Periods of primitive forms

Primitive Cusp Forms

For example, let f11A denote the (unique) level 11 weight 2 cusp form, then both f11A(z) and f11A(2z) are level 22 cusp forms. Similarly, both f14A(z) and f14A(2z) are level 28 cusp forms, and both f15A(z) and f15A(2z) are level 30 cusp forms. None of these are “new” at N = 22, 28 or 30 since they arise from lower levels. Define S k (N) to be the vector space of weight k primitive cusp forms (o...

Closed Geodesics and Periods of Automorphic Forms

We study the detailed structure of the distribution of Eichler-Shimura periods of an automorphic form on a compact hyperbolic surface. We show that these periods do not cluster around the asymptotic period over a homology class discovered by Zelditch. 0. Introduction and Results Let M = H/Γ be a compact hyperbolic surface, where Γ is a discrete subgroup of PSL(2,R) = SL(2,R)/{±I}. Such a surfac...

From Primitive Forms to Frobenius Manifolds

Cohomology of Arithmetic Groups and Periods of Automorphic Forms

(1) Firstly, I gave a brief introduction to the cohomology of arithmetic groups, in particular the fact that the “tempered” (informally: “near the middle dimension”’) part of the cohomology looks like the cohomology of a torus. This corresponds to §2, §3, §4 of these notes. For the reader totally unfamiliar with the cohomology of arithmetic groups, §3 might be the best thing to focus on. (2) Ne...

Primitive forms of meiosis: the possible evolution of meiosis.

Meiosis is a basic process of most eukaryotes, as it forms with conjugation the basis of sexual reproduction. As sex seems to be present in the vast majority of eukaryotes, the origin of meiosis is presently unknown. Protists having optional or alternative sexual and asexual cycles seem to be the best targets for research on the evolution of meiosis. While the budding yeast Saccharomyces cerevi...