Ling Long

Hypergeometric Series, Truncated Hypergeometric Series, and Gaussian Hypergeometric Functions

In this paper, we investigate the relationships among hypergeometric series, truncated hypergeometric series, and Gaussian hypergeometric functions through some families of “hypergeometric” algebraic varieties that are higher dimensional analogues of Legendre curves.

Reciprocity for multirestricted Stirling numbers

Multirestricted Stirling numbers of the second kind count the number of partitions of a given set into a given number of parts, each part being restricted to at most a fixed number of elements. Multirestricted numbers of the first kind are then defined as elements of the matrix inverse to the matrix of corresponding multirestricted numbers of the second kind. The anomalous sign behavior of these latter numbers makes them impervious to combinatorial analysis. In answer to a conjecture that has remained open for several years, we derive a reciprocity law for multirestricted Stirling numbers using algebraic techniques based on polynomial recursions. As corollaries, we obtain new recurrence relations for multirestricted numbers, and a new algebraic derivation of the reciprocity law for Stirling numbers.

Fourier coefficients of noncongruence cuspforms

Given a finite index subgroup of SL2(Z) with modular curve defined over Q, under the assumption that the space of weight k (� 2) cusp forms is 1-dimensional, we show that a form in this space with Fourier coefficients inQ has bounded denominators if and only if it is a congruence modular form.

A supercongruence motivated by the Legendre family of elliptic curves

A new supercongruence associated with a Gaussian hypergeometric series, as well as one of Mortenson’s supercongruences, are established with new congruence relations and the Legendre transforms of certain sequences.

Zeros of some level 2 Eisenstein series

The zeros of classical Eisenstein series satisfy many intriguing properties. Work of F. Rankin and Swinnerton-Dyer pinpoints their location to a certain arc of the fundamental domain, and recent work by Nozaki explores their interlacing property. In this paper we extend these distribution properties to a particular family of Eisenstein series on Γ(2) because of its elegant connection to a classical Jacobi elliptic function cn(u) which satisfies a differential equation. As part of this study we recursively define a sequence of polynomials from the differential equation mentioned above that allows us to calculate zeros of these Eisenstein series. We end with a result linking the zeros of these Eisenstein series to an L-series.

Generalized Legendre curves and quaternionic multiplication

Abstract This paper is devoted to abelian varieties arising from generalized Legendre curves. In particular, we consider their corresponding Galois representations, periods, and endomorphism algebras. For certain one parameter families of 2-dimensional abelian varieties of this kind, we determine when the endomorphism algebra of each fiber defined over the algebraic closure of Q contains a quaternion algebra.

On modular forms for some noncongruence subgroups of SL2(ℤ) II

In this paper we show two classes of noncongruence subgroups satisfy the so-called unbounded denominator property. In particular, we establish our conjecture in [KL08] which says that every type II noncongruence character group of Gamma^0(11) satisfies the unbounded denominator property.

A SHORT PROOF OF MILNE'S FORMULAS FOR SUMS OF INTEGER SQUARES

We give a short proof of Milnes formulas for sums of 4n2 and 4n2 + 4n integer squares using the theory of modular forms. Other identities of Milne are also discussed.

On ℓ-adic representations for a space of noncongruence cuspforms

This paper is concerned with a compatible family of 4-dimensional l-adic representations ρl of GQ := Gal(Q/Q) attached to the space of weight-3 cuspforms S3(Γ) on a noncongruence subgroup Γ ⊂ SL2(Z). For this representation we prove that: 1. It is automorphic: the L-function L(s,ρl∨) agrees with the L-function for an automorphic form for GL4(AQ), where ρl∨ is the dual of ρl. 2. For each prime p≥5 there is a basis hp = {hp+, hp-} of S3(Γ) whose expansion coefficients satisfy 3-term Atkin and Swinnerton-Dyer (ASD) relations, relative to the q-expansion coefficients of a newform f of level 432. The structure of this basis depends on the class of p modulo 12. The key point is that the representation ρl admits a quaternion multiplication structure in the sense of Atkin, Li, Liu, and Long.

Characterization of intersecting families of maximum size in PSL (2, q )

We consider the action of the