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Kimberly Hopkins

Publications and source records attributed to Kimberly Hopkins.

4 recordsLinked to original sources
arXiv2010

Split-CM points and central values of Hecke L-series

Split-CM points are points of the moduli space h_2/Sp_4(Z) corresponding to products $E \times E'$ of elliptic curves with the same complex multiplication. We prove that the number of split-CM points in a given class of h_2/Sp_4(Z) is related to the coefficients of a weight 3/2 modular form studied by Eichler. The main application of this result is a formula for the central value $L(ψ_N, 1)$ of a certain Hecke L-series. The Hecke character $ψ_N$ is a twist of the canonical Hecke character $ψ$ for the elliptic Q-curve A studied by Gross, and formulas for $L(ψ, 1)$ as well as generalizations were proven by Villegas and Zagier. The formulas for $L(\psin, 1)$ are easily computable and numerical examples are given.

math.NT↗
arXiv2009

Bounds on discriminants with one class per genus

We assume a condition on the zeros of Dirichlet L-functions related to the GUE distribution to show there are no discriminants greater than d_{66}\approx 1.9 x 10^{130} with one class per genus.

math.NT↗
arXiv2009

Higher Weight Heegner Points

In this paper we formulate a conjecture which partially generalizes the Gross-Kohnen-Zagier theorem to higher weight modular forms. For f in S_k(N) satisfying certain conditions, we construct a map from the Heegner points of level N to a complex torus defined by f. We define higher weight analogues of Heegner divisors on this torus. We conjecture they all lie on a line, and their positions are given by the coefficients of a certain Jacobi form corresponding to f. In weight 2, our map is the modular parametrization map (restricted to Heegner points), and our conjectures are implied by Gross-Kohnen-Zagier. For any weight, we expect that our map is the Abel-Jacobi map on a certain modular variety, and so our conjectures are consistent with the conjectures of Beilinson-Bloch. We have verified our map is the Abel-Jacobi for weight 4. We provide numerical evidence to support our conjecture for a variety of examples.

math.NT↗