Arithmetic Information Research Articles

A note on the arithmetic of differential equations

In this note we give a method for computing the differential Galois group of some linear second-order ordinary differential equations using arithmetic information, namely the p-curvatures.

A question of Stark

One of the programs of Stark’s conjectures is to nd as many connections as possible between the values that Artin L{functions or their derivatives take (especially at s =0 ) and arithmetic information associated to algebraic number elds. The most rened of Stark’s conjectures involves the values of rst derivatives of L{functions at s =0 :It was recognized early on that the conjecture should be extended to cover cases where the order of vanishing of the L{functions at s =0 is greater than one. In 1980, Stark posed a question along these lines that we will consider in detail here. In particular, we will study his question for relative quadratic extensions and prove that an armative answer to his question exists for all cases considered.

On the Niwa-Shintani theta-kernel lifting of modular forms

Modular forms of half-integral weight are of intrinsic interest: many of the functions of classical number theory transform under a matrix group with half-integral weight. The aim of this paper is to refine some results and techniques which have been introduced to study these functions and the arithmetic information which they contain.

Some Conjectures on Elliptic Curves over Cyclotomic Fields

We give conjectures for the mean values of Hasse-Weil type $L$-functions over cyclotomic fields. In view of the Birch-Swinnerton-Dyer conjectures, this translates to interesting arithmetic information.

Some conjectures on elliptic curves over cyclotomic fields

We give conjectures for the mean values of Hasse-Weil type L L -functions over cyclotomic fields. In view of the Birch-Swinnerton-Dyer conjectures, this translates to interesting arithmetic information.