JournalslemVol. 58, No. 1/2pp. 99–124

Some bounds on the coefficients of covering curves

  • Tom Fisher

    University of Cambridge, United Kingdom
Some bounds on the coefficients of covering curves  cover

Abstract

We compute bounds on the coefficients of the equations defining everywhere locally soluble nn-coverings of elliptic curves over the rationals for nn = 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 33 in the Tate-Shafarevich group.

Cite this article

Tom Fisher, Some bounds on the coefficients of covering curves . Enseign. Math. 58 (2012), no. 1, pp. 99–124

DOI 10.4171/LEM/58-1-4