Ki Ichiro Hashimoto

Zeta Functions of Finite Graphs and Representations of p-Adic Groups

Publisher Summary This chapter focuses on zeta functions of finite graphs and representations of p-adic groups. It discusses two different subjects: first is a combinatorial problem in algebraic graph theory, and the other is arithmetic of discrete subgroups of p-adic groups and their representations. The chapter presents the notation and basic definitions in graph theory. It also presents a generalization of the definition of zeta function. Spectrum of a finite multigraph is analyzed in the chapter. Moreover, the chapter also describes harmonic functions and the Hodge decomposition. The chapter also presents the computation of zeta functions Zx(u) for some well known families of graphs. These computations give many examples of graphs that are not Ramanujan graphs.

Families of cyclic polynomials obtained from geometric generalization of Gaussian period relations

A general method of constructing families of cyclic polynomials over Q with more than one parameter will be discussed, which may be called a geometric generalization of the Gaussian period relations. Using this, we obtain explicit multi-parametric families of cyclic polynomials over Q of degree 3 < e ≤ 7. We also give a simple family of cyclic polynomials with one parameter in each case, by specializing our parameters.

On the Sato-Tate conjecture for QM-curves of genus two

An abelian surface A is called a QM-abelian surface if its endomorphism ring includes an order of an indefinite quaternion algebra, and a curve C of genus two is called a QM-curve if its jacobian variety is a QM-abelian surface. We give a computational result about the distribution of the arguments of the eigenvalues of the Frobenius endomorphisms of QM-abelian surfaces modulo good primes, which supports an analogue of the Sato-Tate Conjecture for such abelian surfaces. We also make some remarks on the field of definition of QM-curves and their endomorphisms.

Selberg-Ihara's Zeta function for p-adic Discrete Groups

This chapter focuses on Selberg–Iharas zeta function for p -adic discrete groups. In [SeI], a zeta function Z Γ ( s ) has been introduced and proved to have many important properties that resemble those of usual L -functions, such as Euler product, functional equation, and analogue of Riemann hypothesis. This function, called with the name of Selberg, is generalized to any discrete subgroup Γ of a semi-simple Lie group of R -rank one. An analogue of Z Γ ( s ) was introduced by Ihara, for a cocompact torsion-free discrete subgroup Γ of PSL(2, K ) or PL(2, K ), where K is a p -adic field. The chapter presents an extension of Iharas results to the case when G is a semi-simple algebraic group over a p -adic field K and Γ is a discrete subgroup of G . The chapter discusses p -adic algebraic groups and the structure of the discrete subgroups Γ .

Noether’s problem and ℚ-generic polynomials for the normalizer of the 8-cycle in ₈ and its subgroups

We study Noethers problem for various subgroups H of the normalizer ot a group C 8 generated by an 8-cycle in S 8 , the symmetric group of degree 8, in three aspects according to the way they act on rational function fields, i.e., Q(X 0 ,..., X 7 ), Q(x 1 ,..., x 4 ), and Q(x, y). We prove that it has affirmative answers for those H containing C 8 properly and derive a Q-generic polynomial with four parameters for each H. On the other hand, it is known in connection to the negative answer to the same problem for C 8 /Q that there does not exist a Q-generic polynomial for C 8 . This leads us to the question whether and how one can describe, for a given field K of characteristic zero, the set of C 8 -extensions L/K. One of the main results of this paper gives an answer to this question.

On Type Numbers of Split Orders of Definite Quaternion Algebras

In this paper, we shall give a new relation between the arithmetic of quaternion algebras and modular forms; we shall express the type numberTq, N of a split order of type (q, N) as the sums of dimensions of some subspaces of the space of cusp forms of weight 2 with respect to Γ0(qN) which are common eigenspaces of Atkin-Lehners involutions.

Involutive modular transformations on the Siegel upper half plane and an application to representations of quadratic forms

Abstract We establish a one-to-one correspondence between the set of conjugacy classes of elliptic transformations in Sp ( n , Z ) which satisfy X 2 + I = 0 (resp. X 2 + X + I = 0) and the set of hermitian forms of rank n over Z [√−1] (resp. Z [(−1 + √−3)/2]) of determinant ±1. As an application, we generalize, to positive symmetric integral matrices S of rank n , the classical fact that any divisor of m 2 + 1 (resp. m + m + 1) can be represented by the quadratic form F ( X , Y ) = X 2 + Y 2 (resp. X 2 + XY + Y 2 ) with relatively prime integers X , Y : Suppose n ≤ 3 (resp. n ≤ 5). Then S can be represented over Z by F ⊗ n ( n copies of F ) if det S is represented by F as above. The proof is based on Siegel-Brauns Mass formula for hermitian forms.

Q-curves with Rational j-invariants and Jacobian Surfaces of GL2-type

We study elliptic curves over quadratic fields with rational j-invariants regarded as Q-curves, in connection with jacobian surfaces of genus two curves over Q. We discuss their minimality as Q-curves, and the classification, as well as the Neben type characters of the associated modular forms. This can be described as the sign change phenomenon by guar-tic twists of curves over Q. We also study their 2-fold covers by genus two curves. Among others, we construct a parametric family {C(j)} of genus two curves over Q covering minimal Q-curves over \( \operatorname{Q} \left( {\sqrt {{j - {{{12}}^{3}}}} } \right) \) with j-invariant j. We find in the twists of {C(j)} concrete equations of curves over Q whose jacobians are isogenous over an extension of Q to Shimura’s abelian surfaces A f attached to normalized eigen forms \( f \in {{S}_{2}}\left( {N,\left( {\frac{N}{*}} \right)} \right) \), whose Fourier coefficients belong to \( \operatorname{Q} \left( {\sqrt {{ - 1}} } \right) \), in all known non CM cases i.e., N = 37, 65, 104, 157, 397, and 877.