On a Refined Stark Conjecture for Function Fields

We prove that a refinement of Stark's Conjecture formulated by Rubin is true up to primes dividing the order of the Galois group, for finite, abelian extensions of function fields over finite fields. We also show that in the case of constant field extensions a statement stronger than Rubin's holds true

Brauer Groups and Tate-Shafarevich Groups

Let XK be a proper, smooth and geometrically connected curve over a global field K. In this paper we generalize a formula of Milne relating the order of the Tate-Shafarevich group of the Jacobian of XK to the order of the Brauer group of a proper regular model of XK. We thereby partially answer a question of Grothendieck

Algebraic cycles on Severi-Brauer schemes of prime degree over a curve

Let $k$ be a perfect field and let $p$ be a prime number different from the characteristic of $k$. Let $C$ be a smooth, projective and geometrically integral $k$-curve and let $X$ be a Severi-Brauer $C$-scheme of relative dimension $p-1$ . In this paper we show that $CH^{d}(X)_{{\rm{tors}}}$ contains a subgroup isomorphic to $CH_{0}(X/C)$ for every $d$ in the range $2\leq d\leq p$. We deduce that, if $k$ is a number field, then $CH^{d}(X)$ is finitely generated for every $d$ in the indicated range.Comment: 6 page

Finiteness theorems for algebraic cycles of small codimension on quadric fibrations over curves

We obtain finiteness theorems for algebraic cycles of small codimension on quadric fibrations X over curves over perfect fields k. For example, if k is finitely generated over Q and the fibration has odd relative dimension at least 11, then CH^{i}(X) is finitely generated for i<=4.Comment: 13 page