A conjecture of Watkins for quadratic twists
HTML articles powered by AMS MathViewer
- by Jose A. Esparza-Lozano and Hector Pasten PDF
- Proc. Amer. Math. Soc. 149 (2021), 2381-2385 Request permission
Abstract:
Watkins conjectured that for an elliptic curve $E$ over $\mathbb {Q}$ of Mordell-Weil rank $r$, the modular degree of $E$ is divisible by $2^r$. If $E$ has non-trivial rational $2$-torsion, we prove the conjecture for all the quadratic twists of $E$ by squarefree integers with sufficiently many prime factors.References
- 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
- Amod Agashe, Kenneth Ribet, and William A. Stein, The Manin constant, Pure Appl. Math. Q. 2 (2006), no. 2, Special Issue: In honor of John H. Coates., 617–636. MR 2251484, DOI 10.4310/PAMQ.2006.v2.n2.a11
- Ahmed Abbes and Emmanuel Ullmo, À propos de la conjecture de Manin pour les courbes elliptiques modulaires, Compositio Math. 103 (1996), no. 3, 269–286 (French, with French summary). MR 1414591
- 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
- Kęstutis Česnavičius, The Manin constant in the semistable case, Compos. Math. 154 (2018), no. 9, 1889–1920. MR 3867287, DOI 10.1112/s0010437x18007273
- Christophe Delaunay, Computing modular degrees using $L$-functions, J. Théor. Nombres Bordeaux 15 (2003), no. 3, 673–682 (English, with English and French summaries). MR 2142230
- 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
- Bas Edixhoven, On the Manin constants of modular elliptic curves, Arithmetic algebraic geometry (Texel, 1989) Progr. Math., vol. 89, Birkhäuser Boston, Boston, MA, 1991, pp. 25–39. MR 1085254, DOI 10.1007/978-1-4612-0457-2_{3}
- G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349–366 (German). MR 718935, DOI 10.1007/BF01388432
- 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
- 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, 2020, www.lmfdb.org
- Ju. I. Manin, Cyclotomic fields and modular curves, Uspehi Mat. Nauk 26 (1971), no. 6(162), 7–71 (Russian). MR 0401653
- B. Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 (1978), no. 2, 129–162. MR 482230, DOI 10.1007/BF01390348
- 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
- Vivek Pal, Periods of quadratic twists of elliptic curves, Proc. Amer. Math. Soc. 140 (2012), no. 5, 1513–1525. With an appendix by Amod Agashe. MR 2869136, DOI 10.1090/S0002-9939-2011-11014-1
- Joseph H. Silverman, Heights and elliptic curves, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 253–265. MR 861979
- 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
- L. Szpiro, Discriminant et conducteur des courbes elliptiques, Astérisque 183 (1990), 7–18 (French). Séminaire sur les Pinceaux de Courbes Elliptiques (Paris, 1988). MR 1065151
- 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
- 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
Additional Information
- Jose A. Esparza-Lozano
- Affiliation: African Institute for Mathematical Sciences, Rue KG590 ST, Kigali, Rwanda
- Address at time of publication: Department of Mathematics, University of Michigan, Ann Arbor, Michigan
- ORCID: 0000-0001-6555-2189
- Email: josealanesparza@gmail.com
- Hector Pasten
- Affiliation: Pontificia Universidad Católica de Chile, Facultad de Matemáticas, 4860 Av. Vicuña Mackenna, Macul, RM, Chile
- MR Author ID: 891758
- Email: hector.pasten@mat.uc.cl
- Received by editor(s): March 25, 2020
- Received by editor(s) in revised form: October 13, 2020
- Published electronically: March 25, 2021
- Additional Notes: The first author was supported by a Carroll L. Wilson Award and the MIT International Science and Technology Initiatives (MISTI) during an academic visit to Pontificia Universidad Católica de Chile.
The second author was supported by FONDECYT Regular grant 1190442. - Communicated by: Matthew Papanikolas
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 149 (2021), 2381-2385
- MSC (2020): Primary 11G05; Secondary 11F11, 11G18
- DOI: https://doi.org/10.1090/proc/15376
- MathSciNet review: 4246791