A random analogue of Gilbreath’s conjecture

On an Archimedean analogue of Tate’s conjecture

We consider an Archimedean analogue of Tate’s conjecture, and verify the conjecture in the examples of isospectral Riemann surfaces constructed by Vignéras and Sunada. We prove a simple lemma in group theory which lies at the heart of T. Sunada’s theorem about isospectral manifolds. r 2002 Elsevier Science (USA). All rights reserved. MSC: primary 58G25; secondary 12A70; 11G30 The aim of this sh...

An Analogue of Artin’s Primitive Root Conjecture

Let S = {a1, a2, . . . , an} be a set of nonzero integers such that for any nonempty subset T of S, the product of all the elements in T is not a perfect square. Then the density of the set of primes p for which the ai’s are quadratic non-residues modulo p, but not primitive roots modulo p, is at least 1 2n(q 1)qm , where m is a non-negative integer with m  n and q is the least odd prime which...

On an Analogue of the James Conjecture

We give a counterexample to the most optimistic analogue (due to Kleshchev and Ram) of the James conjecture for Khovanov-Lauda-Rouquier algebras associated to simply-laced Dynkin diagrams. The first counterexample occurs in type A5 for p = 2 and involves the same singularity used by Kashiwara and Saito to show the reducibility of the characteristic variety of an intersection cohomology D-module...

On an elliptic analogue of Zagier's conjecture.

Modified Proof of a Local Analogue of the Grothendieck Conjecture

A local analogue of the Grothendieck Conjecture is an equivalence of the category of complete discrete valuation fields K with finite residue fields of characteristic p 6= 0 and the category of absolute Galois groups of fields K together with their ramification filtrations. The case of characteristic 0 fields K was considered by Mochizuki several years ago. Then the author proved it by differen...