A conjecture of Watkins for quadratic twists
aa r X i v : . [ m a t h . N T ] F e b A CONJECTURE OF WATKINS FOR QUADRATIC TWISTS
JOSE A. ESPARZA-LOZANO AND HECTOR PASTEN
Abstract.
Watkins conjectured that for an elliptic curve E over Q of Mordell-Weil rank r , themodular degree of E is divisible by 2 r . If E has non-trivial rational 2-torsion, we prove the conjecturefor all the quadratic twists of E by squarefree integers with sufficiently many prime factors. Ranks and modular degree
For an elliptic curve E over Q of conductor N , the modularity theorem [25, 23, 4] gives a non-constant morphism φ E : X ( N ) → E defined over Q where X ( N ) is the modular curve associatedto the congruence subgroup Γ ( N ) ⊆ SL ( Z ). We assume that φ E has minimal degree and that itmaps the cusp i ∞ to the neutral point of E . These requirements uniquely determine φ E up to sign.The modular degree of E is m E = deg φ E and it has profound arithmetic relevance; for instance,polynomial bounds for its size in terms of N are essentially equivalent to the abc conjecture [11, 17].The 2-adic valuation is denoted by v . Motivated by numerical data, Watkins [24] conjecturedthat v ( m E ) for an elliptic curve E is closely related to the Mordell-Weil rank of E over Q . Conjecture 1.1 (Watkins) . For every elliptic curve E over Q we have rank E ( Q ) ≤ v ( m E ) . Dummigan [8] showed that part of the conjecture would follow from strong R = T conjectures.Also, large part of Watkins’ conjecture is proved for elliptic curves of odd modular degree [5, 26,12, 13], although it is not known whether there exist infinitely many elliptic curves of this kind [21].The goal of this note is to prove Watkins’ conjecture unconditionally in several new cases. Letus introduce some notation. For an elliptic curve E and a fundamental (quadratic) discriminant D , the quadratic twist of E by D is denoted by E ( D ) . The Manin constant of E is denoted by c E (cf. Section 2.3). The number of distinct prime factors of an integer n is ω ( n ). Theorem 1.2.
Let E be an elliptic curve over Q of conductor N with non-trivial rational -torsion. Assume that E has minimal conductor among its quadratic twists. If D is a fundamentaldiscriminant with ω ( D ) ≥ ω ( N ) − v ( m E /c E ) , then Watkins’ conjecture holds for E ( D ) . The quantity 6 + 5 ω ( N ) − v ( m E /c E ) is effectively computable and it can be read from existingtables of elliptic curves when N is not too large, see for instance [14].For a positive integer A , it is a standard result of analytic number theory that the number ofpositive integers n up to x having ω ( n ) ≤ A is O ( x (log log x ) A − / log x ). We deduce: Corollary 1.3.
Let E be an elliptic curve over Q with non-trivial rational -torsion. There is aneffective constant κ ( E ) depending only on E such that the number of fundamental discriminants D with | D | ≤ x such that Watkins’ conjecture fails for E ( D ) is bounded by O (cid:0) x (log log x ) κ ( E ) / log x (cid:1) . Let us remark that in the cases where we prove Watkins’ conjecture our argument actually showsthat v ( m E ( D ) ) bounds the 2-Selmer rank, which is a stronger version of Watkins’ conjecture. Date : February 9, 2021.2010
Mathematics Subject Classification.
Primary 11G05; Secondary 11F11, 11G18.
Key words and phrases.
Elliptic curve, rank, modularity.J. E.-L. was supported by a Carroll L. Wilson Award and the MIT International Science and Technology Initiatives(MISTI). H. P. was supported by FONDECYT Regular grant 1190442. . Preliminaries
Faltings height.
Let E be an elliptic curve over Q . We denote by ω E a global Neron dif-ferential for E ; it is unique up to sign. The Faltings height of E (over Q ) is defined as certainArakelov degree [10], which in our case takes the simpler form [19](2.1) h ( E ) = −
12 log i Z E ( C ) ω E ∧ ω E ! . Ramanujan’s cusp form is ∆( z ) = q Q ∞ n =1 (1 − q n ) where q = exp(2 πiz ), defined on the upperhalf plane h = { z ∈ C : ℑ ( z ) > } . The modular j -function is normalized as j ( z ) = q − + 744 + ... The global minimal discriminant of E is denoted by ∆ E . If τ E ∈ h satisfies that j ( τ E ) is the j -invariant of E , then the Faltings height admits the expression [22, 19](2.2) h ( E ) = 112 (cid:0) log | ∆ E | − log (cid:12)(cid:12) ∆( τ E ) ℑ ( τ E ) (cid:12)(cid:12)(cid:1) − log(2 π ) . Given elliptic curves E , E over Q , let us define δ ( E , E ) = exp(2 h ( E ) − h ( E )). Lemma 2.1 (Variation of h ( E ) under quadratic twist) . Let E be an elliptic curve over Q and let E be a quadratic twist of E . Then δ ( E , E ) is a rational number and it satisfies | v ( δ ( E , E )) | ≤ .Proof. We use (2.2) for both E and E . The elliptic curves are isomorphic over C , so we can take τ E = τ E which gives δ ( E , E ) = | ∆ E / ∆ E | / . The result follows from explicit formulas for thevariation of the minimal discriminant under quadratic twists, cf. Proposition 2.4 in [18]. (cid:3) Petersson norm.
For a positive integer N , let S ( N ) be the space of weight 2 cuspidalholomorphic modular forms for the congruence subgroup Γ ( N ) acting on h . Given f ∈ S ( N ), itsFourier expansion is f ( z ) = a ( f ) q + a ( f ) q + ... where q = exp(2 πiz ) and the numbers a n ( f ) arethe Fourier coefficients of f . The Petersson norm of f relative to Γ ( N ) is defined by k f k N = Z Γ ( N ) \ h | f ( z ) | dx ∧ dy ! / , z = x + iy ∈ h . The norm depends on the choice of N in the following sense: If N | M and f ∈ S ( N ), then wecertainly have f ∈ S ( M ), and k f k M = [Γ ( N ) : Γ ( M )] · k f k N .We need some additional notation. For an elliptic curve E over Q of conductor N we denoteby f E ∈ S ( N ) the Hecke newform attached to E by the modularity theorem, normalized by a ( f E ) = 1. The modular form f E is characterized by the following property: If p is a primeof good reduction for E and we define a p ( E ) = p + 1 − E ( F p ), then a p ( f E ) = a p ( E ). For afundamental discriminant D , let P ( D, N ) be the set of primes p with p | D and p ∤ N . Lemma 2.2 (Variation of the Petersson norm under quadratic twist) . Let E be an elliptic curveover Q and let D be a fundamental discriminant. Let N and N ( D ) be the conductors of E and E ( D ) respectively, and assume that N | N ( D ) . Then k f E ( D ) k N ( D ) / k f E k N ∈ Q × and we have v ( k f E ( D ) k N ( D ) / k f E k N ) + 1 ≥ X p ∈ P ( D,N ) v (( p − p + 1 − a p ( E ))( p + 1 + a p ( E ))) . Proof.
The quadratic Dirichlet character attached to D has conductor | D | . The result follows fromthe precise formula given in Theorem 1 of [7] when one only keeps the contribution of p = 2 andthe primes p ∈ P ( D, N ) —the product of the latter primes is denoted by D in loc. cit. (cid:3) We remark that the terms ( p − p + 1 − a p ( E ))( p + 1 + a p ( E )) have a clear conceptual origin;they come from Euler factors of the imprimitive symmetric square L -function L (Sym f E , s ) thatare removed by twisting, and L (Sym f E ,
2) is (up to a mild factor) equal to k f E k N . See [27, 7, 24]. .3. Manin constant.
Given an elliptic curve E over Q of conductor N , we have that φ ∗ E ω E is aregular differential on X ( N ) = Γ ( N ) \ h ∪ { cusps } . More precisely(2.3) φ ∗ E ω E = 2 πic E f E ( z ) dz where c E is a rational number uniquely defined up to sign. We assume that the signs of φ E and ω E are chosen such that c E >
0. It follows from (2.1) and (2.3) that (cf. [22, 19])(2.4) m E = 4 π c E k f E k N exp(2 h ( E )) . The quantity c E is called the Manin constant, and a fundamental fact is Lemma 2.3 (cf. [9]) . The Manin constant c E is an integer. We recall that Manin [15] conjectured that if E is a strong Weil curve in the sense that m E isminimal within the isogeny class of E , then c E = 1. See [16, 3, 2, 6] and the references therein.3. Consequences for Watkins’ conjecture
Lemma 3.1.
Let E be an elliptic curve over Q of conductor N and suppose that E has minimalconductor among its quadratic twists. Let D be a fundamental discriminant. Then v ( m E ( D ) ) ≥ v ( m E /c E ) − X p ∈ P ( D,N ) v (( p − p + 1 − a p ( E ))( p + 1 + a p ( E ))) . Proof.
Applying (2.4) to E and E ( D ) we find m E ( D ) m E = c E ( D ) c E · k f E ( D ) k N ( D ) k f E k N · δ ( E ( D ) , E ) . The result follows from lemmas 2.1, 2.2, and 2.3. (cid:3)
Proposition 3.2.
Let E be an elliptic curve over Q of conductor N with non-trivial rational -torsion and suppose that E has minimal conductor among its quadratic twists. Let D be a fun-damental discriminant. We have v ( m E ( D ) ) ≥ ω ( D ) + v ( m E /c E ) − (7 + 3 ω ( N )) .Proof. As E ( Q )[2] is non-trivial and it maps injectively into E ( F p ) for every prime p ∤ N , we have p + 1 ≡ a p ( E ) mod 2 for these primes. We get v ( m E ( D ) ) ≥ v ( m E /c E ) − · P ( D, N ) fromLemma 3.1, and the result follows from P ( D, N ) ≥ ω ( D ) − ω (2 N ) ≥ ω ( D ) − ω ( N ) − (cid:3) The following upper bound for the Mordell-Weil rank is standard and it comes from a bound fora 2-isogeny Selmer rank (cf. Section X.4 in [20]; see also [1]).
Lemma 3.3.
Let E be an elliptic curve over Q of conductor N with non-trivial rational -torsion.Then rank E ( Q ) ≤ ω ( N ) − .Proof of Theorem 1.2. Since E ( D ) [2] ≃ E [2] as Galois modules and E has non-trivial rational 2-torsion, we can use Lemma 3.3 for E ( D ) , which givesrank E ( D ) ( Q ) ≤ ω ( N ( D ) ) − ≤ ω ( D ) + ω ( N )) − . If Watkins’ conjecture fails for E ( D ) , then Proposition 3.2 would give2( ω ( D ) + ω ( N )) − ≥ v ( m E ( D ) ) + 1 ≥ ω ( D ) + v ( m E /c E ) − − ω ( N ) . This is not possible when ω ( D ) ≥ ω ( N ) − v ( m E /c E ). (cid:3) Acknowledgments
The first author was supported by a Carroll L. Wilson Award and the MIT International Scienceand Technology Initiatives (MISTI) during an academic visit to Pontificia Universidad Cat´olica deChile. The second author was supported by FONDECYT Regular grant 1190442. eferences [1] J. Aguirre, A. Lozano-Robledo, J. Peral, Elliptic curves of maximal rank . Proceedings of the ”Segundas Jornadasde Teor´ıa de N´umeros”, 1-28, Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoamericana, Madrid, 2008.[2] A. Agashe, K. Ribet, W. Stein,
The Manin constant . Pure Appl. Math. Q. 2 (2006), no. 2, Special Issue: In honorof John H. Coates. Part 2, 617-636.[3] A. Abbes, E. Ullmo, `A propos de la conjecture de Manin pour les courbes elliptiques modulaires . Compositio Math.103 (1996), no. 3, 269-286.[4] C. Breuil, B. Conrad, F. Diamond, R. Taylor,
On the modularity of elliptic curves over Q: wild 3-adic exercises .J. Amer. Math. Soc. 14 (2001), no. 4, 843-939.[5] F. Calegari, M. Emerton,
Elliptic curves of odd modular degree . Israel J. Math. 169 (2009), 417-444.[6] K. Cesnavicius,
The Manin constant in the semistable case . Compos. Math. 154 (2018), no. 9, 1889-1920.[7] C. Delaunay,
Computing modular degrees using L-functions . J. Th´eor. Nombres Bordeaux 15 (2003), no. 3, 673-682.[8] N. Dummigan,
On a conjecture of Watkins.
J. Th´eor. Nombres Bordeaux 18 (2006), no. 2, 345-355.[9] B. Edixhoven,
On the Manin constants of modular elliptic curves . Arithmetic algebraic geometry (Texel, 1989),25-39, Progr. Math., 89, Birkh¨auser Boston, Boston, MA, 1991.[10] G. Faltings,
Endlichkeitss¨atze f¨ur abelsche Variet¨aten ¨uber Zahlk¨orpern . Invent. Math. 73 (1983), no. 3, 349-366.[11] G. Frey,
Links between solutions of A-B=C and elliptic curves . Number theory (Ulm, 1987), 31-62, Lecture Notesin Math., 1380, Springer, New York, 1989.[12] M. Kazalicki, D. Kohen,
On a special case of Watkins’ conjecture . Proc. Amer. Math. Soc. 146 (2018), no. 2,541-545.[13] M. Kazalicki, D. Kohen,
Corrigendum to ”On a special case of Watkins’ conjecture”.
Proc. Amer. Math. Soc.147 (2019), no. 10, 4563.[14] The LMFDB Collaboration,
The L-functions and Modular Forms Database . (2020) [15] J. Manin,
Cyclotomic fields and modular curves . (Russian) Uspehi Mat. Nauk 26 (1971), no. 6(162), 7-71.[16] B. Mazur,
Rational isogenies of prime degree (with an appendix by D. Goldfeld). Invent. Math. 44 (1978), no.2, 129-162.[17] R. Murty,
Bounds for congruence primes . Automorphic forms, automorphic representations, and arithmetic (FortWorth, TX, 1996), 177-192, Proc. Sympos. Pure Math., 66, Part 1, Amer. Math. Soc., Providence, RI, 1999.[18] V. Pal,
Periods of quadratic twists of elliptic curves . With an appendix by Amod Agashe. Proc. Amer. Math.Soc. 140 (2012), no. 5, 1513-1525.[19] J. Silverman,
Heights and elliptic curves . Arithmetic geometry (Storrs, Conn., 1984), 253-265, Springer, NewYork, 1986.[20] J. Silverman,
The arithmetic of elliptic curves . Second edition. Graduate Texts in Mathematics, 106. Springer,Dordrecht, 2009. xx+513 pp. ISBN: 978-0-387-09493-9.[21] W. Stein, M. Watkins,
Modular parametrizations of Neumann-Setzer elliptic curves . Int. Math. Res. Not. 2004,no. 27, 1395-1405.[22] L. Szpiro,
Discriminant et conducteur des courbes elliptiques . S´eminaire sur les Pinceaux de Courbes Elliptiques(Paris, 1988). Ast´erisque No. 183 (1990), 7-18.[23] R. Taylor, A. Wiles,
Ring-theoretic properties of certain Hecke algebras . Ann. of Math. (2) 141 (1995), no. 3,553-572.[24] M. Watkins,
Computing the modular degree of an elliptic curve . Experiment. Math. 11 (2002), no. 4, 487-502.[25] A. Wiles,
Modular elliptic curves and Fermat’s last theorem . Ann. of Math. (2) 141 (1995), no. 3, 443-551.[26] S. Yazdani,
Modular abelian varieties of odd modular degree . Algebra Number Theory 5 (2011), no. 1, 37-62.[27] D. Zagier,
Modular parametrizations of elliptic curves . Canad. Math. Bull. 28 (1985), no. 3, 372-384.
African Institute for Mathematical SciencesRue KG590 ST, Kigali, Rwanda
Current address : Department of MathematicsUniversity of Michigan, Ann Arbor, USA
Email address , J. Esparza-Lozano: [email protected]
Pontificia Universidad Cat´olica de ChileFacultad de Matem´aticas4860 Av. Vicu˜na Mackenna, Macul, RM, Chile
Email address , H. Pasten: [email protected]@mat.uc.cl