Narrow Results

We study the behavior of conductors of $L$-functions associated to certain Weil–Deligne representations under twisting. For each global field $K$ we prove a sharp upper bound for the conductor of the Rankin–Selberg $L$-function $L(A⊠B,s)$ where $A,B/K$ are abelian varieties.

It is shown that Watt's new mean value theorem on sums of character sums can be included in the method described in the author's recent work [6] to show that the number of Carmichael numbers up to x exceeds x^⅓ for all large x. This is done by comparing the application of Watt's original version of his mean value theorem [8] to the problem of primes in short intervals [3] with the problem of finding "small" primes in an arithmetic progression.

We obtain new upper bounds for the mean squared modulus of sums ∑_n∈ℕA_nχ(n), where the sequence (A_n) is fixed and the variable χ belongs to the set of non-principal Dirichlet characters for some modulus q. It is assumed that, for some M, some complex sequence (c_m) satisfying c_m = 0 for m ∉ (M/2,M], and some α(x) and β(y) (smooth functions with compact support), one has A_n = ∑_{uvm = n} α(u)β(v)c_m (n ∈ ℕ). There is a natural analogy between the bounds obtained and bounds on mean values of Dirichlet polynomials previously obtained by Deshouillers and Iwaniec. Our proofs make use of results from the spectral theory of automorphic functions, including the bound of Kim and Sarnak for the eigenvalues of Hecke operators acting on certain spaces of Maass cusp forms. The results depend on the size of P, the largest prime factor of q, and improve as log_q(P) is diminished. In separate work, Harman has given an application of our results to the theory of Carmichael numbers.

The original Sato–Tate Conjecture concerns the angle distribution of the eigenvalues arisen from non-CM elliptic curves. In this paper, we formulate an analogue of the Sato–Tate Conjecture on automorphic forms of (GL_n) and, under a holomorphic assumption, prove that the distribution is either uniform or the generalized Sato–Tate measure.

In a previous paper, the second author proved that the equation

We prove some curious identities for generating functions for values of L-functions. It is shown how to obtain generating functions for values of L-functions using a slightly different approach, resulting in some new q-series identities.

We establish an asymptotic for the first moment of Hecke L-series associated to canonical characters on imaginary quadratic fields. This provides another proof and improves recent results by Masri and Kim–Masri–Yang.

We propose a candidate, which we call the fractional Galois ideal after Snaith's fractional ideal, for replacing the classical Stickelberger ideal associated to an abelian extension of number fields. The Stickelberger ideal can be seen as gathering information about those L-functions of the extension which are non-zero at the special point s = 0, and was conjectured by Brumer to give annihilators of class-groups viewed as Galois modules. An earlier version of the fractional Galois ideal extended the Stickelberger ideal to include L-functions with a simple zero at s = 0, and was shown by the present author to provide class-group annihilators not existing in the Stickelberger ideal. The version presented in this paper deals with L-functions of arbitrary order of vanishing at s = 0, and we give evidence using results of Popescu and Rubin that it is closely related to the Fitting ideal of the class-group, a canonical ideal of annihilators.

Finally, we prove an equality involving Stark elements and class-groups originally due to Büyükboduk, but under a slightly different assumption, the advantage being that we need none of the Kolyvagin system machinery used in the original proof.

In this paper, we precise the asymptotic behavior of Newton polygons of L-functions associated to character sums, coming from certain n variable Laurent polynomials. In order to do this, we use the free sum on convex polytopes. This operation allows the determination of the limit of generic Newton polygons for the sum Δ = Δ₁⊕Δ₂ when we know the limit of generic Newton polygons for each factor. To our knowledge, these are the first results concerning the asymptotic behavior of Newton polygons for multivariable polynomials when the generic Newton polygon differs from the combinatorial (Hodge) polygon associated to the polyhedron.

Let F ∈ S_k(Sp(2g, ℤ)) be a cuspidal Siegel eigenform of genus g with normalized Hecke eigenvalues μ_F(n). Suppose that the associated automorphic representation π_F is locally tempered everywhere. For each c > 0, we consider the set of primes p for which |μ_F(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 μ_F(p) ≥ c.

In this work we prove congruences between special values of L-functions of elliptic curves with CM that seem to play a central role in the analytic side of the non-commutative Iwasawa theory. These congruences are the analog for elliptic curves with CM of those proved by Kato, Ritter and Weiss for the Tate motive. The proof is based on the fact that the critical values of elliptic curves with CM, or what amounts to the same, the critical values of Grössencharacters, can be expressed as values of Hilbert–Eisenstein series at CM points. We believe that our strategy can be generalized to provide congruences for a large class of L-values.

We prove, for all quadratic and a wide range of multi-quadratic extensions of global fields, a result concerning the annihilation as Galois modules of ideal class groups by explicit elements constructed from the values of higher-order derivatives of Dirichlet L-functions. This result simultaneously refines Rubin's integral version of Stark's Conjecture and provides evidence for the relevant case of the Equivariant Tamagawa Number Conjecture of Burns and Flach.

The author gives an integral representation for the twisted tensor L-function of a cuspidal, globally generic automorphic representation of GSp₄ over a quadratic extension E of a number field F with trivial central character. He proves the Euler product factorization of the global integral; computes the unramified L-factor via explicit branching from GL₄ to Sp₄ and shows it is equal to the normalized unramified local integral; and proves the absolute convergence and nonvanishing of all local integrals.

This paper is to show a non-vanishing property of the derivative of certain L-functions. For certain primitive holomorphic Hilbert modular forms, if the central critical value of the standard L-function does not vanish, then neither does its derivative. This is a generalization of a result by Gun, Murty and Rath in the case of elliptic modular forms. Some applications in transcendental number theory deduced from this result are discussed as well.

Let Γ be a Fuchsian group of the first kind which has a (regular) cusp at ∞ such as Γ₀(N). In this paper we use Kohnen's kernel function for Γ to analytically continue L(s, f) to the region Re(s) > m/2, where f is a cuspidal modular form of weight m ≥ 5 attached to Γ. We study non-vanishing of L(s, f) using our integral non-vanishing criterion.

We describe the solutions of the linear equation aX + bY = cZ in the class of Dirichlet series with functional equation. Proofs are based on the properties of certain nonlinear twists of the L-functions.

Let $f$ be a normalized Maass cusp form for ${\mathrm{\text{SL}}}_2(Z)$. For $\frac{1}{2}<\sigma <1$, we define $m(\sigma )$$(\ge 2)$ as the supremum of all numbers $m$ such that

Let $\chi$ be a primitive Dirichlet character of conductor $q$ and let us denote by $L(z,\chi )$ the associated $L$-series. In this paper, we provide an explicit upper bound for $|L(1,\chi )|$ when $\chi$ is a primitive even Dirichlet character with $\chi (2)=1$.

In this paper, we introduce arithmetic Heilbronn characters that generalize the notion of the classical Heilbronn characters, and discuss several properties of these characters. This formalism has several arithmetic applications. For instance, we obtain the holomorphy of suitable quotients of L-functions attached to elliptic curves, which is predicted by the Birch–Swinnerton–Dyer conjecture, and the non-existence of simple zeros or poles in such quotients.

Let $L$ be a degree-$2$$L$-function associated to a Maass cusp form. We explore an algorithm that evaluates $t$ values of $L$ on the critical line in time $O(t^{1+𝜖})$. We use this algorithm to rigorously compute an abundance of consecutive zeros and investigate their distribution.