Some bounds on the coefficients of covering curves

Abstract

We compute bounds on the coefficients of the equations defining everywhere locally soluble $n$-coverings of elliptic curves over the rationals for $n$ = 2,3,4. Our proofs use recent work of the author with Cremona and Stoll on the minimisation of genus one curves, together with standard results from the geometry of numbers. We use the same methods to give a criterion (satisfied by only a finite number of "small'' elliptic curves) for ruling out the existence of elements of order $3$ in the Tate-Shafarevich group.