论文标题
关于算术曲线量降低算术曲线的无效性
On the acyclicity of reductions of elliptic curves modulo primes in arithmetic progressions
论文作者
论文摘要
令$ e $为$ \ mathbb {q} $定义的椭圆曲线，对于$ e $ $ e $的prime $ p $，让$ \ tilde {e} _p $表示$ e $ e $ e $ modulo $ p $。受Artin原始根构想的椭圆曲线类似物的启发，S。Lang和H. Trotter于1977年，J-P。 C. Hooley的塞雷改编方法证明了用于Primes $ p \ leq x $的GRH条件渐近公式，该组$ \ tilde {e} _p（\ Mathbb {f} _p）$是循环的。最近，Akbal和Gülo$ \ breve {\ text {g}} $ lu考虑了$ \ tilde {e} _p（\ mathbb {f} _p）$在额外限制下$ p $在arithmetic进程中的环环问题。在本说明中，我们研究了哪个问题的问题$ a \ bmod n $具有一个属性，对于几乎有限的许多素数$ p \ equiv a \ bmod n $，$ \ tilde {e} _p（\ m athbb {f} Gülo$ \ breve {\ text {g}} $ lu在此问题上。
Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and, for a prime $p$ of good reduction for $E$ let $\tilde{E}_p$ denote the reduction of $E$ modulo $p$. Inspired by an elliptic curve analogue of Artin's primitive root conjecture posed by S. Lang and H. Trotter in 1977, J-P. Serre adapted methods of C. Hooley to prove a GRH-conditional asymptotic formula for the number of primes $p \leq x$ for which the group $\tilde{E}_p(\mathbb{F}_p)$ is cyclic. More recently, Akbal and Gülo$\breve{\text{g}}$lu considered the question of cyclicity of $\tilde{E}_p(\mathbb{F}_p)$ under the additional restriction that $p$ lie in an arithmetic progression. In this note, we study the issue of which arithmetic progressions $a \bmod n$ have the property that, for all but finitely many primes $p \equiv a \bmod n$, the group $\tilde{E}_p(\mathbb{F}_p)$ is not cyclic, answering a question of Akbal and Gülo$\breve{\text{g}}$lu on this issue.
