Special values of multiple Hecke L-functions

The purpose of this study is to clarify the structure of the space generated by the special values of multiple Hecke L-functions for the full modular group SL ₂(Z). In this paper we describe the explicit relations among the values by using Manin??s results, and show the numerical example in case of the Ramanujan ??-function.     

Special representations of rank one orthogonal groups

We study the image of the theta correspondence from to a rank one orthogonal group (over a number field). The image consists of cusp forms, the Fourier coefficients of which satisfy a certain invariance property. We show that this property characterizes the image. The proof requires first an analogous local statement (almost everywhere) and then a use of certain Rankin-Selberg integrals. Partially supported by the Bat-Sheva de Rothschild Fund for the Advancement of Science and Technolog.     

L-functions and theta correspondence for classical groups

Using the doubling method of Piatetski-Shapiro and Rallis, we develop a theory of local factors of representations of classical groups and apply it to give a necessary and sufficient condition for nonvanishing of global theta liftings in terms of analytic properties of the L-functions and local theta correspondence.     

Higher regulators and values of L-functions

In the work conjectures are formulated regarding the value of L-functions of motives and some computations are presented corroborating them.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki (Noveishie Dostizheniya), Vol. 24, pp. 181–238, 1984.     

On the mean values of Dirichlet L-functions

We study the 2kth power moment of Dirichlet L-functions L(s,) at the centre of the critical strip , where the average is over all primitive characters (mod q). We extend to this case the hybrid Euler–Hadamardproduct results of Gonek, Hughes and Keating for the Riemannzeta function. This allows us to recover conjectures for themoments based on random matrix models, incorporating the arithmeticalterms in a natural way.     

On discrete mean values of Dirichlet L-functions

Czechoslovak Mathematical Journal - Let χ be a nonprincipal Dirichlet character modulo a prime number p ? 3 and let $${\mathfrak{a}_{\cal X}}: = {1 \over 2}\left( {1{ - _{\cal X}}\left(...     

On the special values of automorphic L-functions

Let π be a cuspidal representation of $\text{GL}{}_{\mathrm{n}}\text{(}\text{A}{}_{\mathrm{<mtext>Q</mtext>}}\text{)}$ with non-vanishing cohomology and denote by L(π,s) its L-function. Under a certain local non-vanishing assumption, we prove the rationality of the values of L(π?χ,0) for characters χ, which are critical for π. Note that conjecturally any motivic L-function should coincide with an automorphic L-function on GL_n; hence, our result corresponds to a conjecture of Deligne for motivic L-functions. To cite this article: J. Mahnkopf, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On special values of certain Dirichlet L-functions

Let r _k (n) denote the number of ways n can be expressed as a sum of k squares. Recently, S. Cooper (Ramanujan J. 6:469–490, [2002]), conjectured a formula for r ₉(t), t≡5 (mod 8), r ₁₁(t), t≡7 (mod 8), where t is a square-free positive integer. In this note we observe that these conjectures follow from the works of Lomadze (Akad. Nauk Gruz. Tr. Tbil. Mat. Inst. Razmadze 17:281–314, [1949]; Acta Arith. 68(3):245–253, [1994]). Further we express r ₉(t), r ₁₁(t) in terms of certain special values of Dirichlet L-functions. Combining these two results we get expressions for these special values of Dirichlet L-functions involving Jacobi symbols.      

Syntomic regulators and special values of p-adic L-functions

In this paper p-adic analogs of the Lichtenbaum Conjectures are proven for abelian number fields F and odd prime numbers p, which generalize Leopoldt's p-adic class number formula, and express special values of p-adic L-functions in terms of orders of K-groups and higher p-adic regulators. The approach uses syntomic regulator maps, which are the p-adic equivalent of the Beilinson regulator maps. They can be compared with étale regulators via the Fontaine-Messing map, and computations of Bloch-Kato in the case that p is unramified in F lead to results about generalized Coates-Wiles homomorphisms and cyclotomic characters. Oblatum 14-V-96 & 9-X-97     

On linear combinations of special values of L-functions

Using a specific evaluation method for L-functions, it is shown that character values and weighted averages of Gauss sums can be well approximated by linear combinations of the algebraic parts of special values of L-functions under correct parity conditions. This is a surprising universal property of these special values since the coefficients of the resulting combinations turn out to be independent of the character but its parity.     

Critical values of L-functions on GSp 2 × Gl 2

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. Supported by the von Neumann Fund at the Institute for Advanced Study, Princeton     

Three-variable Mahler measures and special values of L-functions of modular forms

The Ramanujan Journal - In this paper we study the Mahler measures of two families of Laurent polynomials. We prove several three-variable Mahler measure formulas initially conjectured by D. Samart.     

Liberation of orthogonal Lie groups

We show that under suitable assumptions, we have a one-to-one correspondence between classical groups and free quantum groups, in the compact orthogonal case. We classify the groups under correspondence, with the result that there are exactly 6 of them: O_n, S_n, H_n, B_n, , . We investigate the representation theory aspects of the correspondence, with the result that for O_n, S_n, H_n, B_n, this is compatible with the Bercovici–Pata bijection. Finally, we discuss some more general classification problems in the compact orthogonal case, notably with the construction of a new quantum group.