Quartic residues and binary quadratic forms

Let be a prime, m∈Z and . In this paper we obtain a general criterion for m to be a quartic residue in terms of appropriate binary quadratic forms. Let d>1 be a squarefree integer such that , where is the Legendre symbol, and let ε_d be the fundamental unit of the quadratic field . Since 1942 many mathematicians tried to characterize those primes p so that ε_d is a quadratic or quartic residue . In this paper we will completely solve these open problems by determining the value of , where p is an odd prime, and . As an application we also obtain a general criterion for , where {u_n(a,b)} is the Lucas sequence defined by and .     

When subset-sums do not cover all the residues modulo p

Let . We prove that a subset of , where p is a prime number, with cardinality larger than such that its subset sums do not cover has an automorphic image which is rather concentrated; more precisely, there exists s prime to p such that

     

Congruences involving Bernoulli and Euler numbers

Let [x] be the integral part of x. Let p>5 be a prime. In the paper we mainly determine , , and in terms of Euler and Bernoulli numbers. For example, we have

     

Abelian subgroups of any order in class groups of global function fields

Let be a finite field with q elements, and T a transcendental element over . In this paper, we construct infinitely many real function fields of any fixed degree over with ideal class numbers divisible by any given positive integer greater than 1. For imaginary function fields, we obtain a stronger result which shows that for any relatively prime integers m and n with m,n>1 and relatively prime to the characteristic of , there are infinitely many imaginary fields of fixed degree m such that the class group contains a subgroup isomorphic to .     

On a theorem of Chowla

Let be a finite field with q=p^felements, where p is a prime number and f is a positive integer. For a nonprincipal multiplicative character χ and a nontrivial additive character ψ on , it is well known that Gauss sum has absolute value . In this paper, we investigate when is a root of unity.     

The distribution of patterns of inverses modulo a prime

We investigate the distribution of the numbers x∈[1,p] for which all lie in a subset of the set of multiplicative inverses modulo a prime p. Here the a_i are integers coprime to p and the numbers are distinct .     

The structure of analytic τ-sheaves

Let R be a complete discrete valuation -algebra whose residue field is algebraic over , and let K denote its fraction field. In this paper, we study the structure of τ-sheaves M without good reduction on the curve , seen as a rigid analytic space. One motivation is the Tate uniformization theorem for t-motives of Drinfeld modules, which we want to extend to general τ-sheaves. On the other hand, we are interested in the action of inertia on a generic Tate module T_?(M) of M.For a given τ-sheaf M on , we prove the existence of a maximal model for M on , an R-model of , and, over a finite separable extension R′ of R, of nondegenerate models for M.We prove the following ‘semistability’ theorem: there exists a finite extension K′ of K, a nonempty open subscheme C′⊂C, and a filtration

     

Uniform bounds on the number of rational points of a family of curves of genus 2

We exhibit a genus-2 curve defined over which admits two independent morphisms to a rank-1 elliptic curve defined over . We describe completely the set of -rational points of the curve and obtain a uniform bound on the number of -rational points of a rational specialization of the curve for a certain (possibly infinite) set of values . Furthermore, for this set of values we describe completely the set of -rational points of the curve . Finally, we show how these results can be strengthened assuming a height conjecture of Lang.     

Kloosterman's uniformly distributed sequence

Applying the theory of uniform distribution, especially the Erdös-Turán-Koksma inequality and the Koksma-Hlawka inequality, to the two-dimensional Kloosterman sequence , j=1,2,…,?(n) (where , and ?(n) is the Euler function) we find an estimation for the discrepancy of this sequence and an error term for the Kth moment, K=1,2,…, of the sequence of distances as

     

Volumes and diophantine inequalities associated with decomposable forms

For homogeneous decomposable forms in n variables with real coefficients, we consider the associated volume of all real solutions to the inequality . We relate this to the number of integral solutions to the Diophantine inequality in the case where F has rational coefficients. We find quantities which bound the volume and which yield good upper bounds on the number of solutions to the Diophantine inequality in the rational case.     

An analogue of Hilbert's 10th problem for fields of meromorphic functions over non-Archimedean valued fields

Let K be a complete and algebraically closed valued field of characteristic 0. We prove that the set of rational integers is positive existentially definable in the field of meromorphic functions on K in the language of rings augmented by a constant symbol for the independent variable z and by a symbol for the unary relation “the function x takes the value 0 at 0”. Consequently, we prove that the positive existential theory of in the language is undecidable. In order to obtain these results, we obtain a complete characterization of all analytic projective maps (over K) from an elliptic curve minus a point to , for any elliptic curve defined over the field of constants.