arXiv subjects
Publications and source records attributed to Darrin Doud.
Let $ρ$ be a two-dimensional even Galois representation which is induced from a character $χ$ of odd order of the absolute Galois group of a real quadratic field. After imposing some additional conditions on $χ$, we attach $ρ$ to a Hecke eigenclass in the cohomology of ${\rm GL}(2,\mathbb Z)$ with coefficients in a certain infinite-dimensional vector space over a field of characteristic not equal to 2.
In 1992, Avner Ash and Mark 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.
We prove the following theorem: Let $\bar\F_p$ be an algebraic closure of a finite field of characteristic $p$. Let $ρ$ be a continuous homomorphism from the absolute Galois group of $\Q$ to $\GL(3,\bar\F_p)$ which is isomorphic to a direct sum of a character and a two-dimensional odd irreducible representation. Under the condition that the conductor of $ρ$ is squarefree, we prove that $ρ$ is attached to a Hecke eigenclass in the homology of an arithmetic subgroup $Γ$ of $\GL(3,\Z)$. In addition, we prove that the coefficient module needed is, in fact, predicted by a conjecture of Ash, Doud, Pollack, and Sinnott.
We study a canonical basis for spaces of weakly holomorphic modular forms of weights 12, 16, 18, 20, 22, and 26 on the full modular group. We prove a relation between the Fourier coefficients of modular forms in this canonical basis and a generalized Ramanujan tau-function, and use this to prove that these Fourier coefficients are often highly divisible by 2.
We examine the Fourier coefficients of modular forms in a canonical basis for the spaces of weakly holomorphic modular forms of weights 4, 6, 8, 10, and 14, and show that these coefficients are often highly divisible by the primes 2, 3, and 5.
In this paper, we use a series of targeted Hunter searches to prove that the minimal prime conductor of an even icosahedral Galois representation is 1951. In addition, we give a complete list of all even icosahedral Galois representations of prime conductor less than 10,000.
In this paper we extend a conjecture of Ash and Sinnott relating niveau one Galois representation to the mod p cohomology of congruence subgroups of SL(n,Z) to include Galois representations of higher niveau. We then present computational evidence for our conjecture in the case n=3 in the form of three-dimensional Galois representations which appear to correspond to cohomology eigenclasses as predicted by the conjecture. Our examples include Galois representations with nontrivial weight and level, as well as irreducible three-dimensional representations which are in no obvious way related to lower dimensional representations. In addition, we prove that certain symmetric square representations are actually attached to cohomology eigenclasses predicted by the conjecture.