Hecke L-functions Research Articles

Arithmetic of Hecke L-functions of quadratic extensions of totally real fields

Deep work by Shintani in the 1970's describes Hecke L-functions associated to narrow ray class group characters of totally real fields F in terms of what are now known as Shintani zeta functions. However, for [F:Q]=n≥3, Shintani's method was ineffective due to its crucial dependence on abstract fundamental domains for the action of totally positive units of F on R+n, so-called Shintani sets. These difficulties were recently resolved in independent work of Charollois, Dasgupta, and Greenberg and Diaz y Diaz and Friedman. For those narrow ray class group characters whose conductor is an inert rational prime in a totally real field F with narrow class number 1, we obtain a natural combinatorial description of these sets, allowing us to obtain a simple description of the associated Hecke L-functions. As a consequence, we generalize earlier work of Girstmair, Hirzebruch, and Zagier, that offer combinatorial class number formulas for imaginary quadratic fields, to real and imaginary quadratic extensions of totally real number fields F with narrow class number 1. For CM quadratic extensions of F, our work may be viewed as an effective affirmative answer to Hecke's Conjecture that the relative class number has an elementary arithmetic expression in terms of the relative discriminant.

The twelfth moment of Hecke L-functions in the weight aspect

The twelfth moment of Hecke L-functions in the weight aspect

Ratios conjecture for quadratic Hecke L-functions in the Gaussian field

We develop the L-functions ratios conjecture with one shift in the numerator and denominator in certain ranges for the family of quadratic Hecke L-functions in the Gaussian field using multiple Dirichlet series under the generalized Riemann hypothesis. We also obtain an asymptotical formula for the first moment of central values of the same family of L-functions, obtaining an error term of size O(X1/2+ε)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(X^{1/2+\\varepsilon })$$\\end{document}.

Subconvexity of a double Dirichlet series over the Gaussian field

Abstract We establish a subconvexity bound for a double Dirichlet series involving with the quadratic Hecke L-functions over the Gaussian field.

Hecke L-functions of certain subextensions in an extraspecial extension

Hecke L-functions of certain subextensions in an extraspecial extension

Zero-free regions for spectral averages of Hecke L-functions

We obtain an explicit zero-free region for a weighted sum of [Formula: see text]-functions over the orthogonal basis of Hecke eigen cusp forms of a large integral weight for the full modular group. We also estimate the number of such forms whose [Formula: see text] value does not vanish at a given point inside this zero-free region.

First moment of central values of quadratic Hecke L-functions in the Gaussian field

We evaluate the smoothed first moment of central values of a family of quadratic Hecke [Formula: see text]-functions in the Gaussian field using the method of double Dirichlet series. The asymptotic formula we obtain has an error term of size [Formula: see text] under the generalized Riemann hypothesis. The same approach also allows us to obtain asymptotic formulas for all [Formula: see text], [Formula: see text] for a smoothed double character sum involving [Formula: see text], where [Formula: see text] denotes the quadratic symbol in the Gaussian field.

Notes on the arithmetic of Hecke L-functions

This is an expository article that concerns the various related notions of algebraic idèle-class characters, the Größencharaktere of Hecke, and cohomological automorphic representations of \(\text {GL}(1)\), all under the general title of algebraic Hecke characters. The first part of the article systematically lays the foundations of algebraic Hecke characters. The only pre-requisites are: basic algebraic number theory, familiarity with the adelic language, and basic sheaf theory. Observations that play a crucial role in the arithmetic of automorphic L-functions are also discussed. The second part of the article, on the ratios of successive critical values of the Hecke L-function attached to an algebraic Hecke character, concerns certain variations on a theorem of Harder [6], especially drawing attention to a delicate signature that apparently has not been noticed before.

Simultaneous non-vanishing of quadratic Dirichlet L-functions and twists of Hecke L-functions

Simultaneous non-vanishing of quadratic Dirichlet L-functions and twists of Hecke L-functions

The weak Gram law for Hecke L -functions

We generalize a theorem by Titchmarsh about the mean value of Hardy’s Z-function at the Gram points to the Hecke L-functions, which in turn implies the weak Gram law for them. Instead of proceeding analogously to Titchmarsh with an approximate functional equation we employ a different method using contour integration.

P-adic polylogarithms and p-adic Hecke L-functions for totally real fields

Abstract The purpose of this article is to newly define the p-adic polylogarithm as an equivariant class in the cohomology of a certain infinite disjoint union of algebraic tori associated to a totally real field. We will then express the special values of p-adic L-functions interpolating nonpositive values of Hecke L-functions of the totally real field in terms of special values of these p-adic polylogarithms.

LOWER-ORDER TERMS OF THE ONE-LEVEL DENSITY OF A FAMILY OF QUADRATIC HECKE -FUNCTIONS

AbstractIn this paper, we study lower-order terms of the one-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 are consistent with each other when the Fourier transforms of the test functions are supported in $(-2, 2)$ .

On the Distribution of αp Modulo One in Quadratic Number Fields

Abstract We investigate the distribution of αp modulo one in quadratic number fields 𝕂 with class number one, where p is restricted to prime elements in the ring of integers of 𝕂. Here we improve the relevant exponent 1/4 obtained by the first- and third-named authors for imaginary quadratic number fields [On the distribution of αp modulo one in imaginary quadratic number fields with class number one, J. Théor. Nombres Bordx. 32 (2020), no. 3, 719–760]) and by the first- and second-named authors for real quadratic number fields [Diophantine approximation with prime restriction in real quadratic number fields, Math. Z. (2021)] to 7/22. This generalizes a result of Harman [Diophantine approximation with Gaussian primes, Q. J. Math. 70 (2019), no. 4, 1505–1519] who obtained the same exponent 7/22 for ℚ (i) by extending his method which gave this exponent for ℚ [On the distribution of αp modulo one. II, Proc. London Math. Soc. 72, (1996), no. 3, 241–260]. Our proof is based on an extension of Harman’s sieve method to arbitrary number fields. Moreover, we need an asymptotic evaluation of certain smooth sums over prime ideals appearing in the above-mentioned work by the first- and second-named authors, for which we use analytic properties of Hecke L-functions with Größencharacters.

P-adic Measures for Reciprocals of L-functions of Totally Real Number Fields

We generalize the work of Gelbart, Miller, Pantchichkine, and Shahidi on constructing p-adic measures to the case of totally real fields K. This measure is the Mellin transform of the reciprocal of the p-adic L-function which interpolates the special values at negative integers of the Hecke L-function of K. To define this measure as a distribution, we study the non-constant terms in the Fourier expansion of a particular Eisenstein series of the Hilbert modular group of K. Proving the distribution is a measure requires studying the structure of the Iwasawa algebra.

Moments and One level density of certain unitary families of HeckeL-functions

In this paper, we study the moments of central values of certain unitary families of Hecke L-functions of the Gaussian field, and establish quantitative non-vanishing result for their central values. We also establish a one level density result for the low-lying zeros of these families of Hecke L-functions.

One level density of low-lying zeros of quadratic Hecke L-functions to prime moduli

In this paper, we study the one level density of low-lying zeros of a family of quadratic Hecke L-functions to prime moduli over the Gaussian field under the generalized Riemann hypothesis (GRH) and the ratios conjecture. As a corollary, we deduce that at least 75% of the members of this family do not vanish at the central point under GRH.

Lower order terms for the one-level density of a symplectic family of Hecke L-functions

In this paper we apply the L-function Ratios Conjecture to compute the one-level density for a symplectic family of L-functions attached to Hecke characters of infinite order. When the support of the Fourier transform of the corresponding test function f reaches 1, we observe a transition in the main term, as well as in the lower order term. The transition in the lower order term is in line with behavior recently observed by D. Fiorilli, J. Parks, and A. Södergren in their study of a symplectic family of quadratic Dirichlet L-functions [11,12]. We then directly calculate main and lower order terms for test functions f such that supp(fˆ)⊂[−α,α] for some α<1, and observe that this unconditional result is in agreement with the prediction provided by the Ratios Conjecture. As the analytic conductor of these L-functions grow twice as large (on a logarithmic scale) as the cardinality of the family in question, this is the optimal support that can be expected with current methods. Finally, as a corollary, we deduce that, under GRH, at least 75% of these L-functions do not vanish at the central point.

ONE-LEVEL DENSITY OF LOW-LYING ZEROS OF QUADRATIC HECKE L-FUNCTIONS OF IMAGINARY QUADRATIC NUMBER FIELDS

AbstractIn this paper, we prove a one level density result for the low-lying zeros of quadratic Hecke L-functions of imaginary quadratic number fields of class number 1. As a corollary, we deduce, essentially, that at least $(19-\cot (1/4))/16 = 94.27\ldots \%$ of the L-functions under consideration do not vanish at 1/2.

Moments of central values of cubic Hecke L-functions of $${\mathbb {Q}}(i)$$

In this paper, we study moments of central values of cubic Hecke L-functions in $${\mathbb {Q}}(i),$$ and establish quantitative non-vanishing result for those values.

Simultaneous nonvanishing of products of L-functions associated to elliptic cusp forms

A generalized Riemann hypothesis states that all zeros of the completed Hecke L-function L⁎(f,s) of a normalized Hecke eigenform f on the full modular group should lie on the vertical line Re(s)=k2. It was shown in [6] that there exists a Hecke eigenform f of weight k such that L⁎(f,s)≠0 for sufficiently large k and any point on the line segments Im(s)=t0,k−12<Re(s)<k2−ϵ,k2+ϵ<Re(s)<k+12, for any given real number t0 and a positive real number ϵ. This paper concerns the non-vanishing of the product L⁎(f,s)L⁎(f,w)(s,w∈C) on average.