International Journal of Number Theory

In 1992, Ash and McConnell presented computational evidence of a connection between three-dimensional Galois representations and certain arithmetic cohomology classes. For some examples, they were unable to determine the attached representation. For several Hecke eigenclasses (including one for which Ash and McConnell did not find the Galois representation), we find a Galois representation which appears to be attached and show strong evidence for the uniqueness of this representation. The techniques that we use to find defining polynomials for the Galois representations include a targeted Hunter search, class field theory and elliptic curves.

Some theta function identities are proved and used to give formulae for the number of representations of a positive integer by certain quaternary forms x² + ey² + fz² + gt² with e, f, g ∈ {1, 2, 4, 8}.

Let n be a positive integer and φ(n) denotes the Euler phi function. It is well known that the power sum of n can be evaluated in closed form in terms of n. Also, the sum of all those φ(n) positive integers that are coprime to n and not exceeding n, is expressible in terms of n and φ(n). Although such results already exist in literature, but here we have presented some new analytical results in these connections. Some functional and integral relations are derived for the general power sums.

Define the height function h(a) = min{k + (ka mod p) : k = 1, 2, …, p - 1} for a ∈ {0, 1, …, p - 1.} It is proved that the height has peaks at p, (p + 1)/2, and (p + c)/3, that these peaks occur at a = [p/3], (p - 3)/2, (p - 1)/2, [2p/3], p - 3, p - 2, and p - 1, and that h(a) ≤ p/3 for all other values of a.

Let Γ ⊂ PSL(2, ℝ) be a discrete and finite covolume subgroup. We suppose that the modular curve is "embedded" in a Hilbert modular surface , where Γ_K is the Hilbert modular group associated to a real quadratic field K. We define a sequence of restrictions (ρ_n)_n∈ℕ of Hilbert modular forms for Γ_K to modular forms for Γ. We denote by M_{k1, k2}(Γ_K) the space of Hilbert modular forms of weight (k₁, k₂) for Γ_K. We prove that ∑_n∈ℕ ∑_{k1, k2∈ℕ} ρ_n(M_{k1, k2}(Γ_K)) is a subalgebra closed under Rankin–Cohen brackets of the algebra ⊕_k∈ℕ M_k(Γ) of modular forms for Γ.

In a recent paper, Thorne [5] proved the existence of arbitrarily long strings of consecutive primes in arithmetic progressions in the polynomial ring 𝔽_q[t]. Here we extend this result to show that given any k there exists a string of k consecutive primes of degree D in arithmetic progression for all sufficiently large D.

The aim of this article is to give a concise algebraic treatment of the modular symbols formalism, generalized from modular curves to Hecke triangle surfaces. A sketch is included of how the modular symbols formalism gives rise to the standard algorithms for the computation of holomorphic modular forms. Precise and explicit connections are established to the cohomology of Hecke triangle surfaces and group cohomology. A general commutative ring is used as coefficient ring in view of applications to the computation of modular forms over rings different from the complex numbers.

Fix m greater than one and let E be an elliptic curve over Q with complex multiplication. We formulate conjectures on the density of primes p (congruent to one modulo m) for which the pth Fourier coefficient of E is an mth power modulo p; often these densities differ from the naive expectation of 1/m. We also prove our conjectures for m dividing the number of roots of unity lying in the CM field of E; the most involved case is m = 4 and complex multiplication by Q(i).

Fix a prime N, and consider the action of the Hecke operator T_N on the space of modular forms of full level and varying weight κ. The coefficients of the matrix of T_N with respect to the basis {E₄ⁱ E₆^j | 4i + 6j = κ} for can be combined for varying κ into a generating function F_N. We show that this generating function is a rational function for all N, and present a systematic method for computing F_N. We carry out the computations for N = 2, 3, 5, and indicate and discuss generalizations to spaces of modular forms of arbitrary level.

We show that the prime divisors of a random polynomial in 𝔽_q[t] are typically "Poisson distributed". This result is analogous to the result in ℤ of Granville [1]. Along the way, we use a sieve developed by Granville and Soundararajan [2] to give a simple proof of the Erdös–Kac theorem in the function field setting. This approach gives stronger results about the moments of the sequence {ω(f)}_f∈𝔽q[t] than was previously known, where ω(f) is the number of prime divisors of f.

The Fourier coefficients of classical modular forms of negative weights have been determined for the case for which F(τ) belongs to a subgroup of the full modular group [9]. In this paper, we determine the Fourier coefficients of generalized modular forms of negative weights using the circle method.

A set of real nth roots that is pairwise linearly independent over the rationals must also be linearly independent. We show how this result may be extended to more general fields.

The original Sato–Tate Conjecture concerns the angle distribution of the eigenvalues arisen from non-CM elliptic curves. In this paper, we formulate an analogue of the Sato–Tate Conjecture on automorphic forms of (GL_n) and, under a holomorphic assumption, prove that the distribution is either uniform or the generalized Sato–Tate measure.