Elliptic curves with all quadratic twists of positive rank
aa r X i v : . [ m a t h . N T ] J a n . ELLIPTIC CURVES WITH ALL QUADRATIC TWISTS OFPOSITIVE RANK
TIM AND VLADIMIR DOKCHITSER
Imagine you had an elliptic curve
E/K with everywhere good reduction, definedover a number field K that has no real and an odd number r of complex places. Thenthe global root number w ( E/K ) is ( − r = −
1, and it becomes ( − r = +1 overevery quadratic extension of K . As the root number is the sign in the (conjectural)functional equation for the L -function of E , the Birch–Swinnerton-Dyer conjecturepredicts that the Mordell-Weil rank of E goes up in every quadratic extension of K .Equivalently, every quadratic twist of E/K has positive rank, a behaviour that doesnot occur over Q (and would contradict Goldfeld’s “ / average rank” conjecture).These curves do exist . For example, the elliptic curve over Q E : y = x + x − x − − and acquires everywhere good reduction over any cubic ex-tension of Q which is totally ramified at 11. So one may take K = Q ( ζ , √ m )or K = Q ( √− m ) for any positive m , coprime to 11. (Those who prefer abelianextensions can take E = 1849C1 and K to be the degree 6 field inside Q ( ζ ).)It is even easier to construct curves all of whose quadratic twists have rootnumber +1. For example, E : y = x + x − x − (1369E1)has discriminant 37 and has everywhere good reduction over K = Q ( √− K . (Such a field K exists for everyelliptic curve with integral j -invariant.) In view of the Birch–Swinnerton-Dyerconjecture, we expect E to have even rank over every extension of K , but it is notat all clear how to prove it for this or any other non-CM elliptic curve .Let us say that an elliptic curve E/K is lawful if w ( E/K ′ ) = 1 for every quadraticextension K ′ /K , and chaotic otherwise. Equivalently, E is lawful if and only if allof its quadratic twists have the same root number as E/K . Depending on whetherthis root number is +1 or −
1, let us call the curve lawful good or lawful evil . Thus,conjecturally the rank of a lawful evil curve increases in every quadratic extension .Another way of looking at our first example is that, say, for K = Q ( √−
11) thepolynomial x + x − x − all values in K ∗ /K ∗ . Generally, one mightconjecture that a square-free cubic f ( x ) ∈ K [ x ] takes “0%”, “50%” or all possiblevalues in K ∗ /K ∗ depending on whether the curve y = f ( x ) is lawful good, chaoticor lawful evil over K . So Goldfeld’s conjecture fails over number fields. That there are curves all of whose quadratictwists must have positive rank was observed in [6] and is already implicit in [9]. We also get, viaWeil restriction, abelian varieties over Q all of whose quadratic twists must have positive rank. Similarly, as Karl Rubin remarked to us, there are fields K such that w ( E/K ) = 1 for every elliptic curve E defined over Q ; for instance Q ( i, √
17) is such a field. This is also implied by the conjectural finiteness of X , under mild restrictions on E at v | Can one prove (unconjecturally) that such square-free cubics exist, over some K ? Notethat there cannot be a non-constant parametric solution ( x ( t ) , y ( t )) to ty = f ( x ), for otherwise t ( x ( t ) , ty ( t )) would be a non-constant map P → E . Classification.
In our examples, we had the unnecessarily strong assumption thatthe curve has everywhere good reduction. Recall that the global root number w ( E/K ) is the product of local root numbers w ( E/K v ) over all places v of K . Thecondition that w ( E/K ′ ) = 1 for every quadratic extension K ′ /K is easily seen tobe equivalent to w ( E/K ′ v ) being 1 for every quadratic extension K ′ v /K v , for all v .For instance, this local condition fails for real places but holds for complex placesand primes of good reduction for E .If E is an elliptic curve over a local field k , let us also say that E/k is lawful if w ( E/k ′ ) = 1 for every quadratic extension k ′ /k , and chaotic otherwise. Dependingon whether w ( E/k ) is +1 or −
1, call the curve lawful good or lawful evil . Thereader should be warned that in the local setting lawfulness is not equivalent to theinvariance of the root number under quadratic twists.As mentioned above, a curve over a number field K is lawful if and only if it islawful over every completion of K . Whether it is good or evil is determined by theparity of lawful evil places. For instance, if K has no real and r complex places,an elliptic curve E/K with everywhere good reduction is lawful; it is lawful evil ifand only if r is odd.It turns out that if E/K is lawful then w ( E/F ) = w ( E/K ) [ F : K ] for every ex-tension F/K . In particular, a lawful evil curve
E/K must acquire points of infiniteorder over any extension of even degree, while a lawful good curve should have evenrank over every extension of K , as in the example of 1369E1 above. Generally, Theorem 1.
For an elliptic curve E over a number field K , the following conditionsare equivalent: (1) E/K is lawful, (2)
E/K v is lawful for all places v of K , (3) w ( E/F ) = w ( E/K ) [ F : K ] for every finite extension F/K , (4) K has no real places, and E acquires everywhere good reduction over anabelian extension of K , (5) K has no real places, and for all primes p and all places v ∤ p of K , theaction of Gal( ¯ K v /K v ) on the Tate module T p ( E ) is abelian (“fake CM”) . This is a corollary of the following local statement.
Theorem 2.
For an elliptic curve E over a local non-Archimedean field k of char-acteristic 0, the following conditions are equivalent: (1) E/k is lawful, (2) w ( E/F ) = w ( E/k ) [ F : k ] for every finite extension F/k , (3) E acquires good reduction over an abelian extension of k , (4) For some (any) p different from the residue characteristic of k , the actionof Gal(¯ k/k ) on T p ( E ) is abelian.Proof. (2) ⇒ (1) is obvious, and (3) ⇔ (4) is a simple consequence of the criterionof N´eron–Ogg–Shafarevich in the case of potentially good reduction and the theoryof the Tate curve in the potentially multiplicative case. (4) ⇒ (2) is an elementarycomputation using the formula w ( χ ) w ( ¯ χ ) = χ ( −
1) ([8] 3.4.7). As for (1) ⇒ (4), if k has odd residue characteristic or E has potentially multiplicative reduction, this fol-lows from the formulae for the local root numbers of Rohrlich [7] and Kobayashi [5]. Recall that E has CM over K if and only if the action of the global Galois group Gal( ¯
K/K )on T p ( E ) is abelian; in this case it is obvious that E has even rank over every extension of K . LLIPTIC CURVES WITH ALL QUADRATIC TWISTS OF POSITIVE RANK 3
In residue characteristic 2, computing root numbers is a nightmare. Fortunately,Waldspurger has proved this in general on the other side of the local Langlandscorrespondence: a special or supercuspidal representation π of PGL ( k ) has rootnumber − . (cid:3) An explicit classification of lawful elliptic curves in terms of their j -invariant hasbeen given by Connell, see [1] Prop. 6. We end with an alternative classificationin terms of the Kodaira symbols, which also specifies whether the curve is good orevil. It is easily deduced from the formulae for the local root numbers of Rohrlich,Kobayashi and the authors. Classification 3.
Let k be a local field of characteristic 0, and E/k an ellipticcurve. By [7] Thm. 2, • If E has good reduction, then E is lawful good. • If k ∼ = C , then E is lawful evil. • If k ∼ = R or E has multiplicative or potentially multiplicative reduction,then E is chaotic.Next, suppose E/k has additive potentially good reduction and k has odd residuecharacteristic. Let ∆ be the minimal discriminant of E of valuation δ , and write (cid:0) · k (cid:1) for the quadratic residue symbol on k ∗ . From [7] Thm. 2 and [5] Thm. 1.1, • E is lawful good if it has type I ∗ and (cid:0) − k (cid:1) = 1,type III , III ∗ with (cid:0) − k (cid:1) = 1 and (cid:0) − k (cid:1) = 1, ortype II , II ∗ , IV , IV ∗ with ∆ ∈ k ∗ and (cid:0) − k (cid:1) δ/ = 1. • E is lawful evil if it has type I ∗ and (cid:0) − k (cid:1) = − , III ∗ with (cid:0) − k (cid:1) = 1 and (cid:0) − k (cid:1) = −
1, ortype II , II ∗ , IV , IV ∗ with ∆ ∈ k ∗ and (cid:0) − k (cid:1) δ/ = − • E is chaotic in all other cases.(When k has residue characteristic ≥
5, Kodaira types II, III, IV, I ∗ , IV ∗ , III ∗ , II ∗ correspond to δ = 2 , , , , , ,
10, respectively. Also for 3 ∤ δ , ∆ ∈ k ∗ if and onlyif (cid:0) − k (cid:1) = 1 in this case, see [5] 1.2.)Finally, if E/k has additive potentially good reduction and k has residue char-acteristic 2, let c , c and ∆ be the standard invariants of some model of E , andset γ ( x ) = x − c x − c x − c ∈ k [ x ]. From [3] Prop. 2 and Lemma 3, E/k islawful if and only if • √− ∈ k and γ ( x ) is reducible, or • √− k , √ ∆ ∈ k , and one of the irreducible factors of γ ( x ) becomesreducible over k ( √− E is lawful good if and only if − γ ( x ) to k ([3] Prop. 4b).Note from the classification that if E/K has semistable reduction at placesabove 2, it becomes lawful over some quadratic extension of K if and only if it If E has potentially good reduction and the Galois action on T p ( E ) is non-abelian, the dual V = ( T p ( E ) ⊗ Q p ) ∗ is an irreducible representation of the Weil group of k . Twisting V by the squareroot of the cyclotomic character (this does not change the root number) gives a representationthat corresponds via local Langlands to a supercuspidal representation π of PGL ( k ). By [9]Prop. 16, there is a quadratic extension k ′ /k such that w ( π/k ′ ) = −
1. Because the local Langlandscorrespondence is known for PGL ( k ) (see [4]) and it preserves root numbers and takes restrictionto base change, we get w ( E/k ′ ) = − TIM AND VLADIMIR DOKCHITSER has integral j -invariant. (If v | E/K v has additive potentially good reduction, E stays chaotic in all quadratic extensions of K v if and only the inertia group at v in K ( E [3]) /K is SL ( F ).) In this way, it is easy to construct lawful evil curvesover imaginary quadratic fields. For example, E : y + xy = x − x − x − Q ( i ), so its rank should go up in every extension of Q ( i ) of evendegree. Acknowledgements.
We would like to thank David Rohrlich for suggesting touse Waldspurger’s results, and Andrew Granville for his interesting comments. Thefirst author is supported by a Royal Society University Research Fellowship.
References [1] I. Connell, Good reduction of elliptic curves in abelian extensions, J. Reine Angew. Math. 436(1993), 155–175.[2] T. Dokchitser, V. Dokchitser, On the Birch–Swinnerton-Dyer quotients modulo squares, 2006,arxiv: math.NT/0610290.[3] T. Dokchitser, V. Dokchitser, Root numbers of elliptic curves in residue characteristic 2,Bull. London Math. Soc. 40 (2008), 516–524.[4] H. Jacquet, R. Langlands, Automorphic forms on GL(2), LNM 114, Springer 1970.[5] S. Kobayashi, The local root number of elliptic curves with wild ramification, Math. Ann. 323(2002), 609–623.[6] D. Rohrlich, Nonvanishing of L -functions and structure of Mordell-Weil groups, J. ReineAngew. Math. 417 (1991), 1–26.[7] D. Rohrlich, Galois Theory, elliptic curves, and root numbers, Compos. Math. 100 (1996),311–349.[8] J. Tate, Number theoretic background, in: Automorphic forms, representations and L-functions, Part 2 (ed. A. Borel and W. Casselman), Proc. Symp. in Pure Math. 33 (AMS,Providence, RI, 1979) 3-26.[9] J.-L. Waldspurger, Correspondences de Shimura et quaternions, Forum Math. 3 (1991), 219–307. Robinson College, Cambridge CB3 9AN, United Kingdom
E-mail address : [email protected] Gonville & Caius College, Cambridge CB2 1TA, United Kingdom
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