Watkins’s conjecture for quadratic twists of Elliptic Curves with Prime Power Conductor
Jerson Caro1
1 Department of Mathematics & Statistics, Boston University, 665 Commonwealth Avenue, Boston, MA 02215, USA
Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 2, pp. 647-663

Watkins’s conjecture asserts that the rank of an elliptic curve is upper bounded by the $2$-adic valuation of its modular degree. We show that this conjecture is satisfied when $E$ is any quadratic twist of an elliptic curve with a rational point of order $2$ 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 $y^2=x^3-dx$, with $d$ a fourth power free integer.

La conjecture de Watkins affirme que le rang d’une courbe elliptique est borné par la valuation $2$-adique de son degré modulaire. Nous montrons que cette conjecture est vraie lorsque $E$ est une tordue quadratique d’une courbe elliptique avec un point rationnel d’ordre $2$ et de conducteur une puissance de nombre premier, en particulier pour les courbes elliptiques associées aux nombres congruents. De plus, nous donnons une borne inférieure pour le congruence number des courbes elliptiques de la forme $y^2=x^3-dx$, où $d$ est un entier sans facteur biquadratique.

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DOI : 10.5802/jtnb.1335
Classification : 11G05, 11G18, 11G40
Keywords: Watkins’s conjecture, elliptic curves, modular degree, Mordell–Weil rank

Jerson Caro 1

1 Department of Mathematics & Statistics, Boston University, 665 Commonwealth Avenue, Boston, MA 02215, USA
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Jerson Caro. Watkins’s conjecture for quadratic twists of Elliptic Curves with Prime Power Conductor. Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 2, pp. 647-663. doi: 10.5802/jtnb.1335

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