We obtain a new bound for the first eigenvalue of the Laplacian for Bianchi manifolds by the method of Luo, Rudnick and Sarnak. We use a recent result of Kim on symmetric power $L$-functions. The key idea is to take tensor products of zeta functions, and we report on our recent developments on Kurokawa's multiple zeta functions.

A simple proof of the validity of Cornacchia's algorithm for solving the diophantine equation $x^2 + dy^2 = m$ is presented. Furthermore, the special case $d=1$ is solved completely.

Let $L(s,\phi)$ be an automorphic $L$-function for a Bianchi group defined via an imaginary quadratic field with discriminant $d < 0$. We give an upper bound for the absolute value of $L(s, \phi)$ in terms of $\Im(s)$, the Laplace eigenvalue and the discriminant $d$. The bound as $d \to \infty$ presents a new aspect in the study of $L$-functions.

Let $h \geq 2$ be an integer and $F$ a formal group over $\mathbf{Z}_p$ of Honda type $p + X^h$. The aim of this paper is to calculate the index of the image of the norm of $F$ in the local cyclotomic fields by following Kobayashi's method ([1]). Here we use the property of certain subgroups which we call norm subgroups.

We announce a generalization of the reduction theorem for 0-parameter solutions of the traditional (i.e., second order) Painlevé equations with a large parameter to those of some higher order Painlevé equations, i.e., each member of the Painlevé hierarchies $(P_J)$ ($J = \mbox{I}$, II-1 and II-2) discussed in [KKNT]. Thus the scope of applicability of the reduction theorem ([KT1, KT2]) has been substantially enlarged; only six equations were covered by our previous result, while the result reported here applies to infinitely many equations.

By using norm estimates of the pre-Schwarzian derivatives for certain analytic functions defined by a nonlinear integral transform, we shall give several interesting geometric properties of the integral transform.

In this paper we characterize all totally $\eta$-umbilic real hypersurfaces $M$'s in complex projective or complex hyperbolic spaces by using an inequality related to the shape operator $A$ of $M$.

We calculate derivatives of multiple sine functions to investigate coefficients appearing in the addition type formula. We present explicit expressions and we obtain an interesting modular form.

We construct a $q$-analogue of the Mahler measure using a $q$-analogue of the logarithm. We present some basic examples, where we obtain $q$-analogues of special values of zeta functions. We calculate also the classical limit and the crystal limit.

We describe a classification of regular holonomic $\mathcal{D}$-modules whose characteristic variety is the union of conormal bundles to the orbits of the linear group. We show that the category of such objects is equivalent to the one of graded modules of finite type over an algebra of invariant differential operators with polynomial coefficients.

The choice of an elliptic curve for the implementation of an elliptic curve cryptosystem requires counting the number of points on such a curve over a finite field. An improvement of Schoof's algorithm for counting the number of rational points on an elliptic curve defined over a finite field takes advantage of some factor of the division polynomials. In this paper, we study the possible factorisations of such division polynomials.