The arithmetic of vector-valued modular forms on Γ0(2)

Let [Formula: see text] denote an irreducible two-dimensional representation of [Formula: see text] The collection of vector-valued modular forms for [Formula: see text], which we denote by [Formula: see text], form a graded and free module of rank two over the ring of modular forms on [Formula: see text], which we denote by [Formula: see text] For a certain class of [Formula: see text], we prove that if [Formula: see text] is any vector-valued modular form for [Formula: see text] whose component functions have algebraic Fourier coefficients then the sequence of the denominators of the Fourier coefficients of both component functions of [Formula: see text] is unbounded. Our methods involve computing an explicit basis for [Formula: see text] as a [Formula: see text]-module. We give formulas for the component functions of a minimal weight vector-valued form for [Formula: see text] in terms of the Gaussian hypergeometric series [Formula: see text], a Hauptmodul of [Formula: see text], and the Dedekind [Formula: see text]-function.

Values of Gaussian hypergeometric series and their connections to algebraic curves

In this paper, we explicitly evaluate certain special values of [Formula: see text] hypergeometric series. These evaluations are based on some summation transformation formulas of Gaussian hypergeometric series. We find expressions of the number of points on certain algebraic curves over [Formula: see text] in terms of Gaussian hypergeometric series, which play the vital role in deducing the transformation results.

Character sums, Gaussian hypergeometric series, and a family of hyperelliptic curves

We study the character sums [Formula: see text] [Formula: see text] where [Formula: see text] is the quadratic character defined over [Formula: see text]. These sums are expressed in terms of Gaussian hypergeometric series over [Formula: see text]. Then we use these expressions to exhibit the number of [Formula: see text]-rational points on families of hyperelliptic curves and their Jacobian varieties.