Root numbers and parity of ranks of elliptic curves

Abstract

The purpose of the paper is to complete several global and local results concerning parity of ranks of elliptic curves. Primarily, we show that the Shafarevich-Tate conjecture implies the parity conjecture for all elliptic curves over number fields, give a formula for local and global root numbers of elliptic curves and complete the proof of a conjecture of Kramer and Tunnell in characteristic 0. The method is to settle the outstanding local formulae by deforming from local fields to totally real number fields and then using global parity results.

25 pages

References in corpus (7)

• Ranks of twists of elliptic curves and Hilbert's Tenth Problem
• Regulator constants and the parity conjecture
• On the Birch-Swinnerton-Dyer quotients modulo squares
• Parity of ranks for elliptic curves with a cyclic isogeny
• Self-duality of Selmer groups
• Root numbers of elliptic curves in residue characteristic 2
• Elliptic curves with all quadratic twists of positive rank

Cited by in corpus (9)

• Notes on the Parity Conjecture
• Hasse principle for generalised Kummer varieties
• Growth of Sha in towers for isogenous curves
• The 2-parity conjecture for elliptic curves with isomorphic 2-torsion
• Geometry of the del Pezzo surface y^2=x^3+Am^6+Bn^6
• Heegner points at Eisenstein primes and twists of elliptic curves
• A note on the Mordell-Weil rank modulo n
• Finite quotients of Z[C_n]-lattices and Tamagawa numbers of semistable abelian varieties
• On 2-Selmer ranks of quadratic twists of elliptic curves