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Search: MSC category 11M41 ( Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72} )

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1. CJM Online first

Salazar, Daniel Barrera; Williams, Chris
$P$-adic $L$-functions for GL$_2$
Since Rob Pollack and Glenn Stevens used overconvergent modular symbols to construct $p$-adic $L$-functions for non-critical slope rational modular forms, the theory has been extended to construct $p$-adic $L$-functions for non-critical slope automorphic forms over totally real and imaginary quadratic fields by the first and second authors respectively. In this paper, we give an analogous construction over a general number field. In particular, we start by proving a control theorem stating that the specialisation map from overconvergent to classical modular symbols is an isomorphism on the small slope subspace. We then show that if one takes the modular symbol attached to a small slope cuspidal eigenform, then one can construct a ray class distribution from the corresponding overconvergent symbol, that moreover interpolates critical values of the $L$-function of the eigenform. We prove that this distribution is independent of the choices made in its construction. We define the $p$-adic $L$-function of the eigenform to be this distribution.

Keywords:automorphic form, GL(2), p-adic L-function, L-function, modular symbol, overconvergent, cohomology, automorphic cycle, control theorem, L-value, distribution
Categories:11F41, 11F67, 11F85, 11S40, 11M41

2. CJM 2013 (vol 65 pp. 1320)

Taniguchi, Takashi; Thorne, Frank
Orbital $L$-functions for the Space of Binary Cubic Forms
We introduce the notion of orbital $L$-functions for the space of binary cubic forms and investigate their analytic properties. We study their functional equations and residue formulas in some detail. Aside from their intrinsic interest, the results from this paper are used to prove the existence of secondary terms in counting functions for cubic fields. This is worked out in a companion paper.

Keywords:binary cubic forms, prehomogeneous vector spaces, Shintani zeta functions, $L$-functions, cubic rings and fields
Categories:11M41, 11E76

3. CJM 2012 (vol 65 pp. 1201)

Cho, Peter J.; Kim, Henry H.
Application of the Strong Artin Conjecture to the Class Number Problem
We construct unconditionally several families of number fields with the largest possible class numbers. They are number fields of degree 4 and 5 whose Galois closures have the Galois group $A_4, S_4$ and $S_5$. We first construct families of number fields with smallest regulators, and by using the strong Artin conjecture and applying zero density result of Kowalski-Michel, we choose subfamilies of $L$-functions which are zero free close to 1. For these subfamilies, the $L$-functions have the extremal value at $s=1$, and by the class number formula, we obtain the extreme class numbers.

Keywords:class number, strong Artin conjecture
Categories:11R29, 11M41

4. CJM 2011 (vol 65 pp. 22)

Blomer, Valentin; Brumley, Farrell
Non-vanishing of $L$-functions, the Ramanujan Conjecture, and Families of Hecke Characters
We prove a non-vanishing result for families of $\operatorname{GL}_n\times\operatorname{GL}_n$ Rankin-Selberg $L$-functions in the critical strip, as one factor runs over twists by Hecke characters. As an application, we simplify the proof, due to Luo, Rudnick, and Sarnak, of the best known bounds towards the Generalized Ramanujan Conjecture at the infinite places for cusp forms on $\operatorname{GL}_n$. A key ingredient is the regularization of the units in residue classes by the use of an Arakelov ray class group.

Keywords:non-vanishing, automorphic forms, Hecke characters, Ramanujan conjecture
Categories:11F70, 11M41

5. CJM 2010 (vol 63 pp. 241)

Essouabri, Driss; Matsumoto, Kohji; Tsumura, Hirofumi
Multiple Zeta-Functions Associated with Linear Recurrence Sequences and the Vectorial Sum Formula
We prove the holomorphic continuation of certain multi-variable multiple zeta-functions whose coefficients satisfy a suitable recurrence condition. In fact, we introduce more general vectorial zeta-functions and prove their holomorphic continuation. Moreover, we show a vectorial sum formula among those vectorial zeta-functions from which some generalizations of the classical sum formula can be deduced.

Keywords:Zeta-functions, holomorphic continuation, recurrence sequences, Fibonacci numbers, sum formulas
Categories:11M41, 40B05, 11B39

6. CJM 2010 (vol 62 pp. 1155)

Young, Matthew P.
Moments of the Critical Values of Families of Elliptic Curves, with Applications
We make conjectures on the moments of the central values of the family of all elliptic curves and on the moments of the first derivative of the central values of a large family of positive rank curves. In both cases the order of magnitude is the same as that of the moments of the central values of an orthogonal family of $L$-functions. Notably, we predict that the critical values of all rank $1$ elliptic curves is logarithmically larger than the rank $1$ curves in the positive rank family. Furthermore, as arithmetical applications, we make a conjecture on the distribution of $a_p$'s amongst all rank $2$ elliptic curves and show how the Riemann hypothesis can be deduced from sufficient knowledge of the first moment of the positive rank family (based on an idea of Iwaniec)

Categories:11M41, 11G40, 11M26

7. CJM 2008 (vol 60 pp. 1406)

Ricotta, Guillaume; Vidick, Thomas
Hauteur asymptotique des points de Heegner
Geometric intuition suggests that the N\'{e}ron--Tate height of Heegner points on a rational elliptic curve $E$ should be asymptotically governed by the degree of its modular parametrisation. In this paper, we show that this geometric intuition asymptotically holds on average over a subset of discriminants. We also study the asymptotic behaviour of traces of Heegner points on average over a subset of discriminants and find a difference according to the rank of the elliptic curve. By the Gross--Zagier formulae, such heights are related to the special value at the critical point for either the derivative of the Rankin--Selberg convolution of $E$ with a certain weight one theta series attached to the principal ideal class of an imaginary quadratic field or the twisted $L$-function of $E$ by a quadratic Dirichlet character. Asymptotic formulae for the first moments associated with these $L$-series and $L$-functions are proved, and experimental results are discussed. The appendix contains some conjectural applications of our results to the problem of the discretisation of odd quadratic twists of elliptic curves.

Categories:11G50, 11M41

8. CJM 2007 (vol 59 pp. 673)

Ash, Avner; Friedberg, Solomon
Hecke $L$-Functions and the Distribution of Totally Positive Integers
Let $K$ be a totally real number field of degree $n$. We show that the number of totally positive integers (or more generally the number of totally positive elements of a given fractional ideal) of given trace is evenly distributed around its expected value, which is obtained from geometric considerations. This result depends on unfolding an integral over a compact torus.

Keywords:Eisenstein series, toroidal integral, Fourier series, Hecke $L$-function, totally positive integer, trace
Categories:11M41, 11F30, , 11F55, 11H06, 11R47

9. CJM 2006 (vol 58 pp. 3)

Ben Saïd, Salem
The Functional Equation of Zeta Distributions Associated With Non-Euclidean Jordan Algebras
This paper is devoted to the study of certain zeta distributions associated with simple non-Euclidean Jordan algebras. An explicit form of the corresponding functional equation and Bernstein-type identities is obtained.

Keywords:Zeta distributions, functional equations, Bernstein polynomials, non-Euclidean Jordan algebras
Categories:11M41, 17C20, 11S90

10. CJM 2005 (vol 57 pp. 494)

Friedlander, John B.; Iwaniec, Henryk
Summation Formulae for Coefficients of $L$-functions
With applications in mind we establish a summation formula for the coefficients of a general Dirichlet series satisfying a suitable functional equation. Among a number of consequences we derive a generalization of an elegant divisor sum bound due to F.~V. Atkinson.

Categories:11M06, 11M41

11. CJM 2005 (vol 57 pp. 328)

Kuo, Wentang; Murty, M. Ram
On a Conjecture of Birch and Swinnerton-Dyer
Let \(E/\mathbb{Q}\) be an elliptic curve defined by the equation \(y^2=x^3 +ax +b\). For a prime \(p, \linebreak p \nmid\Delta =-16(4a^3+27b^2)\neq 0\), define \[ N_p = p+1 -a_p = |E(\mathbb{F}_p)|. \] As a precursor to their celebrated conjecture, Birch and Swinnerton-Dyer originally conjectured that for some constant $c$, \[ \prod_{p \leq x, p \nmid\Delta } \frac{N_p}{p} \sim c (\log x)^r, \quad x \to \infty. \] Let \(\alpha _p\) and \(\beta _p\) be the eigenvalues of the Frobenius at \(p\). Define \[ \tilde{c}_n = \begin{cases} \frac{\alpha_p^k + \beta_p^k}{k}& n =p^k, p \textrm{ is a prime, $k$ is a natural number, $p\nmid \Delta$} . \\ 0 & \text{otherwise}. \end{cases}. \] and \(\tilde{C}(x)= \sum_{n\leq x} \tilde{c}_n\). In this paper, we establish the equivalence between the conjecture and the condition \(\tilde{C}(x)=\mathbf{o}(x)\). The asymptotic condition is indeed much deeper than what we know so far or what we can know under the analogue of the Riemann hypothesis. In addition, we provide an oscillation theorem and an \(\Omega\) theorem which relate to the constant $c$ in the conjecture.

Categories:11M41, 11M06

12. CJM 2005 (vol 57 pp. 267)

Conrad, Keith
Partial Euler Products on the Critical Line
The initial version of the Birch and Swinnerton-Dyer conjecture concerned asymptotics for partial Euler products for an elliptic curve $L$-function at $s = 1$. Goldfeld later proved that these asymptotics imply the Riemann hypothesis for the $L$-function and that the constant in the asymptotics has an unexpected factor of $\sqrt{2}$. We extend Goldfeld's theorem to an analysis of partial Euler products for a typical $L$-function along its critical line. The general $\sqrt{2}$ phenomenon is related to second moments, while the asymptotic behavior (over number fields) is proved to be equivalent to a condition that in a precise sense seems much deeper than the Riemann hypothesis. Over function fields, the Euler product asymptotics can sometimes be proved unconditionally.

Keywords:Euler product, explicit formula, second moment
Categories:11M41, 11S40

13. CJM 1997 (vol 49 pp. 641)

Burris, Stanley; Compton, Kevin; Odlyzko, Andrew; Richmond, Bruce
Fine spectra and limit laws II First-order 0--1 laws.
Using Feferman-Vaught techniques a condition on the fine spectrum of an admissible class of structures is found which leads to a first-order 0--1 law. The condition presented is best possible in the sense that if it is violated then one can find an admissible class with the same fine spectrum which does not have a first-order 0--1 law. If the condition is satisfied (and hence we have a first-order %% 0--1 law)

Categories:03N45, 11N45, 11N80, 05A15, 05A16, 11M41, 11P81

14. CJM 1997 (vol 49 pp. 468)

Burris, Stanley; Sárközy, András
Fine spectra and limit laws I. First-order laws
Using Feferman-Vaught techniques we show a certain property of the fine spectrum of an admissible class of structures leads to a first-order law. The condition presented is best possible in the sense that if it is violated then one can find an admissible class with the same fine spectrum which does not have a first-order law. We present three conditions for verifying that the above property actually holds. The first condition is that the count function of an admissible class has regular variation with a certain uniformity of convergence. This applies to a wide range of admissible classes, including those satisfying Knopfmacher's Axiom A, and those satisfying Bateman and Diamond's condition. The second condition is similar to the first condition, but designed to handle the discrete case, {\it i.e.}, when the sizes of the structures in an admissible class $K$ are all powers of a single integer. It applies when either the class of indecomposables or the whole class satisfies Knopfmacher's Axiom A$^\#$. The third condition is also for the discrete case, when there is a uniform bound on the number of $K$-indecomposables of any given size.

Keywords:First order limit laws, generalized number theory
Categories:O3C13, 11N45, 11N80, 05A15, 05A16, 11M41, 11P81

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