Geometry and Analysis of Automorphic Forms of Several Variables

The following sections are included:

• Preface

• Program of symposium

• Contents

Let N ≡ 1(mod 4) be a square-free positive integer, let ε be the primitive quadratic character of conductor N, and let f ∈ S₂(Γ₀(N), ε) be a newform with fourier coefficients in a quadratic field. Shimura associates to f an elliptic curve E defined over the real quadratic field which is isogenous over F to its Galois conjugate. Let M/F be a quadratic extension of F which is neither CM nor totally real, and denote by E_M the twist of E with respect to M/F. The first main result of this article is that, if L′(E_M/F, 1) ≠ 0, then E_M(F) has rank one and III(E/F) is finite. The proof rests on a fundamental result⁴² of Yuan, Zhang and Zhang and on an explicit Heegner point in E_M(F) arising from the modular parametrization X₁(N) → E. We formulate a conjecture relating this Heegner point to Stark-Heegner points⁸ arising from ATR cycles of real dimension one on Hilbert modular surfaces, and present some numerical evidence for this conjecture.

In this paper we show that there exists the supremum of Newton polygons of p-divisible groups with a given p-kernel type, and provide an algorithm determining it.

In this paper, we give several necessary conditions for a holomorphic Siegel modular form on Sp₂(ℤ) to be obtained as a Borcherds lift. As an application, we show that Siegel Eisenstein series are not Borcherds lifts. We also give a condition satisfied by the weight of a Borcherds lift.

We introduce the explicit formulas of archimedean Whittaker functions on GL(3) and their application to archimedean zeta integrals.

In this article, we present arithmetic properties of p-adic elliptic logarithmic functions and p-adic elliptic polylogarithmic functions defined by using a formal group on elliptic curves.

The following sections are included:

• Introduction

• Spherical function ω_T (x; s) on and X_T

• Functional equations, possible zeros and poles

• Explicit formulas

• Spherical Fourier transform on

• An application to hermitian Siegel series

• References

We compute explicitly the Fourier transforms of the invariant distributions associated by Arthur to weighted orbital integrals on the real symplectic group of rank two. These distributions are the main terms on the geometric side of the invariant trace formula.

We give a generalization of the Atkin-Lehner theory to Siegel modular forms of level N and define new forms in this sense. As an application, we show that any non-zero new form in the above sense has a non-zero Fourier coefficient at a primitive half-integral symmetric matrix.

The following sections are included:

• Introduction

• The Matsushima isomorphism and related results

• Basic rudiments on Siegel modular varieties

• Example: The Matsushima isomorphism for compact quotients of Sp(2, R)

• Analytic aspect: Spherical functions

• Cohomology groups of Siegel modular groups of genus two

• The second cohomology group of Siegel modular varieties

• References

In this paper, we first present the basic ideas of the method of determining reducibility or irreducibility of parabolically induced representations of classical p-adic groups using Jacquet modules. After that we explain the construction of irreducible square integrable representations by considering characteristic examples. We end with a brief presentation of the classification of irreducible square integrable representations of these groups modulo cuspidal data, which was obtained jointly with C. Mœglin.

In the last half of 20th century, various generalizations of Hermite's constant and Voronoï's theorem were studied by many authors. In this paper, we give an account of a recent development concerning Voronoï's theorem.