Torsion and exceptional units
Streszczenie
Let $E/\mathbb Q$ be an elliptic curve which has everywhere semistable reduction. We first prove that if $E(\mathbb Q)_{\mathrm{tors}} $ contains an element of order $N\geq 3$, then there exists a prime $p$ for which $E/\mathbb Q$ has split multiplicative reduction modulo $p$, thereby establishing a conjecture of Agashe. We then consider generalizations over number fields inspired by this result. Fix a degree $d$, and consider all number fields $K$ of degree $d$. Fix a prime $N > 2d+1$, and consider all elliptic curves which have a $K$-rational torsion point of order $N$ and such that the Tamagawa number $c(E/K)$ is coprime to $N$. For $d=1,2,3$, we show that there exist only finitely many degree $d$ fields $K$ and finitely many such elliptic curves $E/K$. Partial results are also obtained for $d=4,5$, and we conjecture that the statement holds when $d=6,7$. Fields $K$ which support such elliptic curves are very structured, and we show in particular that their Lenstra constant $M(K)$ is bounded below by $(N-1)/2$ when $N\leq 23$. We conjecture that this statement holds for any prime $N$.
