Motivated by arithmetic applications, we introduce the notion of a partial zeta function which generalizes the classical zeta function of an algebraic variety defined over a finite field. We then explain two approaches to the general structural properties of the partial zeta function in the direction of the Weil-type conjectures. The first approach, using an inductive fibred variety point of view, shows that the partial zeta function is rational in an interesting case, generalizing Dwork's rationality theorem. The second approach, due to Faltings, shows that the partial zeta function is always nearly rational. © 2000 Academic Press.

We prove that the partial zeta function introduced in [9] is a rational function, generalizing Dwork's rationality theorem.

The ψ-operator for (φ{symbol}, Γ)-modules plays an important role in the study of Iwasawa theory via Fontaine's big rings. In this note, we prove several sharp estimates for the ψ-operator in the cyclotomic case. These estimates immediately imply a number of sharp p-adic combinatorial congruences, one of which extends the classical congruences of Fleck (1913) and Weisman [Some congruences for binomial coefficients, Michigan Math. J. 24 (1977) 141-151]. © 2005 Elsevier Inc. All rights reserved.

In this article, we introduce a systematic new method to investigate the conjectural p-adic meromorphic continuation of Professor Bernard Dwork's unit root zeta function attached to an ordinary family of algebraic varieties defined over a finite field of characteristic p.

In this paper, we completely determine the trivial factors for L-functions of symmetric products of Kloosterman sheaves. © 2007 Elsevier Inc. All rights reserved.

In this note an improvement of Katz's bound on the number of elements in a finite field with given trace and norm is given. The improvement is obtained by reducing the problem to estimating the number of rational points on certain toric Calabi-Yau hypersurfaces, and then to use detailed cohomological calculations by Rojas-Leon and the second author for such toric hypersurfaces. © 2010 Walter de Gruyter Berlin · New York.

It has been proved that the maximum likelihood decoding problem of Reed-Solomon codes is NP-hard. However, the length of the code in the proof is at most polylogarithmic in the size of the alphabet. For the complexity of maximum likelihood decoding of the primitive Reed-Solomon code, whose length is one less than the size of alphabet, the only known result states that it is at least as hard as the discrete logarithm in some cases where the information rate unfortunately goes to zero. In this paper, it is proved under a well known cryptography hardness assumption that: 1) There does not exist a randomized polynomial time maximum likelihood decoder for the Reed-Solomon code family [q, k(q)]q, where k(x) is any function in Z+ → Z + computable in time xO(1) satisfying √x ≤ k(x)≤ x - √x. 2) There does not exist a randomized polynomial time boundeddistance decoder for primitive Reed-Solomon codes at distance 2/3 + ε of the minimum distance for any constant 0 < ∈ < 1/3. In particular, this rules out the possibility of a polynomial time algorithm for maximum likelihood decoding problem of primitive Reed-Solomon codes of any rate under the assumption. © 2010 IEEE.

Let F be a field. We show that certain subrings contained between the polynomial ring F[X] = F[X1, ⋯ ,Xn] and the power series ring F[X][[Y ]] = F[X1, ⋯ ,Xn][[Y ]] have Weierstrass Factorization, which allows us to deduce both unique factorization and the Noetherian property. These intermediate subrings are obtained from elements of F[X][[Y ]] by bounding their total X-degree above by a positive real-valued monotonic up function λ on their Y -degree. These rings arise naturally in studying the p-adic analytic variation of zeta functions over finite fields. Future research into this area may study more complicated subrings in which Y = (Y1, ⋯ , Ym) has more than one variable, and for which there are multiple degree functions, λ1, ⋯ , λm. Another direction of study would be to generalize these results to k-affinoid algebras. © 2010 American Mathematical Society.