Smorodinsky’s conjecture on rank-one mixing

Rank One Case of Dwork’s Conjecture

In the higher rank paper [17], we reduced Dwork’s conjecture from higher rank case over any smooth affine variety X to the rank one case over the simplest affine space A. In the present paper, we finish our proof by proving the rank one case of Dwork’s conjecture over the affine space A, which is called the key lemma in [17]. The key lemma had already been proved in [16] in the special case whe...

The Rank One Abelian Stark Conjecture

Rank One Power Weakly Mixing Nonsingular Transformations

We show that Chacon's nonsingular type III transformation T , 0 < 1, is power weakly mixing, i.e., for all sequences of nonzero integers fk1; : : : ; krg, T k1 : : : T kr is ergodic. We then show that in in nite measure, this condition is not implied by in nite ergodic index (having all nite Cartesian products ergodic), and that in nite ergodic index does not imply 2-recurrence.

RANK ONE CASE OF DWORK ' S CONJECTURE 3 Theorem 1

1. Introduction In the higher rank paper 17], we reduced Dwork's conjecture from higher rank case over any smooth aane variety X to the rank one case over the simplest aane space A n. In the present paper, we nish our proof by proving the rank one case of Dwork's conjecture over the aane space A n , which is called the key lemma in 17]. The key lemma had already been proved in 16] in the specia...

Course and Project Description: The rank one abelian Stark conjecture

Let K/F denote an abelian extension of number fields with associated rings of integers OK and OF . Let S denote a finite set of places of F containing the archimedean places and those which ramify in K. Assume that S contains at least one place v that splits completely in K and that |S| ≥ 2. For each ideal n ⊂ OF not divisible by a prime that ramifies in K, we denote by σn the associated Froben...