Galois Representations with Non-Surjective Traces

David, Chantal ; Kisilevsky, Hershy ; Pappalardi, Francesco

Canadian journal of mathematics, Tome 51 (1999) no. 5, pp. 936-951

Résumé

Let $E$ be an elliptic curve over $\mathbb{Q}$ , and let $r$ be an integer. According to the Lang-Trotter conjecture, the number of primes $p$ such that ${{a}_{p}}\left( E \right)=r$ is either finite, or is asymptotic to ${{C}_{E,r}}\sqrt{x}/\log x$ where ${{C}_{E,r}}$ is a non-zero constant. A typical example of the former is the case of rational $\ell $ -torsion, where ${{a}_{p}}\left( E \right)=r$ is impossible if $r\equiv 1\,\left( \bmod \,\ell\right)$ . We prove in this paper that, when $E$ has a rational $\ell $ -isogeny and $\ell \ne 11$ , the number of primes $p$ such that ${{a}_{p}}\left( E \right)\equiv r\,\left( \bmod \,\ell\right)$ is finite (for some $r$ modulo $\ell $ ) if and only if $E$ has rational $\ell $ -torsion over the cyclotomic field $\mathbb{Q}\left( {{\zeta }_{\ell }} \right)$ . The case $\ell =11$ is special, and is also treated in the paper. We also classify all those occurences.

Détail

DOI : 10.4153/CJM-1999-041-0

Mots-clés : 14H52

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