Generalised form of a conjecture of Jacquet and a local consequence

Details Generalised form of a conjecture of Jacquet and a local consequence Generalised form of a conjecture of Jacquet and a local consequence

2006-06-21

arxiv.org/abs/math/0606515v2

Mathematics

Number Theory

20 pages. Typos corrected and some minor changes. To appear in Journal fuer die Reine und Angewandte Mathematik

Scientific paper

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.

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