Torsion points of elliptic curves with good reduction

Abstract

We consider the torsion points of elliptc curves over certain number fields with good reduction everywhere.1

Points of order $13$ on elliptic curves

“…We will now describe how to obtain a model of the above form. Let F=Q[x, y]/(y 2 − f ) be the function field of X 1 (13) and let e 0 denote the hyperelliptic involution of F . The fixed field of e 0 is generated by the image of x in F. An elliptic involution of G = Aut(F ) is an involution different from e 0 [12].…”

“…If E has (potentially) multiplicative reduction at p then x must reduce (mod p) to a cusp Q of X 1 (13). The class of (x − Q) is a K-rational point T on J 1 (13), and hence is torsion. The point T generates a finite flat subgroup scheme C of J.…”

“…Because C is étale it must already be 0 in characteristic 0, i.e. (x − Q) is the divisor of a function on the genus 2 curve X 1 (13). This is clearly impossible.…”

“…Thus, E has good reduction at all primes of residue characteristic different from 13, and at the prime above 13 E has good reduction or potentially good reduction. If E has good reduction at 13 then it cannot exist by [13]. If E has potentially good reduction then we use Cremona's algorithm (which we could also use in the case of good reduction) to find the finite set of possible E, and check that none of them possess a K-rational point of order 13.…”

“…Here we return to the prime 13, the original subject of [9]. The modular curve X 1 (13), whose non-cuspidal points classify isomorphism classes of elliptic curves with a rational torsion point of order 13, has genus 2 so there are infinitely many elliptic curves with a rational point of order 13 defined over quadratic number fields. Bosman, Bruin, Dujella, and Najman [2] have shown that any such curve must be defined over a real quadratic field.…”