Given a Hecke operator which acts on the space of modular forms for $\mathrm{SL}_2(\mathbb{Z})$, are the eigenvalues necessarily distinct?
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4$\begingroup$ I would also like to know if there is a modular form for $\mathrm{SL}_2(\mathbb{Z})$ with a vanishing Hecke eigenvalue. $\endgroup$– GH from MOCommented Aug 28, 2012 at 14:13
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2$\begingroup$ Just to clarify, are you asking if a particular Hecke operator $T_\ell$ acting on the space $M_k(\mathrm{SL}_2(\mathbb{Z}))$ of forms of fixed weight $k$ for the full modular group has distinct eigenvalues (i.e. square-free characteristic polynomial)? $\endgroup$– RamseyCommented Aug 28, 2012 at 14:28
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1$\begingroup$ @Will: That bound is right in weight $2$, but in general the bound for weight $k$ is $2p^{(k-1)/2}$. $\endgroup$– RamseyCommented Aug 28, 2012 at 14:45
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6$\begingroup$ It's conjectured (Maeda's conjecture) that for any prime $p$ and any even weight $k \ge 12$, the characteristic polynomial of $T_p$ acting on $S_k(\operatorname{SL}_2(\mathbb{Z}))$ is irreducible over $\mathbb{Q}$; in particular, it has no multiple roots. Various authors (eg Buzzard, Kleinerman) have verified this for lots of values of $p$ and $k$. If I remember correctly, in all cases that have been tested the stronger statement holds that the char polys all have Galois group equal to the full symmetric group of the appropriate degree. $\endgroup$– David LoefflerCommented Aug 28, 2012 at 15:22
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1$\begingroup$ The way I understand the question: is it true that for any Hecke eigenform $f\in S_k(\operatorname{SL}_2(\mathbb{Z}))$ the Hecke eigenvalues $\lambda_f(p)$, normalized so that they lie between $\pm 2p^{(k-1)/2}$, are all distinct? $\endgroup$– GH from MOCommented Aug 28, 2012 at 15:30
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1 Answer
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Let $f(z)=\sum_{n\geq1}a_f(n)e^{2\pi i n z}$ be a holomorphic cuspidal Hecke eigenform of even weight $k\geq 12$ on the full modular group with $a_f(1)=1$, with $|a_f(p)|\leq 2p^{\frac{k-1}{2}}$. A conjecture of Atkin and Serre proposes that for every prime $p$,
$|a_f(p)|\gg_{\epsilon} p^{\frac{k-3}{2}-\epsilon}$,
from which it would follow by multiplicativity that the map $n\mapsto a_f(n)$ is finite-to-1. In this direction, Murty and Murty (http://www.numdam.org/article/BSMF_1987__115__391_0.pdf) proved that there exists an effective constant $c>0$ such that $|a_f(n)|>(\log n)^c$ when $a_f(n)$ is odd. Their result also extends to $f$ on congruence subgroups. Assuming the generalized Riemann hypothesis for Hecke $L$-functions, one can replace $(\log n)^c$ with $n^c$ on a subset of the integers of natural density 1.
answered Oct 23, 2017 at 13:40