On $p$-degree of elliptic curves

0.18 MB·

by  Jędrzej Garnek

On p-degree of elliptic curves
Jędrzej Garnek
7
1 Inthisnoteweinvestigatethep-degreefunctionofanellipticcurve
0 E/Q . The p-degree measures the least complexity of a non-zero
p
2
p-torsion point on E. We prove some properties of this function

n and compute it explicitly in some special cases.
a
J

0 1. Introduction
3

] Let p = 2,3 be a prime. In this paper we define the p-degree of an elliptic
T 6
curve E over the field Q to be:
p
N
.
h dp(E) = min [Qp(P) : Qp] :P E[p], P = ,
{ ∈ 6 O}
t
a
m where Q (P) denotes the field obtained by adjoining to Q the coordinates
p p
[ of a point P E(Qp). It turns out that the p-degree of an elliptic curve with
good reductio∈n depends only on the reduction modp2:
1
v
2 Theorem 1.1. If the elliptic curves E ,E /Q have good reduction and
1 2 p
2 their reductions to Z/p2 are isomorphic then d (E )= d (E ).
p 1 p 2
8
8
0 Inparticular,curveswithlowp-degreecorrespondtothecanonicallift modp2
. and inthiscasewe canderive anexplicitformula forthe p-degree.Let ord x
1 p
×
0 denote the order of x in the group (Z/p) and a (E) be the trace of Frobe-
p
7
nius endomorphism for an elliptic curve E defined over a finite field.
1
:
v
Theorem 1.2. Let E/Q be an elliptic curve with good reduction. Let us
p
i
X consider the following statements:
r
a (1) d (E) < p 1,
p
−
(2) EFp is ordinary and EZ/p2 is a canonical lift of EFp,
(3) E(Qun)[p]= 0, where Qun is the maximal unramified extension of Q ,
p 6 p p
(4) EFp is ordinary and dp(E) = ordpap(E).
Then (1) implies (2), (2) and (3) are equivalent, and (3) implies (4).
1
On p-degree of elliptic curves 2
Failures of complementary implications are discussed in Remark 3.3.
A result similar to Theorem 1.2 was partially stated already in [3]. Our
proof of Theorem 1.2 uses classification of elliptic curves over finite rings
by the j-invariant and an effective version of Serre-Tate theorem, which we
prove.
Investigatingthep-degreeisespeciallyinterestingwhenE isafixedcurve
over the field of rational numbers Q and p varies over primes. In particular
it is natural to ask about the asymptotic behaviour of the p-degree:
Question 1.3. Does the p-degree of a fixed elliptic curve tend to infinity as
p becomes large?
The authors of [3] predict that for an elliptic curve without complex
multiplication the answer to Question 1.3 is affirmative. They justify this
conjecturebyasimpleheuristicsandanaveragingresult.Theheuristicsgiven
byDavidandWestondoesn’tworkinthecaseofellipticcurveswithCM.We
try to verify Question 1.3 in this case. The first ingredient is Theorem 1.2,
which yields an explicit formula in the case when p is ordinary. For the
remaining primes we apply the following result:
Theorem 1.4. Let E/Q be an elliptic curve withgood supersingular reduc-
p
tion. Then d (E) = p2 1.
p
−
Combining methods mentioned above we get explicit formulas for the
p-degree of elliptic curves with complex multiplication by Gauss and by
Eisenstein integers. Let E denote the elliptic curve given by the Weier-
A,B
strass equation y2 = x3+Ax+B.
Theorem 1.5. Let D Z, D = 0 and assume that p ∤ 6D. Then:
∈ 6
p−1
ordp ( D) 4 (2s) , for p 1 (mod 4),
d (E ) = − · ≡
p D,0
(p2 (cid:16)1, (cid:17) for p 3 (mod 4),
− ≡
p−1
ordp (4D) 6 (2A) , for p 1 (mod 3),
d (E )= − · ≡
p 0,D
(p2 (cid:16)1, (cid:17) for p 2 (mod 3),
− ≡
where:
On p-degree of elliptic curves 3
s is defined for p 1 (mod 4) by the equation
• ≡
p = s2+t2
and conditions 2 ∤ s and s+t 1 (mod 4),
≡
A is defined for p 1 (mod 3) by the equation
• ≡
4p = A2+3B2
and condition A 1 (mod 3).
≡
Theorem 1.5 and results of Cosgrave and Dilcher from [1] allow us to
show that if a certain recurrence sequence contains infinitely many primes,
then Question 1.3 has negative answer for elliptic curves with complex mul-
tiplication by Z[i] (cf. Corollary 4.2).
Finally we compute the p-degree for elliptic curves with multiplicative
reduction, by reducing the problem to an investigation of the Tate curve (cf.
Theorem 4.3).
Outline of the paper. Section 2 provides a quick overview of facts related
to elliptic curves over rings, inculding a classification of elliptic curves over
complete rings by their j-invariant andaneffectiveversion ofSerre-Tate the-
orem. In Section 3 we prove Theorems 1.1 and 1.2 using the theory of formal
groups applied to elliptic curves over finite rings. Finally, in the last section
we investigate curves with good supersingular reduction, complex multipli-
cation and bad multiplicative reduction.
Acknowledgements.Theauthorwouldliketoexpresshissincerethanksto
WojciechGajdaforsuggestingthisproblemandformanystimulatingdiscus-
sions.The author alsothanks Bartosz Naskręckiforvaluable comments. The
author was supported by NCN research grant UMO-2014/15/B/ST1/00128,
the Scholarship of Jan Kulczyk for Graduate Students and by the doctoral
scholarship of Adam Mickiewicz University.
2. Preliminaries
Let R beacommutative unital ring withtrivial Picard group (e.g., alocal or
×
a finite ring), satisfying 6 R . Any elliptic curve over R (as defined in [6])
∈
On p-degree of elliptic curves 4
is isomorphic to a projective scheme of the form:
(2.1) E := Proj(R[x,y,z]/(y2z x3 Axz2 Bz3)),
A,B
− − −
for some A,B R, satisfying: ∆(E ) := 16 (4A3+27B2) R×.
A,B
∈ − · ∈
Moreover, E = E if and only if
A,B ∼ a,b
(2.2) A = u4 a, B = u6 b for some u R×.
· · ∈
Note that R-rational points of Pn are in the correspondence with the points
of the "naive" projective space:
Pn(R) = (x ,...,x ) Rn+1 : x R+...+x R = R ,
naive 0 n 0 n
{ ∈ } ∼
(cid:30)
×
where (x ,...,x ) (y ,...,y ) if and only if x = u y for some u R
0 n 0 n i i
∼ · ∈
and all i. Therefore R-rational points of the Weierstrass curve (2.1) can be
identified with elements of P2(R) that satisfy the Weierstrass equation.
naive
From now on we use the following notation:
K – a field which is complete with respect to a discrete valuation v,
•
R – the valuation ring of v. It is a local ring with principal maximal
•
ideal m = (π),
k := R/m – the residue field of v. We assume that it is perfect and of
•
non-zero characteristic p = 2,3,
6
R := R/mj – a local ring with maximal ideal m := m/mj. For j =
j j
• ∞
we denote: R∞ =R.
Note that if i j, then we can reduce any elliptic curve E/R to E :=
≤ j Ri
E R over R . In this way we obtain an exact sequence:
×Rj i i
(2.3) 0 E(mi) E(R ) E (R ) 0
→ j → j → Ri i →
(surjecivity of the reducbtion follows from Hensel lemma), where E is a
one-parameter formal group over R . By the general theory of formal groups
j
we have the following isomorphism for i> e : b
p−1
(2.4) E(mi)= mi,
j ∼ j
that commutes with the reduction. Note that proofs of [9, Theorem IV.6.4.]
b
and [9, Proposition VII.2.2.] remain valid in this case.
On p-degree of elliptic curves 5
It turns out that elliptic curves over R are essentially classified by their
j
j-invariant and the isomorphism class of reduction to k. The problem occurs
for elliptic curves satisfying j(E ) 0,1728 . To treat this issue we intro-
k
∈ { }
duce the following definition:
Definition 2.1. The type of an elliptic curve E /R is (m,n), if
A,B j
A= πm α, B = πn β for α,β R×
· · ∈ j
and ( ,0) (respectively (0, )), if A= 0 (respectively if B = 0).
∞ ∞
For a curve E of type (m,0) (where 1 m < ) we define:
A,B
≤ ∞
j1(EA,B) := Bα32 (mod mjj−m) ∈ Rj−m.
Analogously we define j (E ) to be A3/β2 for a curve of (0,n)-type.
1 A,B
Thecondition(2.2)assuresthatthetypeofacurve,thej-invariant given
by the usual formula and the invariant j do not depend on the choice of the
1
Weierstrass equation.
Lemma 2.2.
(1) An isomorphism class of a (0,0)-type elliptic curve E/R is uniquely
j
determined by its reduction E and by the j-invariant j(E) R .
k j
∈
(2) Isomorphism class of an elliptic curve E/R of type ( ,0) or (0, )
j
∞ ∞
is determined uniquely by the isomorphism class of E .
k
(3) Assume that 3(p 1). Then the isomorphism class of a curve E/R of
j
| −
type (m,0) (where 1 m < ) is determined uniquely by the isomor-
≤ ∞
phism class of Ek and j1(E) Rj−m.
∈
(4) Assume that 4(p 1). Then the isomorphism class of a curve E/R of
j
| −
type (0,n) (where 1 n < ) is determined uniquely by the isomor-
≤ ∞
phism class of Ek and j1(E) Rj−n.
∈
Proof. (1) Let us assume that elliptic curves E /R and E /R satisfy:
A,B j a,b j
E k = E k,
A,B ×Rj ∼ a,b×Rj
(2.5) j(E ) = j(E ),
A,B a,b
× ×
where A,B,a,b R . Then by (2.2) there exists u k such that
∈ j ∈
A u4 a (mod m ), B u6 b (mod m ).
j j
≡ · ≡ ·
On p-degree of elliptic curves 6
Using (2.5) we obtain:
(2.6) A3 b2 = a3 B2.
· ·
By Hensel lemma the equations:
x4 A a−1 = 0 and x6 B b−1 = 0
− · − ·
haveinR uniquesolutionswhichliftu.Letusdenotethembyu ,u respec-
j 1 2
tively. On the other hand the equality (2.6) implies that both u and u
1 2
satisfy the equation:
0 = x12 (A a−1)3 = x12 (B b−1)2.
− · − ·
Using again Hensel lemma we see that the above equation has a unique solu-
tion in R . Thus u = u and the map (x,y) (u2 x,u3 y) provides an
j 1 2 7→ 1· 1·
isomorphism between E and E .
1 2
(2), (3), (4) are proven in a similar way. The condition 3(p 1) implies that
× | −
µ k , which allows us to "twist" a lift of u by a suitable cube root of
3
uni⊂ty in R . Analogously, if 4(p 1) then µ k×. (cid:3)
j 4
| − ⊂
Remark 2.3. Note that a curve E /k (E /k respectively) is ordinary if
0,b a,0
and only if 3(p 1) (4(p 1) respectively). Othercases will notberelevant
| − | −
for purposes we have in mind.
Let us consider an elliptic curve E over the residue field k. By a theorem of
Serre-Tate ([5, Theorem 1.2.1]) lifts of E to R for j < are determined by
j
lifts of the p-divisible group E [p∞]= (E [pn]) to R∞. Using the previous
k k n j
classification of elliptic curves we prove an effective version of Serre-Tate
theorem:
Theorem 2.4. Let E , E be lifts of an ordinary elliptic curve E/k to R .
1 2 j
If E [pj−1] and E [pj−1] are isomorphic as R -group schemes then E = E .
1 2 j 1 ∼ 2
In order to prove this theorem, we’ll have to switch to the algebraic
closure k. Let Kun be the completion of the maximal unramified extension
Kun/K,withtheringofintegralelementsAandthemaximalidealM = (π).
We denote A :d= A/Mj, M := M/Mj, following our earlier notation.
j j
On p-degree of elliptic curves 7
Lemma 2.5. (A×)pm = (A×)pm+1 for m j 1.
j j ≥ −
Proof. Weuseinductiononj.Forj = 1itisimmediate.Ifa = bpm (A× )pm
∈ j+1
andm j,thenbyinductionhypothesisbpm−1 = cpm +dforsomed Mj .
≥ ∈ j+1
By raising this equality to the p-th power and using Newton binomial theo-
rem, we get a = cpm+1. (cid:3)
Proof of Theorem 2.4. By the Serre-Tate theorem lifts of an elliptic curve
E/k to A are classified by the lifts of its p-divisible group. However, since
j
the residue field of A is algebraically closed, the étale-connected sequence
j
∞
(cf. [8, p. 43]) of any p-divisible group G over the ring A lifting E[p ] must
j
be of the form:
0 µp∞ G Qp/Zp 0.
→ → → →
∞
Thus the lifts of E[p ] to A are classified by:
j
Ext1p−div/Aj(Qp/Zp,µp∞) = l←im−Ext1GS/Aj(Z/pn,µpn),
where p div/A and GS/A denote respectively categories of p-divisible
j j
−
groups and of group schemes over A . The Kummer sequence for flat coho-
j
mology (cf. [7, example II.2.18., p. 66]) gives us an isomorphism
Ext1 (Z/pn,µ )= H1(A ,µ )= A×/(A×)pn.
GS/Aj pn ∼ fl j pn ∼ j j
Finally, by Lemma 2.5 the natural projection
Ext1p−div/Aj(Qp/Zp,µp∞) → Ext1GS/Aj(Z/pj−1,µpj−1)
is an isomorphism. Hence, the lifts of E/k to A are classified by the lifts
j
of E[pj−1]. However, Lemma 2.2 assures us that if E ,E /R are isomorphic
1 2 j
over k and A , then they must be also isomorphic over R . This ends the
j j
proof. (cid:3)
3. Proofs of Theorems 1.1 and 1.2
WeassumethatK/Q isafiniteextensionandsticktothenotationfromthe
p
previous section.Letalson = [K : Q ],d= [k :F ]andebetheramification
p p
degree of K/Qp. For an abelian group M we define rankpM := dimFpM[p].
The following lemma computes rank E (R ) forj big enough and provides
p Rj j
a lower bound in the remaining cases. It is a generalization of [3, Lemma
3.1], which computed rank E (R ) in the unramified case.
p R2 2
On p-degree of elliptic curves 8
Lemma 3.1. If E/Q has good reduction and i > e is an integer, then:
p p−1
rank E (R ) d (j i)+rank E(K), for i j < i+e,
p Rj j ≥ · − p ≤
rank E (R ) = n+rank E(K), for j i+e.
p Rj j p ≥
Proof. Let Ej := E and Ei := E . By applying the snake lemma to the
Rj Ri
sequence (2.3) with multiplication-by-p morphism and using (2.4) we obtain
the following commutative diagram:
0 0 E(K)[p] Ei(R )[p] mi/mi+e
i
0 mi[p] Ej(R )[p] Ei(R )[p] mi/mi+e
j j i j j
with exact rows. The lower row of the diagram induces an exact sequence:
(3.1) 0 mi[p] Ej(R )[p] ker Ei(R )[p] mi/mi+e 0.
→ j → j → i → j j →
(cid:16) (cid:17)
Elementary computations show that:
(Z/p)d·(j−i) for j < i+e
(3.2) mi[p]=
j ∼ (Z/p)n for j i+e
(cid:26) ≥
and that for j i+e the natural projection
≥
(3.3) mi/mi+e mi/mi+e
→ j j
is an isomorphism. We consider the following two cases:
1) if i j < i+e, then by (3.1) and (3.2):
≤
rank Ej(R ) = d (j i)+rank ker(Ei(R )[p] mi/mi+e)
p j · − p i → j j
d (j i)+rank E(K)[p].
p
≥ · −
2) if j i+e, then by (3.3):
≥
ker(Ei(R ) mi/mi+e) = ker(Ei(R ) mi/mi+e) = E(K)[p].
i → j j i → ∼
Thus by (3.1) the sequence
0 mi[p] Ej(R )[p] E(K)[p] 0
→ j → j → →
(cid:3)
is exact and the proof follows now from (3.2).
On p-degree of elliptic curves 9
Lemma 3.1 plays a central role in proofs of Theorems 1.1 and 1.2. Indeed,
the second part of the lemma easily implies Theorem 1.1:
Proof of Theorem 1.1. Letj = e+ e .Notethatj 2eandthuswehave
⌈p−1⌉ ≤
a natural homomorphism:
Z/p2 R .
j
→
Therefore ERj = (EZ/p2)Rj and the proof follows from the second part of
Lemma 3.1. (cid:3)
In order to prove Theorem 1.2 we investigate the p-torsion in fields of low
ramification:
Theorem 3.2. Let K/Q be a finite extension with ramification degree
p
e < p 1. We keep notation introduced in Section 2. For an elliptic curve
−
E/Q with good reduction the following conditions are equivalent:
p
(1) E(K)[p] = 0,
6
(2) E (k)[p] = 0 and rank E (R )[p] = d+1,
k 6 p R2 2
(3) Ek(k)[p] 6= 0 and EZ/p2 is the canonical lift of EFp,
(4) E(K Qun)[p] = 0.
∩ p 6
Proof. (1) (2).By(2.4)itfollowsthatE(K)[p] ֒ E (k)[p],sothatE (k)[p]=0.
k k
⇒ → 6
Therefore the étale-connected sequence of E [p] is of the form:
R2
(3.4) 0 µ E [p] Z/p 0
→ p → R2 → →
and thus:
rank E (R ) rank µ (R )+rank (Z/p)= d+1.
p R2 2 ≤ p p 2 p
Theequality followsfromthefirstpartof Lemma3.1applied to(i,j)=(1,2).
(2) (3). By comparing ranks, we see that the image of µ (R ) in
p 2
⇒
E (R )[p] has index p. Let g E (R )[p] be an element not belonging to
R2 2 ∈ R2 2
the image of µ (R ). Then g corresponds to a morphism Z/p E[p], which
p 2
→
(after an eventual twist by an automorphism of Z/p) provides a section of
the étale-connected sequence (3.4). Thus this sequence splits and E is a
R2
canonical lift of Ek. This implies that EZ/p2 is a canonical lift of EFp.
On p-degree of elliptic curves 10
(3) (4). Let us replace K by K Qun. Since E is a canonical lift
⇒ ∩ p R2
of E :
k
E [p](R ) = µ (R ) Z/p(R ) = (Z/p)d+1.
R2 2 ∼ p 2 ⊕ 2 ∼
Thus by Lemma 3.1 rank E(K) = rank E (R ) d = 1.
The implication (4) p (1) is obviousp. R2 2 − (cid:3)
⇒
Proof of Theorem 1.2. Theorem3.2easilyimplies(1) (2) (3).Theimpli-
cation (3) (4) follows from Theorem 3.2 and [3, Le⇒mma 4⇔.3.]. (cid:3)
⇒
Remark 3.3. The condition (4) of Theorem 1.2 doesn’t imply (3) in gen-
eral. In order to see this consider the curve E /Q . Its division poly-
1,1 5
nomial Ψ factors into two polynomials of degrees 2 and 10. Therefore
5
d (E) 2,4 . Suppose, fora contradiction, that E(Qun)[5] =0. Then The-
5 ∈ { } 5 6
orem 3.2 implies that E(K)[5] = 0 for some unramified extension K/Q of
p
6
degree 4. On the other hand the degree 2 factor of Φ is of the form:
5
≤
x2+ 2 5+4 52+... x+ 2 5−1+2+4 52+53+... .
· · · ·
(cid:0) (cid:1) (cid:0) (cid:1)
One easily checks that roots of the latter polynomial are ramified. This
implies that E(Qun)[5]=0 and d (E)=4 (since d (E)=2 would imply
5 5 5
E(Qun)[5] =0 by Theorem 1.2). Straightforward calculation shows that
5 6
ord a (E) = 4.
5 5
The implication (2) (1) is not true either. Indeed, let E/Q be the
5
⇒
canonical lift of E /F . Then E clearly satisfies the condition (2), but
1,1 5
d (E) = ord a (E) = 4. One constructs counterexamples for other primes
5 5 5
in a similar way.
The Theorem 1.2 shows that the elliptic curves with low p-degree are well
understood.Itmotivatesthefollowingquestion,whichseemstoremainopen:
Question 3.4. What values larger than (p 1) can the p-degree attain for
−
a fixed p? Are there any divisibility conditions on d (E), e.g. d (E)(p2 1)?
p p
| −
4. Supersingular, CM and multiplicative reduction curves
In the final section we compute the p-degree explicitly in some special cases.
We start with elliptic curves with supersingular reduction. Theorem 1.4 will
be proven by means of formal groups.