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Congruences for Fourier coefficients of integer weight modular forms have been the focal point of a number of investigations. In this note we shall exhibit congruences for Fourier coefficients of a slightly different type. Let f(z) = P∞ n=0 a(n)q n be a holomorphic half integer weight modular form with integer coefficients. If ` is prime, then we shall be(More)
the unique normalized cusp form of weight 12 with respect to the full modular group. Although Lehmer’s speculation that τ(n) 6= 0 for every positive n remains open, Serre [S] has made substantial progress on the basic question regarding the number of Fourier coefficients of a modular form which can be zero. He shows (see [p. 179, S]) that τ(n) is non-zero(More)
We investigate conditions which ensure that systems of binomial polynomials with integer coefficients are simultaneously free of large prime factors. In particular, for each positive number ", we show that there are infinitely many strings of consecutive integers of size about n, free of prime factors exceeding n, with the length of the strings tending to(More)
In 1998, the second author raised the problem of classifying the irreducible characters of Sn of prime power degree. Zalesskii proposed the analogous problem for quasi-simple groups, and he has, in joint work with Malle, made substantial progress on this latter problem. With the exception of the alternating groups and their double covers, their work(More)
Let E be an elliptic curve over Q. In 1988, N. Koblitz conjectured a precise asymptotic for the number of primes p up to x such that the order of the group of points of E over Fp is prime. This is an analogue of the Hardy–Littlewood twin prime conjecture in the case of elliptic curves. Koblitz’s Conjecture is still widely open. In this paper we prove that(More)
A variation on the sum-product problem seeks to show that a set which is defined by additive and multiplicative operations will always be large. In this paper, we prove new results of this type. In particular, we show that for any finite set A of positive real numbers, it is true that ∣ ∣ ∣ a + b c + d : a, b, c, d ∈ A }∣ ∣ ∣ ≥ 2|A|2 − 1. As a consequence(More)