User sylvain julien - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T01:00:48Z http://mathoverflow.net/feeds/user/11542 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/50581/automorphisms-of-the-selberg-class Automorphisms of the Selberg class Sylvain JULIEN 2010-12-28T17:47:19Z 2011-03-13T23:31:26Z <p>Hello,</p> <p>assuming Selberg's orthonormality conjecture, let's consider bijective maps $f$ from Selberg's class $\mathcal{S}$ to itself such as:</p> <p>1) $f$ maps a primitive function of $\mathcal{S}$ to a primitive function of $\mathcal{S}$,</p> <p>2) $f$ maps a function of degree $d$ to a function of degree $d$,</p> <p>3) for every $F$ in $\mathcal{S}$, $n_{f(F)}=n_{F}$, where $n_{F}$ is the integer involved in Selberg's conjecture $A$,</p> <p>4) if $F=F_{1}^{e_{1}}.F_{2}^{e_{2}}....F_{k}^{e_{k}}$, then $f(F)=f(F_{1})^{e_{1}}.f(F_{2})^{e_{2}}....f(F_{k})^{e_{k}}$ (so that $f$ is "strongly multiplicative"),</p> <p>5) if $f$ verifies the above conditions, then so does the inverse of $f$.</p> <p>The set of all such maps makes a group for the composition law. I would like to know whether this group is isomorphic to the absolute Galois group of the field of the rational numbers or not.</p> <p>Thanks in advance.</p> http://mathoverflow.net/questions/52438/zeroes-of-complete-l-functions Zeroes of complete L-functions Sylvain JULIEN 2011-01-18T21:51:03Z 2011-01-19T18:15:25Z <p>Hello,</p> <p>Let $F$ and $G$ be two functions belonging in the Selberg class, $\Lambda_{F}$ and $\Lambda_{G}$ the complete L-functions associated to $F$ and $G$. I would like to know whether this assertion is true or not:</p> <p>"$\Lambda_{F}$ and $\Lambda_{G}$ have the same zeroes if and only if $F=G$ or $F=\overline{G}$."</p> <p>Thank you in advance.</p> http://mathoverflow.net/questions/50581/automorphisms-of-the-selberg-class/50748#50748 Comment by Sylvain JULIEN Sylvain JULIEN 2011-02-08T06:42:37Z 2011-02-08T06:42:37Z I was thinking of continuity since the map $f$ preserves the &quot;inner product&quot; defined for $(F,G)\in\mathcal{S}^{2}$ by $\displaystyle{\lim_{x\to\infty}\dfrac{1}{\log\log x}\sum_{p\leq x}\dfrac{a_{p}(F)\overline{a_{p}(G)}}{p}}$, so that $f$ appears to be some kind of an isometry of the Selberg class. And as generally speaking, an isometry is a continuous map... http://mathoverflow.net/questions/50581/automorphisms-of-the-selberg-class/50748#50748 Comment by Sylvain JULIEN Sylvain JULIEN 2011-02-07T17:57:46Z 2011-02-07T17:57:46Z But wouldn't it be possible to consider the conditions above as some kind of continuity assumptions that would make identity and complex conjugation the only possible maps such as condition 4) is true? A bit like the only continuous field automorphisms of $\mathbb{C}$ are namely the identity and the complex conjugation? http://mathoverflow.net/questions/50581/automorphisms-of-the-selberg-class/50748#50748 Comment by Sylvain JULIEN Sylvain JULIEN 2011-01-30T20:36:19Z 2011-01-30T20:36:19Z Could someone give a rigorous proof (always assuming Selberg's orthonormality conjecture) of the fact that there is no other automorphism than the identity and the complex conjugation? Thanks in advance. http://mathoverflow.net/questions/52438/zeroes-of-complete-l-functions/52537#52537 Comment by Sylvain JULIEN Sylvain JULIEN 2011-01-19T21:14:22Z 2011-01-19T21:14:22Z Thank you very much for this wonderful answer. http://mathoverflow.net/questions/52438/zeroes-of-complete-l-functions Comment by Sylvain JULIEN Sylvain JULIEN 2011-01-19T09:46:26Z 2011-01-19T09:46:26Z Zen, I see no typo: can't you see the overline on $G$ ? http://mathoverflow.net/questions/52438/zeroes-of-complete-l-functions Comment by Sylvain JULIEN Sylvain JULIEN 2011-01-18T22:15:49Z 2011-01-18T22:15:49Z When I write &quot;the same zeroes&quot;, I mean &quot;the same zeroes with the same multiplicity for $\lambda_{F}$ and $\Lambda_{G}$&quot;, so that I don't think it's necessary to assume $F$ and $G$ are primitive. http://mathoverflow.net/questions/50581/automorphisms-of-the-selberg-class Comment by Sylvain JULIEN Sylvain JULIEN 2011-01-15T12:13:54Z 2011-01-15T12:13:54Z Yes indeed, it is just the product of the functions $F_i$. http://mathoverflow.net/questions/50581/automorphisms-of-the-selberg-class/50748#50748 Comment by Sylvain JULIEN Sylvain JULIEN 2010-12-30T18:56:53Z 2010-12-30T18:56:53Z I'm not sure to understand your answer, but isn't the complex conjugation such an automorphism of $\mathcal{S}$ ? http://mathoverflow.net/questions/49284/abelian-varieties-and-selberg-class Comment by Sylvain JULIEN Sylvain JULIEN 2010-12-13T22:49:29Z 2010-12-13T22:49:29Z Indeed the word &quot;correspondence&quot; may not fit exactly what I have on my mind, but my English is far from being perfect (I'm French). I didn't know that there were uncountably many functions in S, would you have some reference? By the way, still assuming Selberg's orthonormality conjecture, do both abelian varieties and Selberg's class form semi-simple categories? If so, the concept of functor from abelian varieties into Selberg's class, as you suggested, may be the good way to express my idea.