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

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

Darstellung durch definite ternäre quadratische Formen

AbstractWe study the representation behaviour of a Z-lattice L on a positive definite ternary quadratic space V over Q. As a new tool for this we use the Bruhat-Tits building of the spingroup of the completion of V at a suitable prime p. In Section 2 we show how this can be described in an elementary way as a graph whose vertices are the Zp-maximal lattices on Vp, and in Section 4 we let this graph induce a graph, whose vertices are lattices on V, which differ from L only at the prime p. In Section 3 we investigate which lattices from the graph defined in Section 2 have a given vector in common. The results are used in Sections 5 and 6 to obtain information on the representation behaviour of some special lattices. In Section 5 we get a list of lattices, which represent all numbers they represent locally everywhere; this list contains that given by Watson in [16]. In Section 6 we sharpen a result of Jones and Pall from [6]