Given any integer N>1 prime to 3, we denote by C N the elliptic curve x 3 +y =N. We first study 3-adic valuation of algebraic part value Hasse–Weil L-function L(C ,s) over ℚ at s=1, and exhibit a relation between 3-part its Tate–Shafarevich group number distinct divisors which are inert in imaginary quadratic field K=ℚ(-3). In case where ,1)≠0 is product split primes K, show that order as predi...

Over any field an implicit linear difference equation one can reduce to the usual explicit one, which has infinitely many solutions ~ for each initial value. It is interesting consider over ring, because case of a ring significantly different from one. The previous results on equations rings mostly concern integers and low order equations. In present article high some other classes rings, parti...

In this paper we consider the Newton polygons of L-functions coming from additive exponential sums associated to a polynomial over a finite field Fq. These polygons define a stratification of the space of polynomials of fixed degree. We determine the open stratum: we give the generic Newton polygon for polynomials of degree d ≥ 2 when the characteristic p is greater than 3d, and the Hasse polyn...

p-Adic mathematical physics is a branch of modern mathematical physics based on the application of p-adic mathematical methods in modeling physical and related phenomena. It emerged in 1987 as a result of efforts to find a non-Archimedean approach to the spacetime and string dynamics at the Planck scale, but then was extended to many other areas including biology. This paper contains a brief re...

We introduce the Swan class of an -adic etale sheaf on a variety over a local field. It is a generalization of the classical Swan conductor measuring the wild ramification and is defined as a 0-cycle class supported on the reduction. We establish a Riemann-Roch formula for the Swan class. Let K be a complete discrete valuation field of characteristic 0. We assume that the residue field F is a p...

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 → 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 an element σ ∈ WK , let n(σ) denote the intege...

Let p≥5 be a prime number, F finite field of characteristic p and let χ¯ the mod-p cyclotomic character. ρ¯:GQ→GL2(F) Galois representation such that local ρ¯↾GQp is flat irreducible. Further, assume detρ¯=χ¯. The celebrated theorem Khare Wintenberger asserts if ρ¯ satisfies some natural conditions, there exists normalized Hecke-eigencuspform f=∑n≥1anqn p|p in its Fourier coefficients associate...

1.1. Motivation. In the original work of Katz on p-adic modular forms [Kz], a key insight is the use of Lubin’s work on canonical subgroups in 1-parameter formal groups to define a relative theory of a “canonical subgroup” in p-adic families of elliptic curves whose reduction types are good but not too supersingular. The theory initiated by Katz has been refined in various directions (as in [AG...

Conductor is a numerical invariant of a variety over a local field measuring the wild ramification of the inertia action on the l-adic etale cohomology. In [3], S.Bloch proposes a conjectural formula, Conjecture 1.9, which we call the conductor formula of Bloch. To formulate it, he defines another numerical invariant as the degree of the self-intersection class, which is defined using the local...