Let $E$ be an elliptic curve over $q$, and let $r$ be an integer.According to the Lang-Trotter conjecture, the number of primes $p$such that $a_p(E) = r$ is either finite, or is asymptotic to$C_{E,r} {sqrt{x}} / {log{x}}$ where $C_{E,r}$ is a non-zeroconstant. A typical example of the former is the case of rational$ell$-torsion, where $a_p(E) = r$ is impossible if $r equiv 1pmod{ell}$. We prove in this paper that, when $E$ has a rational$ell$-isogeny and $ell eq 11$, the number of primes $p$ suchthat $a_p(E) equiv r pmod{ell}$ is finite (for some $r$ modulo$ell$) if and only if $E$ has rational $ell$-torsion over thecyclotomic field $q(zeta_ell)$. The case $ell=11$ is special,and is also treated in the paper. We also classify all thoseoccurences.
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机译：令$ E $为$ q $的椭圆曲线，令$ r $为整数。根据Lang-Trotter猜想，素数$ p的素数使得$ a_p（E）= r $是有限的，或渐近到$ C_ {E，r} {sqrt {x}} / {log {x}} $，其中$ C_ {E，r} $是非零常数。前者的一个典型示例是有理$ ell $扭转的情况，其中如果$ r等于1pmod {ell} $，则$ a_p（E）= r $是不可能的。我们在本文中证明，当$ E $具有有理$ ell $-同质性且$ ell eq 11 $时，素数$ p $使得$ a_p（E）等于pmod {ell} $是有限的（对于当且仅当$ E $在环原子场qq（zeta_ell）$上具有合理的$ ell-扭力时，才对某些$ r $进行模数化。 $ ell = 11 $的情况很特殊，在本文中也进行了处理。我们还将所有这些事件分类。
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