423 research outputs found
On a theorem of Ax and Katz
The well-known theorem of Ax and Katz gives a p-divisibility bound for the
number of rational points on an algebraic variety V over a finite field of
characteristic p in terms of the degree and number of variables of defining
polynomials of V. It was strengthened by Adolphson-Sperber in terms of Newton
polytope of the support set G of V. In this paper we prove that for every
generic algebraic variety over a number field supported on G the
Adolphson-Sperber bound can be achieved on special fibre at p for a set of
prime p of positive density in SpecZ. Moreover we show that if G has certain
combinatorial conditional number nonzero then the above bound is achieved at
special fiber at p for all but finitely many primes p.Comment: 11 page
Zeta functions of totally ramified p-covers of the projective line
In this paper we prove that there exists a Zariski dense open subset U
defined over the rationals Q in the space of all one-variable rational
functions with arbitrary k poles of prescribed orders, such that for every
geometric point f in U(Qbar)$, the L-function of the exponential sum of f at a
prime p has Newton polygon approaching the Hodge polygon as p approaches
infinity. As an application to algebraic geometry, we prove that the p-adic
Newton polygon of the zeta function of a p-cover of the projective line totally
ramified at arbitrary k points of prescribed orders has an asymptotic generic
lower bound.Comment: 17 page
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