Watkins’s conjecture for quadratic twists of Elliptic Curves with Prime Power Conductor

Watkins’s conjecture asserts that the rank of an elliptic curve is upper bounded by the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>2</mml:mn> </mml:math> -adic valuation of its modular degree. We show that this conjecture is satisfied when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> </mml:math> is any quadratic twist of an elliptic curve with a rational point of order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>2</mml:mn> </mml:math> and prime power conductor, in particular, for the congruent number elliptic curves. Furthermore, we give a lower bound for the congruence number for elliptic curves of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>y</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>-</mml:mo> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:math> , with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>d</mml:mi> </mml:math> a fourth power free integer.