modular form Fourier coefficients and associated automorphic representation - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T17:35:56Z http://mathoverflow.net/feeds/question/78954 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/78954/modular-form-fourier-coefficients-and-associated-automorphic-representation modular form Fourier coefficients and associated automorphic representation unknown 2011-10-24T05:01:53Z 2011-10-24T07:24:40Z <p>Hi,</p> <p>Let $f$ be a cuspidal modular form of some weight and level $N$. Then it determines an irreducible automorphic representation $\pi = \bigotimes'\pi_p$ of $GL_2(\mathbf Q)$. Let $f = \sum_i a_i q^i$ be its fourier expansion. Then it is known that if $p\nmid N$, then $a_p$ determines $\pi_p$ (it is an unramified principal series). Is it true that $a_p$ determines $\pi_p$ in general? And if so, how?</p> <p>Thanks!</p> http://mathoverflow.net/questions/78954/modular-form-fourier-coefficients-and-associated-automorphic-representation/78959#78959 Answer by Olivier for modular form Fourier coefficients and associated automorphic representation Olivier 2011-10-24T06:47:43Z 2011-10-24T06:47:43Z <p>No. </p> <p>A supercuspidal representation, a Steinberg twisted by a ramified character and a principal series ramified at both characters at $p$ will all have zero $a_{p}$. A reference for this is Jacquet-Langlands LN 114 Proposition 3.5, 3.6. I am also not sure one can distinguish a priori a simply ramified principal series and a Steinberg twisted by an unramified character purely using $a_{p}$, you might need to look at the order of the central character at $p$ to do this (in the latter case, the central character is trivial at $p$ while it is not in the former). </p> http://mathoverflow.net/questions/78954/modular-form-fourier-coefficients-and-associated-automorphic-representation/78962#78962 Answer by David Loeffler for modular form Fourier coefficients and associated automorphic representation David Loeffler 2011-10-24T07:24:40Z 2011-10-24T07:24:40Z <p>Jared Weinstein and I wrote a paper on how to compute $\pi_p$: see <a href="http://dx.doi.org/10.1090/S0025-5718-2011-02530-5%20" rel="nofollow">here</a>. </p> <p>As Olivier says, $a_p$ will often be zero, and in fact if the central character is trivial (or has conductor coprime to $p$) this is always the case when $p^2$ divides the level of $f$. One can get a bit futher by twisting: you can always twist a newform by Dirichlet characters, and Atkin and Li have shown that $\pi_p$ is principal series or Steinberg at $p$ if and only if there is some Dirichlet character $\chi$ such that the twist of $f$ by $\chi$ is a newform with nonzero Fourier coefficient at $p$ (or an oldform attached to such a newform).</p> <p>So that leaves the supercuspidal cases, and here Hecke theory won't help you at all: no matter how you twist your form, the Hecke eigenvalues are all zero. One can actually show (the "local converse theorem") that $\pi_p$ is uniquely determined by the Atkin-Lehner pseudo-eigenvalues of all of the twists of $f$; but it is not so easy to calculate these, or to explicitly identify $\pi_p$ from a list of them once you've done so. In my paper with Jared you can find details of a different approach, using Bushnell and Kutzko's theory of "types", which seems to work quite well.</p> <p>These algorithms are implemented in recent versions of Magma (and should be in Sage fairly shortly as well, once someone gets around to reviewing my code). </p>