Abelian varieties and Selberg class - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T00:13:51Z http://mathoverflow.net/feeds/question/49284 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/49284/abelian-varieties-and-selberg-class Abelian varieties and Selberg class Sylvain JULIEN 2010-12-13T17:39:53Z 2011-04-24T22:22:13Z <p>Hello everyone,</p> <p>I would like to know whether, assuming Selberg's orthonormality conjecture, it would be possible to establish a "natural" correspondence between abelian varieties and functions belonging to the Selberg class S in such a way: 1) One associates to a simple abelian variety a primitive function of S, 2) One associates to an abelian variety of dimension d a function of S of degree d, 3) If V is an abelian variety isogenous to a product of abelian varieties of lower dimensions V_1, V_2, ... V_n, then the function F of S related to V is the product of the F_i where F_i is the function of S related to V_i.</p> <p>Thank you in advance.</p> http://mathoverflow.net/questions/49284/abelian-varieties-and-selberg-class/58362#58362 Answer by Denis Chaperon de LauziÃ¨res for Abelian varieties and Selberg class Denis Chaperon de LauziÃ¨res 2011-03-13T21:02:38Z 2011-03-13T21:02:38Z <p>(Assuming the abelian varieties are supposed to be defined over $\mathbf{Q}$). For every dimension $d\geq 1$, select a cuspidal automorphic representation $f_d$ of $GL(d)$ over the rationals. Then map any simple $A$ of dimension $d$ to $L(f_d,s)$, and extend by multiplicativity using the simple factors of a general $A$ up to isogeny.</p> <p>(Of course this question is utterly artificial; it seems reasonable to think that the only natural $L$-function associated to an abelian variety is its Hasse-Weil $L$-function, which has degree $2\dim(A)$.)</p>