Watkins’s conjecture for elliptic curves with non-split multiplicative reduction

Caro, Jerson; Pasten, Hector

doi:10.1090/proc/15942

Watkins’s conjecture for elliptic curves with non-split multiplicative reduction
HTML articles powered by AMS MathViewer

by Jerson Caro and Hector Pasten PDF

Proc. Amer. Math. Soc. 150 (2022), 3245-3251 Request permission

Abstract:

Let $E$ be an elliptic curve over the rational numbers. Watkins [Experiment. Math. 11 (2002), pp. 487–502 (2003)] conjectured that the rank of $E$ is bounded by the $2$-adic valuation of the modular degree of $E$. We prove this conjecture for semistable elliptic curves having exactly one rational point of order $2$, provided that they have an odd number of primes of non-split multiplicative reduction or no primes of split multiplicative reduction.

References

Amod Agashe, Kenneth A. Ribet, and William A. Stein, The modular degree, congruence primes, and multiplicity one, Number theory, analysis and geometry, Springer, New York, 2012, pp. 19–49. MR 2867910, DOI 10.1007/978-1-4614-1260-1_{2}
Julián Aguirre, Álvaro Lozano-Robledo, and Juan Carlos Peral, Elliptic curves of maximal rank, Proceedings of the “Segundas Jornadas de Teoría de Números”, Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoamericana, Madrid, 2008, pp. 1–28. MR 2603895
A. O. L. Atkin and J. Lehner, Hecke operators on $\Gamma _{0}(m)$, Math. Ann. 185 (1970), 134–160. MR 268123, DOI 10.1007/BF01359701
Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, On the modularity of elliptic curves over $\mathbf Q$: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843–939. MR 1839918, DOI 10.1090/S0894-0347-01-00370-8
Frank Calegari and Matthew Emerton, Elliptic curves of odd modular degree, Israel J. Math. 169 (2009), 417–444. MR 2460912, DOI 10.1007/s11856-009-0017-x
Alina Carmen Cojocaru and Ernst Kani, The modular degree and the congruence number of a weight 2 cusp form, Acta Arith. 114 (2004), no. 2, 159–167. MR 2068855, DOI 10.4064/aa114-2-5
Neil Dummigan, On a conjecture of Watkins, J. Théor. Nombres Bordeaux 18 (2006), no. 2, 345–355 (English, with English and French summaries). MR 2289428, DOI 10.5802/jtnb.548
Neil Dummigan and Srilakshmi Krishnamoorthy, Powers of 2 in modular degrees of modular abelian varieties, J. Number Theory 133 (2013), no. 2, 501–522. MR 2994371, DOI 10.1016/j.jnt.2012.08.024
Jose A. Esparza-Lozano and Hector Pasten, A conjecture of Watkins for quadratic twists, Proc. Amer. Math. Soc. 149 (2021), no. 6, 2381–2385. MR 4246791, DOI 10.1090/proc/15376
Gerhard Frey, Links between solutions of $A-B=C$ and elliptic curves, Number theory (Ulm, 1987) Lecture Notes in Math., vol. 1380, Springer, New York, 1989, pp. 31–62. MR 1009792, DOI 10.1007/BFb0086544
Dale Husemöller, Elliptic curves, 2nd ed., Graduate Texts in Mathematics, vol. 111, Springer-Verlag, New York, 2004. With appendices by Otto Forster, Ruth Lawrence and Stefan Theisen. MR 2024529
Matija Kazalicki and Daniel Kohen, On a special case of Watkins’ conjecture, Proc. Amer. Math. Soc. 146 (2018), no. 2, 541–545. MR 3731689, DOI 10.1090/proc/13759
Matija Kazalicki and Daniel Kohen, Corrigendum to “On a special case of Watkins’ conjecture”, Proc. Amer. Math. Soc. 147 (2019), no. 10, 4563. MR 4002564, DOI 10.1090/proc/14456
The LMFDB Collaboration, The L-functions and modular forms database. 2021, www.lmfdb.org
P. Monsky, Generalizing the Birch-Stephens theorem. I. Modular curves, Math. Z. 221 (1996), no. 3, 415–420. MR 1381589, DOI 10.1007/PL00004518
M. Ram Murty, Bounds for congruence primes, Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996) Proc. Sympos. Pure Math., vol. 66, Amer. Math. Soc., Providence, RI, 1999, pp. 177–192. MR 1703750
Victor V. Prasolov, Polynomials, Algorithms and Computation in Mathematics, vol. 11, Springer-Verlag, Berlin, 2004. Translated from the 2001 Russian second edition by Dimitry Leites. MR 2082772, DOI 10.1007/978-3-642-03980-5
Edward F. Schaefer and Michael Stoll, How to do a $p$-descent on an elliptic curve, Trans. Amer. Math. Soc. 356 (2004), no. 3, 1209–1231. MR 2021618, DOI 10.1090/S0002-9947-03-03366-X
Bennett Setzer, Elliptic curves of prime conductor, J. London Math. Soc. (2) 10 (1975), 367–378. MR 371904, DOI 10.1112/jlms/s2-10.3.367
Joseph H. Silverman and John T. Tate, Rational points on elliptic curves, 2nd ed., Undergraduate Texts in Mathematics, Springer, Cham, 2015. MR 3363545, DOI 10.1007/978-3-319-18588-0
Joseph H. Silverman, The arithmetic of elliptic curves, 2nd ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009. MR 2514094, DOI 10.1007/978-0-387-09494-6
William Stein and Mark Watkins, Modular parametrizations of Neumann-Setzer elliptic curves, Int. Math. Res. Not. 27 (2004), 1395–1405. MR 2052021, DOI 10.1155/S1073792804133916
Richard Taylor and Andrew Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553–572. MR 1333036, DOI 10.2307/2118560
Mark Watkins, Computing the modular degree of an elliptic curve, Experiment. Math. 11 (2002), no. 4, 487–502 (2003). MR 1969641, DOI 10.1080/10586458.2002.10504701
Andrew Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551. MR 1333035, DOI 10.2307/2118559
Soroosh Yazdani, Modular abelian varieties of odd modular degree, Algebra Number Theory 5 (2011), no. 1, 37–62. MR 2833784, DOI 10.2140/ant.2011.5.37
D. Zagier, Modular parametrizations of elliptic curves, Canad. Math. Bull. 28 (1985), no. 3, 372–384. MR 790959, DOI 10.4153/CMB-1985-044-8