Let R be a complete discrete valuation ring of mixed characteristic (0, p) with perfect residue field, K the fraction field of R. Suppose G is a Barsotti-Tate group (p-divisible group) defined over K which acquires good reduction over a finite extension K′ of K. We prove that there exists a constant c ≥ 2 which depends on the absolute ramification index e(K′/Qp) and the height of G such that G ...

Let K be a complete discrete valuation field with finite residue field F of order q. We call such a field a local field. The geometric Frobenius FrF is the inverse of the map a 7→ a in the absolute Galois group GF = Gal(F̄ /F ). The Weil group WK is defined as the inverse image of the subgroup 〈FrF 〉 ⊂ GF by the canonical map GK = Gal(K̄/K) → GF . For a scheme XK of finite type over K, the l-adic...

We study the valuation at an irreducible polynomial v of the v-adic power sum, for exponent k (or −k), of polynomials of a given degree d in Fq[t], as a sequence in d (or k). Understanding these sequences has immediate consequences, via standard Newton polygon calculations, for the zero distribution of the corresponding v-adic Goss zeta functions. We concentrate on v of degree one and two and g...

coming from the cokernels of the rst column. These are cyclic (pro-cyclic in the limit) and are generated by the image of q t. So, lim A(K t)=NA(K n;t) = ?=< (q t ; L t =K t) > where (q t ; L t =K t) is the norm residue symbol of q t for the extension L t =K t. We write Q t for the image of (q t ; L t =K t) under the isomorphism ? ! Z p induced by sending to 1. Note that although Q t is not can...

We define complexes analogous to Goncharov’s complexes for the K–theory of discrete valuation rings of characteristic zero. Under suitable assumptions in K–theory, there is a map from the cohomology of those complexes to the K–theory of the ring. In case the ring is the localization of the ring of integers in a number field, there are no assumptions necessary. We compute the composition of our ...

Let p be a fixed prime. Throughout this paper Z, Zp, Qp, and Cp will, respectively, denote the ring of rational integers, the ring of p-adic rational integers, the field of p-adic rational numbers and the completion of algebraic closure of Qp, cf.[1, 2, 3]. Let vp be the normalized exponential valuation of Cp with |p| = p −vp(p) = p and let a+ pZp = {x ∈ Zp|x ≡ a( mod p N )}, where a ∈ Z lies i...

This work is a part of my research in the area of model theory. First we study a particular L-theory of rings whose models are called p-convexly valued domains (the first-order language L is the language of rings equipped with a linear divisibility predicate and predicates for the nth powers). It consists of a p-adic counterpart of Becker’s convexly oredered valuation rings. Then we are interes...

These are lecture notes from a graduate course on p-adic and motivic integration (given at BGU). Themain topics are: Quantifier elimination in the p-adics, rationality of p-adic zeta functions and their motivic analogues, basic model theory of algebraically closed valued fields, motivic integration following Hrushovski and Kazhdan, application to the Milnor fibration. Background: basic model th...

Let a and q be two units of Zp, q not a root of unity, and let Vq be the closure of the set {aqn | n = 0, 1, 2, ..}. K is a non-archimedean valued field, K contains Qp, and K is complete for the valuation |.|, which extends the p-adic valuation. C(Vq → K) is the Banach space of continuous functions from Vq to K, equipped with the supremum norm. Let E and Dq be the operators on C(Vq → K) defined...

Let f1 (resp. f2) denote two (elliptic) newforms of prime level N , trivial character and weight 2 (resp. k + 2, where k ∈ {8, 12}). We provide evidence for the Bloch-Kato conjecture for the motive M = ρf1⊗ρf2 (−k/2−1) by proving that under some assumptions the p-valuation of the order of the Bloch-Kato Selmer group of M is bounded from below by the p-valuation of the relevant L-value (a specia...