Degree conjecture and automorphic L-functions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T03:14:13Z http://mathoverflow.net/feeds/question/81384 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/81384/degree-conjecture-and-automorphic-l-functions Degree conjecture and automorphic L-functions Sylvain JULIEN 2011-11-19T23:10:23Z 2011-11-21T22:36:35Z <p>Hello,</p> <p>this may be a very naive question, but has the degree conjecture (namely "the degree of any function of the Selberg class is a non negative integer") been proven for automorphic L-functions? Thank you in advance.</p> http://mathoverflow.net/questions/81384/degree-conjecture-and-automorphic-l-functions/81387#81387 Answer by Stefano V. for Degree conjecture and automorphic L-functions Stefano V. 2011-11-20T00:14:58Z 2011-11-20T07:12:28Z <p>To the best of my knowledge, we are very far from proving such a conjecture. If $\mathcal S_d$ is the subclass of the Selberg class $\mathcal S$ consisting of the functions of degree $d$ then it is known that</p> <p>1) $\mathcal S_0={1}$ (Conrey-Ghosh, 1993);</p> <p>2) $\mathcal S_d=\emptyset$ for $0\lt d\lt1$ (Richert, 1957 and others);</p> <p>3) $\mathcal S_1$ consists of the Riemann zeta function $\zeta(s)$ and the shifted Dirichlet $L$-functions $L(s+i\tau,\chi)$ with $\tau\in\mathbb R$ and $\chi$ a primitive character (Kaczorowski-Perelli, 1999);</p> <p>4) $\mathcal S_d=\emptyset$ for $1\lt d\lt2$ (Kaczorowski-Perelli, 2002 and 2011).</p> <p>Apart from these results, I think that nothing has been established in general. A nice survey of the results obtained so far can be found in the introduction to</p> <p>J. Kaczorowski, A. Perelli, "On the structure of the Selberg class, VII: $1\lt d\lt2$", <em>Ann. of Math. (2)</em> <strong>173</strong> (2011), 1397-1441.</p> <p>Note that, in fact, Kaczorowski and Perelli prove their results for functions in the so-called <em>extended Selberg class</em> $\mathcal S^\sharp$, whose elements are not required to satisfy the Ramanujan hypothesis and the Euler product property. </p> http://mathoverflow.net/questions/81384/degree-conjecture-and-automorphic-l-functions/81401#81401 Answer by BR for Degree conjecture and automorphic L-functions BR 2011-11-20T06:53:39Z 2011-11-21T22:36:35Z <p>Going off <a href="http://en.wikipedia.org/wiki/Selberg_class#Basic_properties" rel="nofollow">wikipedia</a>, it is true that automorphic $L$-functions for $GL_n$ over a number field have non-negative integral degree, where by degree I mean the number $2\sum_{i=1}^k \omega_i$, where the $\omega_i$ are the coefficients of $s$ appearing in the gamma factor, which is more-or-less $$L_\infty(s,F)=Q^s\prod_{i=1}^k\Gamma(\omega_i s+\mu_i)$$ We know that for $GL_n(\mathbb R)$ and $GL_n(\mathbb C)$ the $\omega_i$ are either $1/2$ or $1$ (see, e.g., Knapp's "Local Langlands Correspondence: The Archimedean case", in Motives, vol 2), so twice the sum will always be an integer (of course, only a few of these $L$-functions are known to be in Selberg's class).</p> <p>For general $G$, it depends on whether someone has written done the $L$-factors for general real reductive groups. I don't know if this has been done, or if it is technically known but difficult to write out, or if it is not known.</p>