The Conference on L-Functions

The following sections are included:

• Preface

• List of Participants

• Contents

In this paper, we propose a definition of the concept of quantum Maass forms. On the basis of this definition, we discuss the relation with Maass forms and with cohomology groups.

The following sections are included:

• Introduction

• Lecture 1: Galois deformation and -invariant
  • Selmer groups
  • Greenberg's -invariant

• Lecture 2: Elliptic curves with multiplicative reduction
  • Hecke algebras for quaternion algebras
  • Proof of Theorem 2.1

• Lecture 3: -invariants of CM fields
  • Ordinary CM fields and their Iwasawa modules
  • Anticyclotomic Iwasawa modules
  • The -invariants of CM fields

• Appendix: Differential and adjoint square Selmer group

In this short article, we give conjectures on explicit dimensions of Siegel modular forms of degree two of weight 3 in several cases. We also explain much deeper conjectures, a conjecture on correspondence between Siegel modular forms of half-integral weight and integral weight of degree two, and a conjecture on correspondence between the split group of type C₂ and its compact twist, from which the conjectures on dimensions come. Also some relations of these conjectures to super-singular abelian surfaces will be explain. The dimension of the space of Siegel modular forms can be calculated, in principle, either by Riemann-Roch theorem or by Selberg trace formula, as far as the weight is big enough. Of course in actual calculation, there are difficult technical details, and there is no explicit dimension formula for general degree except for conjectural one [12], [13], but at least there is a general way to calculate it, and it is often well handled when the degree is two. On the other hand, if the weight is very small, e.g. if weight (= the representation at the real place) is not integrable, then there is no known general way to calculate it. For example, for Siegel modular forms in the case of degree two, i.e. for automorphic forms of S p(2,R) ( split rank 2), the weight 1, 2, or 3 is a difficult weight of this type. On the other hand, these are interesting weights for geometry. The weight three case corresponds to the canonimal divisor, so this is very interesting case. The weight two case has some conjectural connection with a degree two version of Shimura-Taniyama conjecture as conjectured by Hiroyuki Yoshida. As for weight one, it rarely exists (cf. [15]). This article concerns about weight 3.

The following sections are included:

• Introduction

• An arithmetic formula for Fourier coefficients

• Applications to convolutions

• References

No abstract received.

The following sections are included:

• Introduction

• Symmetric fourth

• Symmetric mth powers

• First occurences of poles of symmetric power L-functions

• Descent to cuspidal representations on classical groups

• Remark on the images of functorial lift

• References

In this paper, we introduce multi-variable zeta-functions of roots, and prove the analytic continuation of them. For the root systems associated with Lie algebras, these functions are also called Witten zeta-functions associated with Lie algebras which can be regarded as several variable generalizations of Witten zeta-functions defined by Zagier. In the case of type A_r, we have already studied some analytic properties in our previous paper. In the present paper, we prove certain functional relations among these functions of types A_r (r = 1,2,3) which include what is called Witten's volume formulas. Moreover we mention some structural background of the theory of functional relations in terms of Weyl groups.

The aim of the present article is to show that the Spectral–Kloosterman and the Kloosterman–Spectral sum formulas are practically equivalent to each other. Previously, only the procedure for deriving the latter from the former was known, which is the inversion due to N.V. Kuznetsov [10][11]. We shall not only prove it in our own way but also establish a reverse procedure. Our motivation is detailed in the third and the final sections. Technically, our discussion is focused on two Bessel transforms which appear in those sum formulas. These transforms, especially their inversions, seem to be best understood as special instances of the Plancherel theorem on Lie groups as discussed by R.W. Bruggeman [2]. We shall,however, employ an alternative argument which appears to demand less background knowledge, yet it has a certain merit in applications of the sum formula. We shall restrict ourselves to the most basic instance offered by the full modular group, solely for the sake of simplicity. Notations are introduced where they are needed for the first time, and will continue to be effective thereafter.

We define a class of functions that contains the Selberg class and which has a natural ring structure.

The following sections are included:

• Introduction

• The idea of the proof

• The frame of the proof

• Proof of Theorem 1

• Proof of Lemma 1

• Proof of Lemma 5

• A question

• References

The Apéry-like numbers J₂(n) introduced in [6] are indispensable to a description of the special value ζ_Q(2) of the spectral zeta function ζ_Q(s) for the non- commutative harmonic oscillator Q. These numbers have remarkable modular form interpretation as the original Apéry numbers possess. In fact, we show that the differential equation satisfied by the generating function w₂(t) of J₂(n) is the Picard-Fuchs equation for the universal family of elliptic curves equipped with rational 4-torsion. The parameter t of this family can be considered as a modular function for the congruent subgroup Γ ₀(4). Further, we see that the function w₂(t) is regarded as a Γ₀(4)-modular form of weight 1 in the variable τ by taking t as the classical Legendre modular function λ(τ).

• High Rank Zetas for Number Fields

• Non-Abelian L-Functions

• Geometric and Analytic Truncations

• Rankin-Selberg & Zagier Method

• High Rank Zetas and Eisenstein Series

• Stability and Distance to Cusps

• Explicit Formulae for Rank Two Zetas

• Zeros of Rank Two Zetas

• A Rank Three Zeta and Its Zeros

• References