Conjecturally, the parity of the Mordell-Weil rank of an elliptic 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 modulo 2 is conjecturally the sum over all
places of K of a function of elliptic curves over local fields. This note shows
that there can be no analogue for the rank modulo 3, 4 or 5, or for the rank
itself. In fact, standard conjectures for elliptic curves imply that there is
no analogue modulo n for any n>2, so this is purely a parity phenomenon.Comment: 7 page

Dokchitser, Tim

Dokchitser, Vladimir

English

arXiv

AbstractConjecturally, the parity of the Mordell–Weil rank of an elliptic 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 modulo 2 is (conjecturally) the sum over all places of K of a function of elliptic curves over local fields. This note shows that there can be no analogue for the rank modulo 3, 4 or 5, or for the rank itself. In fact, standard conjectures for elliptic curves imply that there is no analogue modulo n for any n>2, so this is purely a parity phenomenon

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Journal of Number Theory 131 (2011) 1833–1839Contents lists available at ScienceDirect
Journal of Number Theory
www.elsevier.com/locate/jnt
A note on the Mordell–Weil rank modulo n
Tim Dokchitser a,∗, Vladimir Dokchitser b
a Robinson College, Cambridge CB3 9AN, United Kingdom
b Emmanuel College, Cambridge CB2 3AP, United Kingdom
a r t i c l e i n f o a b s t r a c t
Article history:
Received 3 November 2009
Revised 11 March 2011
Accepted 11 March 2011
Available online 19 May 2011
Communicated by D. Zagier
MSC:
primary 11G05
secondary 11G40
Keywords:
Mordell–Weil rank
Elliptic curves
Conjecturally, the parity of the Mordell–Weil rank of an elliptic
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 modulo 2 is (conjecturally) the sum over all places of K of
a function of elliptic curves over local ﬁelds. This note shows that
there can be no analogue for the rank modulo 3, 4 or 5, or for the
rank itself. In fact, standard conjectures for elliptic curves imply
that there is no analogue modulo n for any n > 2, so this is purely
a parity phenomenon.
© 2011 Elsevier Inc. All rights reserved.
It is a consequence of the Birch–Swinnerton-Dyer conjecture that the parity 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
leads to a conjectural formula of the form
rk E/K ≡
∑
v
λ(E/Kv) mod 2,
where λ is an invariant of elliptic curves over local ﬁelds, and v runs over the places of K . One might
ask whether there is a local expression like this for the rank modulo 3 or modulo 4, or even for the
rank itself. The purpose of 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/nZ-valued invariants, then rk E/K
would be a multiple of n whenever E is deﬁned over Q and K/Q is a Galois extension where every
* Corresponding author.
E-mail addresses: t.dokchitser@dpmms.cam.ac.uk (T. Dokchitser), v.dokchitser@dpmms.cam.ac.uk (V. Dokchitser).0022-314X/$ – see front matter © 2011 Elsevier Inc. All rights reserved.
doi:10.1016/j.jnt.2011.03.010
1834 T. Dokchitser, V. Dokchitser / Journal of Number Theory 131 (2011) 1833–1839place of Q splits into a multiple of n places. However, for small n > 2 it is easy to ﬁnd E and K for
which this property fails (Theorem 2). In fact, if one believes the standard heuristics concerning ranks
of elliptic curves in abelian extensions, it fails for every n > 2 and every E/Q (Theorems 9, 13).
This kind of argument can be used to test whether a global invariant has a chance of being a
sum of local terms. We will apply it to other standard invariants of elliptic curves and show that the
parity of the 2-Selmer rank, the parity of the rank of the p-torsion and the rank of the 2-torsion in
the Tate–Shafarevich group Ш modulo 4 cannot be expressed as a sum of local terms (Theorem 6).
Finally, we will also comment on L-functions all of whose local factors are nth powers and discuss
the parity of 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 local terms. Local data does
determine the rank, see Remark 15.
1. Mordell–Weil rank is not a sum of local invariants
Deﬁnition. Suppose (K , E) → Λ(E/K ) is some global invariant of elliptic curves over number ﬁelds.1
We say it is a sum of local invariants if
Λ(E/K ) =
∑
v
λ(E/Kv),
where λ is some invariant of elliptic curves over local ﬁelds, and the sum is taken over all places
of K .
Implicitly, Λ and λ take values in some abelian group A, usually Z or Z/nZ for some n  2.
Moreover λ(E/Kv) should be 0 for all but ﬁnitely many v .
Example. If the Birch–Swinnerton-Dyer conjecture holds (or if Ш is ﬁnite, see [4]), then the Mordell–
Weil rank modulo 2 is a sum of local invariants with values in Z/2Z. Speciﬁcally, for an ellip-
tic curve E over a local ﬁeld k write w(E/k) = ±1 for its local root number, and deﬁne λ by
(−1)λ(E/k) = w(E/k). Then
rk E/K ≡
∑
v
λ(E/Kv) 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 ∈ {3,4,5} the Mordell–Weil rank modulo n is not a sum of local invariants (with values in
Z/nZ).
Lemma 3. Suppose Λ : (number ﬁelds) → Z/nZ satisﬁes Λ(K ) = ∑v λ(Kv) for some invariant
λ : (local ﬁelds) → Z/nZ. Then Λ(F ) = 0 whenever F/K is a Galois extension of number ﬁelds in which
the number of places above each 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. 
1 Meaning that if K ∼= K ′ and E/K and E ′/K ′ are isomorphic elliptic curves (identifying K with K ′), then Λ(E/K ) = Λ(E ′/K ′).
T. Dokchitser, V. Dokchitser / Journal of Number Theory 131 (2011) 1833–1839 1835Proof of Theorem 2. Take E/Q : y2 = x(x + 2)(x − 3), which is 480a1 in Cremona’s notation. Writing
ζp for a primitive pth root of unity, let
Fn =
⎧⎨
⎩
the degree 9 subﬁeld of Q(ζ13, ζ103) if n = 3,
the degree 25 subﬁeld of Q(ζ11, ζ241) if n = 5,
Q(
√−1,√41,√73) if n = 4.
Because 13 and 103 are cubes modulo one another, and all other primes are unramiﬁed in F3, every
place of Q splits into 3 or 9 places in F3. Similarly F4 and F5 also satisfy the assumptions of Lemma
3 with n = 4,5. Hence, if the Mordell–Weil rank modulo n were a sum of local invariants, it would be
0 ∈ Z/nZ for E/Fn . However, 2-descent shows that rk E/F3 = rk E/F5 = 1 and rk E/F4 = 6 (e.g. using
Magma [1], over all minimal non-trivial subﬁelds of Fn). 
Remark 4. The L-series of the curve E = 480a1 used in the proof over F = F4 = Q(
√−1,√41,√73)
is formally a 4th power, in the sense that each Euler factor is:
L(E/F , s) = ( 11
)4( 1
1−3−2s
)4( 1
1−5−2s
)4( 1
1+14·7−2s+72−4s
)4( 1
1+6·11−2s+112−4s
)4 · · · .
However, it is not a 4th power of an entire function, as it vanishes to order 6 at s = 1. Actually, it is
not even a square of an entire function: it has a simple zero at 1+ 2.1565479 . . . i.
In fact, by construction of F , for any E/Q the L-series L(E/F , s) is formally a 4th power and
vanishes to even order at s = 1 by the functional equation. Its square root has analytic continuation
to a domain including Re s > 32 , Re s <
1
2 and the real axis, and satisﬁes a functional equation s ↔ 2−s,
but it is not clear whether it has an arithmetic meaning.
Lemma 5. Suppose an invariant Λ ∈ Z/2kZ is a sum of local invariants. Let F = K (√α1, . . . ,√αm) be a
multi-quadratic extension in which every prime of K splits into a multiple of 2k primes of F . Then for every
elliptic curve E/K ,
Λ(E/K ) +
∑
D
Λ(ED/K ) = 0,
where the sum is taken over the quadratic subﬁelds K (
√
D) of F/K , and ED denotes the quadratic twist of E
by D.
Proof. In the local expression for the left-hand side of the formula each local term (λ of a given
elliptic curve over a given local ﬁeld) occurs a multiple of 2k times. 
Theorem 6. Each of the following is not a sum of local invariants:
• dimF2Ш(E/K )[2] mod 4,• rk(E/K ) + dimF2Ш(E/K )[2] mod 4,• dimF2 Sel2 E/K mod 2,• dimFp E(K )[p] mod 2 for any prime p.
HereШ is the Tate–Shafarevich group and Sel2 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 : y2 + y = x3 − x (37a1) with K = Q and
F = Q(√−1,√17,√89). The quadratic twists of E by 1,−17,−89,17 · 89 have rank 1, and those
by −1,17,89,−17 · 89 have rank 0; the twist by −17 · 89 has |Ш[2]| = 4 and the other seven have
1836 T. Dokchitser, V. Dokchitser / Journal of Number Theory 131 (2011) 1833–1839trivial Ш[2]. The sum over all twists is therefore 2 mod 4 in both cases, so they are not sums of local
invariants.
For the parity of the 2-Selmer rank and of dim E[2] apply Lemma 3 to E : y2 + xy+ y = x3 +4x−6
(14a1) with K = Q, F = Q(√−1,√17) and n = 2. The 2-torsion subgroup of E/F is of order 2 and its
2-Selmer group over F is of order 8.
Finally, for dimFp E[p] mod 2 for p > 2 take any elliptic curve E/Q with Gal(Q(E[p])/Q) ∼=
GL2(Fp), e.g. E : y2 = x3 − x2 + x (24a4), see [10, 5.7.2]. Let K be the ﬁeld obtained by adjoining
to Q the coordinates of one p-torsion point and F = K (√−1,√17). Because F does not contain the
pth roots of unity, dimFp E(F )[p] = 1. So, by Lemma 3 the parity of this dimension is not a sum of
local invariants. 
Remark 7. The functional equation expresses the parity of the analytic rank as a sum of local in-
variants not only for elliptic curves (or abelian varieties), 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
such expression.
For example, let χ be a non-trivial Dirichlet character of (Z/7Z)× of order 3. Then there is no
function (k, E) → λ(E/k) ∈ Z deﬁned for elliptic curves over local ﬁelds k, such that for all elliptic
curves E/Q,
ords=1 L(E,χ, s) ≡
∑
v
λ(E/Qv) mod 2.
To see this, take
E/Q : y2 + y = x3 + x2 + x (19a3), K = Q, F = Q(√−1,√17)
and apply Lemma 5. The twists of E, E−1 and E17 by χ have analytic rank 0, and that of E−17 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 terms for any n > 2 and any
class of elliptic curves. Theorems 9 and 13 below show that this is a consequence of modularity of
elliptic curves, the known cases of the Birch–Swinnerton-Dyer conjecture and standard conjectures
for analytic ranks of elliptic curves.
Notation. For a prime p we write Σp for the set of all Dirichlet characters of order p. We say that
S ⊂ Σp has density α if
lim
x→∞
{χ : χ ∈ S | N(χ) < x}
{χ : χ ∈ Σp | N(χ) < x} = α,
where N(χ) denotes the conductor of χ .
Conjecture 8 (Weak form of [3, Conj. 1.2]). For p > 2 and every elliptic curve E/Q, those χ ∈ Σp for which
L(E,χ,1) = 0 have density 0 in Σp .
Theorem 9. Let E/Q be an elliptic curve and p an odd prime. Assuming Conjecture 8, there is no function
k → λ(E/k) ∈ Z/pZ deﬁned for local ﬁelds k of characteristic 0, such that for every number ﬁeld K ,
rk E/K ≡
∑
v
λ(E/Kv) mod p.
T. Dokchitser, V. Dokchitser / Journal of Number Theory 131 (2011) 1833–1839 1837Lemma 10. Let p be a prime number and S ⊂ Σp a set of characters of density 0 in Σp . For every d 1 there
is an abelian extension Fd/Q with Galois group G ∼= Fdp , such that no characters of G are in S.
Proof. Without loss of generality, we may assume that if χ ∈ S then χn ∈ S for 1  n < p. When
d = 1, take F1 to be the kernel of any χ ∈ Σp \ S . Now proceed by induction, supposing that Fd−1 is
constructed. Writing Ψ for the set of characters of Gal(Fd−1/Q), the set
Sd =
⋃
ψ∈Ψ
{φψ: φ ∈ S}
still has density 0. Pick any χ ∈ Σp \ Sd , and set Fd to be the compositum of Fd−1 and the degree p
extension of Q cut out by χ . It is easy to check that no character of Gal(Fd/Q) lies in S . 
Proof of Theorem 9. Pick a quadratic ﬁeld Q(
√
D) such that the quadratic twist ED 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(ED ,χ,1) = 0 has density 0. Apply Lemma 10 to S with d = 3. Every place of
Q splits in the resulting ﬁeld F = F3 into a multiple of p places (Ql has no F3p-extensions, so every
prime has to split).
Arguing by contradiction, suppose the rank of E mod p is a sum of local invariants. Because in
F/Q and therefore also in F (
√
D)/Q every place splits into a multiple of p places,
rk E/F ≡ 0 mod p and rk E/F (√D) ≡ 0 mod p
by Lemma 3. Therefore rk ED/F = rk E/F (
√
D) − rk E/F is a multiple of p. On the other hand,
L(ED/F , s) =
∏
χ
L(ED ,χ, s),
the product taken over the characters of Gal(F/Q). By construction, it has a simple zero at s = 1.
Because F is totally real of odd degree over Q, by Zhang’s theorem [12, Thm. A], ED/F has Mordell–
Weil rank 1 ≡ 0 mod p, a contradiction. 
Conjecture 11. (See Goldfeld [6].) For every elliptic curve E/Q, those χ ∈ Σ2 for which ords=1 L(E,χ, s) > 1
have density 0 in Σ2 .
Conjecture 12. Every elliptic curve E/Q has a quadratic twist of Mordell–Weil rank 2.
Theorem 13. Let E/Q be an elliptic curve. Assuming Conjectures 11 and 12, there is no function
k → λ(E/k) ∈ Z/4Z deﬁned for local ﬁelds k of characteristic 0, such that for every number ﬁeld K ,
rk E/K ≡
∑
v
λ(E/Kv) mod 4.
Proof. Let Q(
√
D) be a quadratic ﬁeld such that the quadratic twist E ′ = ED of E by D has Mordell–
Weil rank 2 (Conjecture 12). The set S of those χ ∈ Σ2 for which ords=1 L(E ′,χ, s) > 1 has density 0
(Conjecture 11). Let P be the set of primes where E ′ has bad reduction union {∞}, and apply
Lemma 10 to S with d = 5 + 3|P |. The resulting ﬁeld Fd has a subﬁeld F of degree 25 over Q,
where all places in P split completely: Ql has no F42-extensions, so the condition that a given place
in P splits completely drops the dimension by at most 3. By the same argument, every place of Q
splits in F into a multiple of 4 places.
1838 T. Dokchitser, V. Dokchitser / Journal of Number Theory 131 (2011) 1833–1839Arguing by contradiction, suppose the rank of E mod 4 is a sum of local invariants. Because in
F/Q and therefore also in F (
√
D)/Q every place splits into a multiple of 4 places,
rk E/F ≡ 0 mod 4 and rk E/F (√D) ≡ 0 mod 4
by Lemma 3. Therefore rk 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 numbers and the places in P split in Q(
√
m). (The local root
number is +1 at primes of 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-trivial characters of Gal(F/Q)
have the same analytic rank 0 or 1, by the choice of F . By Kolyvagin’s theorem [7], their Mordell–Weil
ranks are the same as their analytic ranks, and so rk E ′/F is either 2+ 0 or 2+ 31. 
Remark 14. In some cases, it may seem reasonable to try and write some global invariant in Z/nZ as
a sum of local invariants in 1mZ/nZ, i.e. to allow denominators in the local terms. For instance, one
could ask whether the parity of the rank of a cubic twist (as in Remark 7) can be written as a sum of
local invariants of the form a3 mod 2Z.
However, introducing a denominator does not appear to help. First, the prime-to-n part m′ of m
adds no ﬂexibility, as can be seen by multiplying the formula by m′ . (For instance, if there were
a formula for the parity of the rank of a cubic twist as a sum of local terms in a3 mod 2Z, then
multiplying it by 3 would yield a formula for the same parity with local terms in Z/2Z.) As for the
non-prime-to-n part, e.g. the proofs of Theorems 9 and 13 immediately adapt to local invariants in
1
pk
Z/pZ and 1
2k
Z/4Z, by increasing d by k.
Remark 15. The negative results in this paper rely essentially on the fact that we allow only additive
formulae for global invariants in terms of local invariants. Although Theorem 1 shows that there is no
formula of the form
rk E/K =
∑
v
λ(E/Kv),
the Mordell–Weil rank is determined by the set {E/Kv}v of curves over local ﬁelds. In other words,
rk E/K = function({E/Kv}v
)
.
In fact, for any abelian variety A/K the set {A/Kv}v determines the L-function L(A/K , s) which is the
same as L(W /Q, s) where W is the Weil restriction 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 (= rkW /Q).
Acknowledgments
The ﬁrst author is supported by a Royal Society University Research Fellowship. The second author
would like to thank Gonville & Caius College, Cambridge.
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A note on the Mordell–Weil rank modulo n 

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A note on the Mordell-Weil rank modulo n

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