Rank vs. Modular degree for Elliptic curves

3:00 pm, Wednesday, 6th November

Abstract

Watkins conjectured that the rank of an elliptic curve over the rationals is bounded by the 2-adic valuation of its modular degree. Under GRH, we show that in almost all cases the gap between this valuation and the rank grows as fast as the double logarithm of the curve's height. The same result holds under BSD and GRH for curves with a rational 2-torsion. Additionally, we discuss some unconditional results using the classical descent via 2-isogeny.