The Lucas numbers, 2, 1, 3, 4, 7, 11, 18, 29, 47, . . . , named in honor of Edouard Lucas (1842-1891), are defined by L0 = 2, L1 = 1, and Ln = Ln−1 + Ln−2 for n ≥ 2. It is easy to show that, for n ≥… (More)

Given an abelian algebraic group A over a global field F , α ∈ A(F ), and a prime `, the set of all preimages of α under some iterate of [`] generates an extension of F that contains all `-power… (More)

For a prime p, let Z(p) be the smallest positive integer n so that p divides Fn, the nth term in the Fibonacci sequence. Paul Bruckman and Peter Anderson conjectured a formula for ζ(m), the density… (More)

Suppose that p ≡ 1 (mod 4) is a prime, and that OK is the ring of integers of K := Q( √ p). A classical result of Hirzebruch and Zagier asserts that certain generating functions for the intersection… (More)

An eta-quotient of levelN is a modular form of the shape f(z) = ∏ δ|N η(δz) rδ . We study the problem of determining levels N for which the graded ring of holomorphic modular forms for Γ0(N) is… (More)

A cusp form f(z) of weight k for SL2(Z) is determined uniquely by its first ` := dimSk Fourier coefficients. We derive an explicit bound on the nth coefficient of f in terms of its first `… (More)

For k 1, let ∑∞ n=k τk(n)q n = q ∏∞ n=1(1 − q). It follows from Deligne’s proof of the Weil conjectures that there is a constant Ck so that |τk(n)| Ckd(n)n(12k−1)/2. We study the value of Ck as a… (More)

Let f(z) = ∑∞ n=1 af (n)q n ∈ S k (Γ0(N)) be a newform with squarefree level N that does not have complex multiplication. For a prime p, define θp ∈ [0, π] to be the angle for which af (p) = 2p… (More)

In this paper, we study congruences for modular forms of half-integral weight on Γ0(4). Suppose that ` ≥ 5 is prime, that K is a number field, and that v is a prime of K above `. Let Ov denote the… (More)

A certain sequence of weight 1/2 modular forms arises in the theory of Borcherds products for modular forms for SL2(Z). Zagier proved a family of identities between the coefficients of these weight… (More)