A note on the Mordell-Weil rank modulo n
aa r X i v : . [ m a t h . N T ] O c t . A NOTE ON THE MORDELL-WEIL RANK MODULO n TIM AND VLADIMIR DOKCHITSER
Abstract.
Conjecturally, the parity of the Mordell-Weil rank of anelliptic curve over a number field K is determined by its root number.The root number is a product of local root numbers, so the rank modulo2 is (conjecturally) the sum over all places of K of a function of ellipticcurves over local fields. This note shows that there can be no analoguefor the rank modulo 3, 4 or 5, or for the rank itself. In fact, standardconjectures for elliptic curves imply that there is no analogue modulo n for any n >
2, so this is purely a parity phenomenon.Surely, otherwise somebody would have spotted it by now.—
Tom Fisher
It is a consequence of the Birch–Swinnerton-Dyer conjecture that theparity of the Mordell-Weil rank of an elliptic curve E over a number field K is determined by its root number, the sign in the functional equation of the L -function. The root number is a product of local root numbers, which leadsto a conjectural formula of the formrk E/K ≡ X v λ ( E/K v ) mod 2 , where λ is an invariant of elliptic curves over local fields, and v runs overthe places of K . One might ask whether there is a local expression like thisfor the rank modulo 3 or modulo 4, or even for the rank itself. The purposeof this note is to show that, unsurprisingly, the answer is ‘no’.The idea is simple: if the rank modulo n were a sum of local Z /n Z -valuedinvariants, then rk E/K would be a multiple of n whenever E is definedover Q and K/ Q is a Galois extension where every place of Q splits into amultiple of n places. However, for small n > E and K forwhich this property fails (Theorem 2). In fact, if one believes the standardheuristics concerning ranks of elliptic curves in abelian extensions, it failsfor every n > E/ Q (Theorems 9, 13).This kind of argument can be used to test whether a global invariant hasa chance of being a sum of local terms. We will apply it to other standardinvariants of elliptic curves and show that the parity of the 2-Selmer rank,the parity of the rank of the p -torsion and dim F X [2] modulo 4 cannot beexpressed as a sum of local terms (Theorem 6). Finally, we will also commenton L -functions all of whose local factors are n th powers and discuss the parityof the analytic rank for non-self-dual twists of elliptic curves (Remarks 4,7).Our results only prohibit an expression for the rank as a sum of localterms. Local data does determine the rank, see Remark 15. Mathematics Subject Classification. Mordell-Weil rank is not a sum of local invariants
Definition.
Suppose (
K, E ) Λ( E/K ) is some global invariant of ellipticcurves over number fields . We say it is a sum of local invariants ifΛ( E/K ) = X v λ ( E/K v ) , where λ is some invariant of elliptic curves over local fields, and the sum istaken over all places of K .Implicitly, Λ and λ take values in some abelian group A , usually Z . Moreover λ ( E/K v ) should be 0 for all but finitely many v . Example.
If the Birch–Swinnerton-Dyer conjecture holds (or if X is finite,see [4]), then the Mordell-Weil rank modulo 2 is a sum of local invariants withvalues in Z / Z . Specifically, for an elliptic curve E over a local field k write w ( E/k ) = ± λ by ( − λ ( E/k ) = w ( E/k ).Then rk
E/K ≡ X v λ ( E/K v ) mod 2 . An explicit description of local root numbers can be found in [9] and [4].
Theorem 1.
The Mordell-Weil rank is not a sum of local invariants.
This is a consequence of the following stronger statement:
Theorem 2.
For n ∈ { , , } the Mordell-Weil rank modulo n is not a sumof local invariants (with values in Z /n Z ). Lemma 3.
Suppose
Λ : (number fields) → Z /n Z satisfies Λ( K ) = P v λ ( K v ) for some function λ : (local fields) → Z /n Z . Then Λ( F ) = 0 whenever F/K is a Galois extension of number fields in which the number of places aboveeach place of K is a multiple of n .Proof. In the local expression for Λ( F ) each local field occurs a multiple of n times. (cid:3) Proof of Theorem 2.
Take E/ Q : y = x ( x + 2)( x − ζ p for a primitive p th root of unity, let F n = the degree 9 subfield of Q ( ζ , ζ ) if n = 3 , the degree 25 subfield of Q ( ζ , ζ ) if n = 5 , Q ( √− , √ , √
73) if n = 4 . Because 13 and 103 are cubes modulo one another, and all other primes areunramified in F , every place of Q splits into 3 or 9 in F . Similarly F and F also satisfy the assumptions of Lemma 3 with n = 4 ,
5. Hence, ifthe Mordell-Weil rank modulo n were a sum of local invariants, it would be Meaning that if K ∼ = K ′ and E/K and E ′ /K ′ are isomorphic elliptic curves (identi-fying K with K ′ ), then Λ( E/K ) = Λ( E ′ /K ′ ). NOTE ON THE MORDELL-WEIL RANK MODULO n ∈ Z /n Z for E/F n . However, 2-descent shows that rk E/F = rk E/F = 1and rk E/F = 6 (e.g. using Magma [1], over all minimal non-trivial subfieldsof F n ). (cid:3) Remark 4.
The L -series of the curve E = 480a1 used in the proof over F = F = Q ( √− , √ , √
73) is formally a 4th power, in the sense that eachEuler factor is: L ( E/F, s ) = 1 · (cid:0) − − s (cid:1) (cid:0) − − s (cid:1) (cid:0) · − s +7 − s (cid:1) (cid:0) · − s +11 − s (cid:1) ··· . However, it is not a 4th power of an entire function, as it vanishes to order 6at s = 1. Actually, it is not even a square of an entire function: it has asimple zero at 1 + 2 . ... i .In fact, by construction of F , for any E/ Q the L -series L ( E/F, s ) isformally a 4th power and vanishes to even order at s = 1 by the functionalequation. Its square root has analytic continuation to a domain includingRe s > , Re s < and the real axis, and satisfies a functional equation s ↔ − s , but it is not clear whether it has an arithmetic meaning. Lemma 5.
Suppose an invariant Λ ∈ Z / k Z is a sum of local invariants.Let F = K ( √ α , ..., √ α m ) be a multi-quadratic extension in which everyprime of K splits into a multiple of k primes of F . Then for every ellipticcurve E/K , Λ( E/K ) + X D Λ( E D /K ) = 0 , where the sum is taken over the quadratic subfields K ( √ D ) of F/K , and E D denotes the quadratic twist of E by D .Proof. In the local expression for the left-hand side of the formula each localterm ( λ of a given elliptic curve over a given local field) occurs a multipleof 2 k times. (cid:3) Theorem 6.
Each of the following is not a sum of local invariants: • dim F X ( E/K )[2] mod 4 , • rk( E/K ) + dim F X ( E/K )[2] mod 4 , • dim F Sel E/K mod 2 , • dim F p E ( K )[ p ] mod 2 for any prime p .Here X is the Tate-Shafarevich group and Sel is the 2-Selmer group.Proof. The argument is similar to that of Theorem 2:For the first two claims, apply Lemma 5 to E : y + y = x − x (37a1)with K = Q and F = Q ( √− , √ , √ E by1 , − , − , ·
89 have rank 1, and those by − , , , − ·
89 have rank 0;the twist by − ·
89 has | X [2] | = 4 and the other seven have trivial X [2].The sum over all twists is therefore 2 mod 4 in both cases, so they are notsums of local invariants. TIM AND VLADIMIR DOKCHITSER
For the parity of the 2-Selmer rank and of dim E [2] apply Lemma 3 to E : y + xy + y = x + 4 x − K = Q , F = Q ( √− , √
17) and n = 2. The 2-torsion subgroup of E/F is of order 2 and its 2-Selmer groupover F is of order 8.Finally, for dim F p E [ p ] mod 2 for p > E/ Q withGal( Q ( E [ p ]) / Q ) ∼ = GL ( F p ), e.g. E : y = x − x + x (24a4), see [10] 5.7.2.Let K be the field obtained by adjoining to Q the coordinates of one p -torsionpoint and F = K ( √− , √ F does not contain the p th roots ofunity, dim F p E ( F )[ p ] = 1. So, by Lemma 3 the parity of this dimension isnot a sum of local invariants. (cid:3) Remark 7.
The functional equation expresses the parity of the analyticrank as a sum of local invariants not only for elliptic curves (or abelian vari-eties), but also for their twists by self-dual Artin representations. However,for the parity of the rank of non-self-dual twists there is presumably no suchexpression.For example, let χ be a non-trivial Dirichlet character of ( Z / Z ) × oforder 3. Then there is no function ( k, E ) λ ( E/k ) ∈ Z defined for ellipticcurves over local fields k , such that for all elliptic curves E/ Q ,ord s =1 L ( E, χ, s ) ≡ X v λ ( E/ Q v ) mod 2 . To see this, take E/ Q : y + y = x + x + x (19a3) , K = Q , F = Q ( √− , √ E, E − and E by χ have analyticrank 0, and that of E − has analytic rank 1, adding up to an odd number.2. Expectations
We expect the Mordell-Weil rank modulo n not to be a sum of local termsfor any n > Notation.
For a prime p we write Σ p for the set of all Dirichlet charactersof order p . We say that S ⊂ Σ p has density α iflim x →∞ { χ : χ ∈ S | N ( χ ) < x }{ χ : χ ∈ Σ p | N ( χ ) < x } = α, where N ( χ ) denotes the conductor of χ . Conjecture 8 (Weak form of [3] Conj. 1.2) . For p > and every ellipticcurve E/ Q , those χ ∈ Σ p for which L ( E, χ,
1) = 0 have density 0 in Σ p . NOTE ON THE MORDELL-WEIL RANK MODULO n Theorem 9.
Let E/ Q be an elliptic curve and p an odd prime. AssumingConjecture 8, there is no function k λ ( E/k ) ∈ Z /p Z defined for localfields k of characteristic 0, such that for every number field K , rk E/K ≡ X v λ ( E/K v ) mod p. Lemma 10.
Let p be a prime number and S ⊂ Σ p a set of characters ofdensity 0 in Σ p . For every d ≥ there is an abelian extension F d / Q withGalois group G ∼ = F dp , such that no characters of G are in S .Proof. Without loss of generality, we may assume that if χ ∈ S then χ n ∈ S for 1 ≤ n < p . When d = 1, take F to be the kernel of any χ ∈ Σ p \ S .Now proceed by induction, supposing that F d − is constructed. Writing Ψfor the set of characters of Gal( F d − / Q ), the set S d = [ ψ ∈ Ψ { φψ : φ ∈ S } still has density 0. Pick any χ ∈ Σ p \ S d , and set F d to be the compositumof F d − and the degree p extension of Q cut out by χ . It is easy to checkthat no character of Gal( F d / Q ) lies in S . (cid:3) Proof of Theorem 9.
Pick a quadratic field Q ( √ D ) such that the quadratictwist E D of E by D has analytic rank 1, which is possible by [2, 8, 11].By Conjecture 8, the set S of Dirichlet characters χ of order p such that L ( E D , χ,
1) = 0 has density 0. Apply Lemma 10 to S with d = 3. Everyplace of Q splits in the resulting field F = F into a multiple of p places( Q l has no F p -extensions, so every prime has to split).Arguing by contradiction, suppose the rank of E mod p is a sum of localinvariants. Because in F/ Q and therefore also in F ( √ D ) / Q every placesplits into a multiple of p places,rk E/F ≡ p and rk E/F ( √ D ) ≡ p by Lemma 3. Therefore rk E D /F = rk E/F ( √ D ) − rk E/F is a multipleof p . On the other hand, L ( E D /F, s ) = Y χ L ( E D , χ, s ) , the product taken over the characters of Gal( F/ Q ). By construction, it hasa simple zero at s = 1. Because F is totally real of odd degree over Q , byZhang’s theorem [12] Thm. A, E D /F has Mordell-Weil rank 1 p ,a contradiction. (cid:3) Conjecture 11 (Goldfeld [6]) . For every elliptic curve E/ Q , those χ ∈ Σ for which ord s =1 L ( E, χ, s ) > have density 0 in Σ . Conjecture 12.
Every elliptic curve E/ Q has a quadratic twist of Mordell-Weil rank 2. TIM AND VLADIMIR DOKCHITSER
Theorem 13.
Let E/ Q be an elliptic curve. Assuming Conjectures 11and 12, there is no function k λ ( E/k ) ∈ Z / Z defined for local fields k of characteristic 0, such that for every number field K , rk E/K ≡ X v λ ( E/K v ) mod 4 . Proof.
Let Q ( √ D ) be a quadratic field such that the quadratic twist E ′ = E D of E by D has Mordell-Weil rank 2 (Conjecture 12). The set S of those χ ∈ Σ for which ord s =1 L ( E ′ , χ, s ) > P be the set of primes where E ′ has bad reduction union {∞} , andapply Lemma 10 to S with d = 5+ 3 | P | . The resulting field F d has a subfield F of degree 2 over Q , where all places in P split completely: Q l has no F -extensions, so the condition that a given place in P splits completelydrops the dimension by at most 3. By the same argument, every place of Q splits in F into a multiple of 4 places.Arguing by contradiction, suppose the rank of E mod 4 is a sum of localinvariants. Because in F/ Q and therefore also in F ( √ D ) / Q every placesplits into a multiple of 4 places,rk E/F ≡ E/F ( √ D ) ≡ E ′ /F = rk E/F ( √ D ) − rk E/F is a multiple of 4.Now we claim that E ′ /F has rank 2 or 33, yielding a contradiction.Let Q ( √ m ) ⊂ F be a quadratic subfield. The root number of E ′ over Q ( √ m ) is 1, because the root number is a product of local root numbersand the places in P split in Q ( √ m ). (The local root number is +1 at primesof good reduction.) So L ( E ′ / Q ( √ m ) , s ) = L ( E ′ / Q , s ) L ( E ′ m / Q , s )vanishes to even order at s = 1. Hence the 31 twists of E ′ by the non-trivialcharacters of Gal( F/ Q ) have the same analytic rank 0 or 1, by the choiceof F . By Kolyvagin’s theorem [7], their Mordell-Weil ranks are the same astheir analytic ranks, and so rk E ′ /F is either 2 + 0 or 2 + 31. (cid:3) Remark 14.
In some cases, it may seem reasonable to try and write someglobal invariant in Z /n Z as a sum of local invariants in m Z /n Z , i.e. to allowdenominators in the local terms. For instance, one could ask whether theparity of the rank of a cubic twist (as in Remark 7) can be written as a sumof local invariants of the form a mod 2 Z .However, introducing a denominator does not appear to help. First, theprime-to- n part m ′ of m adds no flexibility, as can be seen by multiplyingthe formula by m ′ . (For instance, if there were a formula for the parityof the rank of a cubic twist as a sum of local terms in a mod 2 Z , thenmultiplying it by 3 would yield a formula for the same parity with local termsin Z / Z .) As for the non-prime-to- n part, e.g. the proofs of Theorems 9and 13 immediately adapt to local invariants in p k Z /p Z and k Z / Z , byincreasing d by k . NOTE ON THE MORDELL-WEIL RANK MODULO n Remark 15.
The negative results in this paper rely essentially on the factthat we allow only additive formulae for global invariants in terms of localinvariants. Although Theorem 1 shows that there is no formula of the formrk
E/K = X v λ ( E/K v ) , the Mordell-Weil rank is determined by the set { E/K v } v of curves over localfields. In other words, rk E/K = function( { E/K v } v ) . In fact, for any abelian variety
A/K the set { A/K v } v determines the L -function L ( A/K, s ) which is the same as L ( W/ Q , s ) where W is the Weilrestriction of A to Q . By Faltings’ theorem [5] the L -function recovers W up to isogeny, and hence also recovers the rank rk A/K (= rk W/ Q ). Acknowledgements.
The first author is supported by a Royal SocietyUniversity Research Fellowship. The second author would like to thankGonville & Caius College, Cambridge.
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