Generalised form of a conjecture of Jacquet and a local consequence

Following the work of Harris and Kudla we prove a more general form of a conjecture of Jacquet relating the non-vanishing of a certain period integral to non-vanishing of the central critical value of a certain $L$-function. As a consequence we deduce certain local results about the existence of $GL_2(k)$-invariant linear forms on irreducible, admissible representations of $GL_2({\Bbb K})$ for ${\Bbb K}$ a commutative semi-simple cubic algebra over a non-archimedean local field $k$ in terms of certain local epsilon factors which were proved only in certain cases by the first author in his earlier work. This has been achieved by globalising a locally distinguished representation to a globally distinguished representation, a result of independent interest.Comment: 20 pages. Typos corrected and some minor changes. To appear in Journal fuer die Reine und Angewandte Mathemati

A decomposition of the Fourier-Jacobi coefficients of Klingen Eisenstein series

We investigate the relation between Klingen's decomposition of the space of Siegel modular forms and Dulinski's analogous decomposition of the space of Jacobi forms.Comment: Summary of a talk at the RIMS workshop "Automorphic Forms and Related Topics", February 2017, Kyot

Tri-quotient maps become inductively perfect with the aid of consonance and continuous selections

AbstractGeneralizing the result of Arhangel'skii that each open map with Čech-complete domain is compact-covering, it is proved that every tri-quotient map with consonant domain is harmonious, thus compact-covering, and its range is consonant. The latter constitutes a strong answer to a question of Nogura and Shakhmatov. Conditions for harmonious maps to be inductively perfect, or countable compact-covering and for countable compact-covering maps to be harmonious are given. They extend theorems of Just and Wicke

Averages of Fourier coefficients of Siegel modular forms and representation of binary quadratic forms by quadratic forms in four variables

Let $-d$ be a a negative discriminant and let $T$ vary over a set of representatives of the integral equivalence classes of integral binary quadratic forms of discriminant $-d$. We prove an asymptotic formula for $d \to \infty$ for the average over $T$ of the number of representations of $T$ by an integral positive definite quaternary quadratic form and obtain results on averages of Fourier coefficients of linear combinations of Siegel theta series. We also find an asymptotic bound from below on the number of binary forms of fixed discriminant $-d$ which are represented by a given quaternary form. In particular, we can show that for growing $d$ a positive proportion of the binary quadratic forms of discriminant $-d$ is represented by the given quaternary quadratic form.Comment: v5: Some typos correcte