Timeline for Automorphisms of the Selberg class
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 8, 2011 at 6:42 | comment | added | Sylvain JULIEN | I was thinking of continuity since the map $f$ preserves the "inner product" 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... | |
| Feb 7, 2011 at 19:18 | comment | added | GH from MO | I don't think so. Automorphic L-functions occur rather discretely among all possible Dirichlet series. For example, there are only countably many cusp forms on SL(2,Z)\H. This is similar to the zeros of the Riemann zeta function: although we expect this set to be algebraically independent, it is still a God-given countable set. That's the beauty of it! | |
| Feb 7, 2011 at 17:57 | comment | added | Sylvain JULIEN | 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? | |
| Jan 31, 2011 at 9:45 | comment | added | GH from MO | I think this will be very hard to do. As far as I know we only have conjectures about the lack of algebraic relations between the Euler coefficients of a general automorphic L-function, e.g. an L-function associated to a Maass cusp form on SL(2,Z). | |
| Jan 30, 2011 at 20:36 | comment | added | Sylvain JULIEN | 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. | |
| Jan 22, 2011 at 17:43 | vote | accept | Sylvain JULIEN | ||
| Dec 30, 2010 at 19:04 | comment | added | GH from MO | You are right, this makes the dual $L$-function which enters the functional equation. So my guess is that this is the only nontrivial automorphism. My original answer tried to emphasize that most automorphic $L$-functions have highly transcendental Dirichlet coefficients, at least this is what we conjecture. Few automorphic $L$-functions (e.g. those of geometric type) have algebraic coefficients, but this is an exception. | |
| Dec 30, 2010 at 18:56 | comment | added | Sylvain JULIEN | I'm not sure to understand your answer, but isn't the complex conjugation such an automorphism of $\mathcal{S}$ ? | |
| Dec 30, 2010 at 18:38 | history | answered | GH from MO | CC BY-SA 2.5 |