning to it. We define a new invariant for M, theHurwitz constant of M. It measures how well all geodesic lines starting from the cusp are approximated by ones returning to it. In the case of constant curvature, we express the Hurwitz constant in terms of lengths of closed geodesics and their depths outside the cusp neighborhood. Using the cut locus of the cusp, we define an explicit approximation sequence for a geodesic line starting from the cusp and explore its properties. We prove that the modular once-punctured hyperbolic torus has the minimum Hurwitz constant in its moduli space.

In this paper we investigate the properties of the automorphic (standard) L-function on \(GSp_2 \times Gl_2\) for the case, where the automorphic form comes from a holomorphic Siegel modular form and a holomorphic elliptic modular form. To cover arbitrary weights, we use two different types of differential operators.

1.2 Indeed, Rankin and Selberg applied their method to the more general situation of holomorphic cusp forms of type (F', r, v), where F ' is of finite index in F, r > 0 is the weight and v is a compatible multiplier system. Let f ( z ) be such a cusp form, [F 'F ' ] = m and M1 . . . . . Mm be representatives of distinct right cosets. Then the vector-valued function F ( z ) = ((clz + d l ) ~ f ( M l z ) . . . . . (cruz + d,,)-rf(Mmz)) ~ (T means transpose, (c~,dl) is the second row of M~) is of type (F, r, E) where E is a unitary m-dimensional multiplier system. So, the concept of vector-valued automorphic forms for F is intimately related to the investigation of automorphic forms for subgroups of finite index in V (cf. [13]).