The ring of quasimodular forms for a cocompact group

We describe the additive structure of the graded ring of quasimodular forms over any discrete and cocompact group Γ⊂PSL(2,R). We show that this ring is never finitely generated. We calculate the exact number of new generators in each weight k. This number is constant for k sufficiently large and equals where I and are the ideals of modular forms and quasimodular forms, respectively, of positive weight. We show that is contained in some finitely generated ring of meromorphic quasimodular forms with i.e., the same order of growth as      

Correspondence among Eisenstein series of weight 2

There is a known correspondence among modular forms, Jacobi forms and Siegel modular forms of genus 2. In this paper we show this correspondence can be extended to non-holomorphic Eisenstein series, in particular, among , E2,1(τ,z;δ;0), and .     

The twelfth moment of central values of Hecke series

Let H_j(s) be the Hecke L-function attached to the Maass wave form for the jth eigenvalue of the hyperbolic Laplacian acting in the Hilbert space of automorphic functions for the full modular group. The following mean value estimate for the central values is proved:

     

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

     

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

     

The distribution of prime ideals in a real quadratic field with units having a given index in the residue class field

Let k be a real quadratic number field and the ring of integers and the group of units in k. Denote by the subgroup represented by elements of E of for a prime ideal in k. We show that for a given positive rational integer a, the set of prime numbers p for which the residual index of for lying above p is equal to a has a natural density c under the Generalized Riemann Hypothesis. Moreover, we give the explicit formula of c and conditions to c=0.     

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 .     

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.     

Simultaneous Diophantine approximation on manifolds and Hausdorff dimension

Let M be an m-dimensional, C^k manifold in , for any , and for any τ>0 let

     

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.     

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.