Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
A local-global principle for isogenies
between elliptic curves over number
fields
Samuele Anni
Universiteit Leiden - Université Bordeaux 1
Grenoble, 28th Journées Arithmétiques;
4th July 2013
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
1
Local-to-global principles
Local-to-global principle for elliptic curves: TORSION
Local-to-global principle for elliptic curves: ISOGENIES
Preliminaries and introduction
Local-to-global principles
Local-to-global principle for elliptic curves: TORSION
Local-to-global principle for elliptic curves: ISOGENIES
2
Exceptional pairs
3
New results
4
Finiteness result
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Local-to-global principles
Local-to-global principle for elliptic curves: TORSION
Local-to-global principle for elliptic curves: ISOGENIES
Local-global problems: from knowledge about local structures to
knowledge about global structures.
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Local-to-global principles
Local-to-global principle for elliptic curves: TORSION
Local-to-global principle for elliptic curves: ISOGENIES
Local-global problems: from knowledge about local structures to
knowledge about global structures.
In this context: from p-adic information global information. For example:
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Local-to-global principles
Local-to-global principle for elliptic curves: TORSION
Local-to-global principle for elliptic curves: ISOGENIES
Local-global problems: from knowledge about local structures to
knowledge about global structures.
In this context: from p-adic information global information. For example:
Local-to-global principles for quadratic forms.
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Local-to-global principles
Local-to-global principle for elliptic curves: TORSION
Local-to-global principle for elliptic curves: ISOGENIES
Local-global problems: from knowledge about local structures to
knowledge about global structures.
In this context: from p-adic information global information. For example:
Local-to-global principles for quadratic forms.
Theorem of Hasse-Minkowski
Two quadratic forms with coefficients in Q which are equivalent over Q p
for all prime numbers p and over R are equivalent over Q .
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Local-to-global principles
Local-to-global principle for elliptic curves: TORSION
Local-to-global principle for elliptic curves: ISOGENIES
Local-to-global principle for elliptic curves: TORSION
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Local-to-global principles
Local-to-global principle for elliptic curves: TORSION
Local-to-global principle for elliptic curves: ISOGENIES
Local-to-global principle for elliptic curves: TORSION
Let ` be a prime. Katz in 1981 studied the local-global principle for
`-torsion for elliptic curves.
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Local-to-global principles
Local-to-global principle for elliptic curves: TORSION
Local-to-global principle for elliptic curves: ISOGENIES
Local-to-global principle for elliptic curves: TORSION
Let ` be a prime. Katz in 1981 studied the local-global principle for
`-torsion for elliptic curves.
Theorem (Katz)
Let E be an elliptic curve over a number field K . If E has non-trivial
`-torsion locally at a set of primes with density one then E is
K -isogenous to an elliptic curve which has non-trivial `-torsion over K .
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Local-to-global principles
Local-to-global principle for elliptic curves: TORSION
Local-to-global principle for elliptic curves: ISOGENIES
Local-to-global principle for elliptic curves: TORSION
Let ` be a prime. Katz in 1981 studied the local-global principle for
`-torsion for elliptic curves.
Theorem (Katz)
Let E be an elliptic curve over a number field K . If E has non-trivial
`-torsion locally at a set of primes with density one then E is
K -isogenous to an elliptic curve which has non-trivial `-torsion over K .
He proved this by reducing the problem to a purely group-theoretic
statement.
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Local-to-global principles
Local-to-global principle for elliptic curves: TORSION
Local-to-global principle for elliptic curves: ISOGENIES
Local-to-global principle for elliptic curves: ISOGENIES
Definition
Let E be an elliptic curve defined on a number field K , and let ` be a
prime number. If p is a prime of K where E has good reduction, p not
dividing `, we say that E admits an `-isogeny locally at p if the Néron
model of E over the ring of integer of Kp admits an `-isogeny.
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Local-to-global principles
Local-to-global principle for elliptic curves: TORSION
Local-to-global principle for elliptic curves: ISOGENIES
Local-to-global principle for elliptic curves: ISOGENIES
Definition
Let E be an elliptic curve defined on a number field K , and let ` be a
prime number. If p is a prime of K where E has good reduction, p not
dividing `, we say that E admits an `-isogeny locally at p if the Néron
model of E over the ring of integer of Kp admits an `-isogeny.
If E admits an `-isogeny over K , then E necessarily admits an `-isogeny
locally at every prime of good reduction. The converse statement has
been recently studied by Sutherland (arxiv, November 2011).
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Local-to-global principles
Local-to-global principle for elliptic curves: TORSION
Local-to-global principle for elliptic curves: ISOGENIES
Question
Let E be an elliptic curve defined over a number field K , and let ` be a
prime number, if E admits an `-isogeny locally at a set of primes with
density one then does E admit an `-isogeny over K ?
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Local-to-global principles
Local-to-global principle for elliptic curves: TORSION
Local-to-global principle for elliptic curves: ISOGENIES
Question
Let E be an elliptic curve defined over a number field K , and let ` be a
prime number, if E admits an `-isogeny locally at a set of primes with
density one then does E admit an `-isogeny over K ?
Theorem (Sutherland)
Let E be an elliptic curve
over a number field K and let ` be a
qdefined
−1
prime number. Assume
`∈
/ K , and suppose E /K admits an
`
`-isogeny locally at a set of primes with density one.
Then E admits an `-isogeny over a quadratic extension of K .
Moreover, if ` ≡ 1 mod 4 or ` < 7, E admits an `-isogeny defined over K .
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
1
Preliminaries and introduction
2
Exceptional pairs
3
New results
4
Finiteness result
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Definition
Let K be a number field, let E be an elliptic curve over K and ` a prime
number, a pair (`, j(E )) is said to be exceptional for K if E /K admits
an `-isogeny locally everywhere but not over K .
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Definition
Let K be a number field, let E be an elliptic curve over K and ` a prime
number, a pair (`, j(E )) is said to be exceptional for K if E /K admits
an `-isogeny locally everywhere but not over K .
Sutherland proved the following result:
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Definition
Let K be a number field, let E be an elliptic curve over K and ` a prime
number, a pair (`, j(E )) is said to be exceptional for K if E /K admits
an `-isogeny locally everywhere but not over K .
Sutherland proved the following result:
Theorem (Sutherland)
The pair (7, 2268945/128) is the only exceptional pair for Q .
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Theorem (Sutherland)
Let E be an elliptic curve
over a number field K and let ` be a
qdefined
−1
prime number. Assume
`∈
/ K , and suppose E /K admits an
`
`-isogeny locally at a set of primes with density one.
Then E admits an `-isogeny over a quadratic extension of K .
Moreover, if ` ≡ 1 mod 4 or ` < 7, E admits an `-isogeny defined over K .
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Let E be an elliptic curve defined over a number field K , then on the
`-torsion points of E [`] (Q ) there is a Gal(Q /K )-action. Then there
exist a Galois representation ρE ,` :
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Let E be an elliptic curve defined over a number field K , then on the
`-torsion points of E [`] (Q ) there is a Gal(Q /K )-action. Then there
exist a Galois representation ρE ,` :
ρE ,`
Gal(Q /K ) −→ Aut(E [`]) ∼
= GL2 (F` ).
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Let E be an elliptic curve defined over a number field K , then on the
`-torsion points of E [`] (Q ) there is a Gal(Q /K )-action. Then there
exist a Galois representation ρE ,` :
ρE ,`
Gal(Q /K ) −→ Aut(E [`]) ∼
= GL2 (F` ).
Remark
Let (`, j(E )) be an exceptional pair for the number field K and let
G = ρE ,` (Gal(Q/K )). Then G is a subgroup of GL2 (F` ) such that
|P1 (F` )g | > 0 for all g ∈ G but |P1 (F` )G | = 0.
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Given an elliptic curve E , defined over a number field K , the
compatibility between ρE ,` and the Weil pairing on E [`] implies that:
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Given an elliptic curve E , defined over a number field K , the
compatibility between ρE ,` and the Weil pairing on E [`] implies that:
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Given an elliptic curve E , defined over a number field K , the
compatibility between ρE ,` and the Weil pairing on E [`] implies that:
ζ` is in K if and only if G is contained in SL2 (F` );
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Given an elliptic curve E , defined over a number field K , the
compatibility between ρE ,` and the Weil pairing on E [`] implies that:
ζ` is in K if and only if G is contained in SL2 (F` );
H, the q
projective image of G , is contained in SL2 (F` )/ {±1} if and
−1
only if
` is in K .
`
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Given an elliptic curve E , defined over a number field K , the
compatibility between ρE ,` and the Weil pairing on E [`] implies that:
ζ` is in K if and only if G is contained in SL2 (F` );
H, the q
projective image of G , is contained in SL2 (F` )/ {±1} if and
−1
only if
` is in K .
`
Remark
The solution q
to the local-global principle about `-isogenies over K
−1
depends on
` belonging to K .
`
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
First case
Second case
1
Preliminaries and introduction
2
Exceptional pairs
3
New results
First case
Second case
4
Finiteness result
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Let us assume that
q
−1
`
First case
Second case
`∈
/ K.
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Let us assume that
q
−1
`
First case
Second case
`∈
/ K.
Lemma (Sutherland)
Let G be a subgroup of GL2 (F` ) whose image H in PGL2 (F` ) is not
contained in SL2 (F` )/ {±1}. Suppose |P1 (F` )g | > 0 for all g ∈ G but
|P1 (F` )G | = 0.
Then ` ≡ 3 mod 4 and the following holds:
(1) H is dihedral of order 2n, where n > 1 is an odd divisor of (`−1)/2;
(2) G is properly contained in the normalizer of a split Cartan subgroup;
(3) P1 (F` )/G contains an orbit of size 2.
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
First case
Second case
Proposition (A.)
q
−1
Let (`, j(E )) be an exceptional pair for the number field K with
`
`
√
not belonging to K . Then E admits an `-isogeny over K ( −`) (and
actually, two such isogenies).
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
First case
Second case
Main Theorem (A.)
Let (`, j(E )) be an q
exceptional pair for the number field K of degree d
−1
over Q , such that
`∈
/ K . Then
`
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
First case
Second case
Main Theorem (A.)
Let (`, j(E )) be an q
exceptional pair for the number field K of degree d
−1
over Q , such that
`∈
/ K . Then ` ≡ 3 mod 4 and
`
7 ≤ ` ≤ 6d+1.
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
First case
Second case
Main Theorem (A.)
Let (`, j(E )) be an q
exceptional pair for the number field K of degree d
−1
over Q , such that
`∈
/ K . Then ` ≡ 3 mod 4 and
`
7 ≤ ` ≤ 6d+1.
Remark
This theorem implies the result obtained by Sutherland in the case
K = Q.
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
First case
Second case
Main Theorem (A.)
Let (`, j(E )) be an q
exceptional pair for the number field K of degree d
−1
over Q , such that
`∈
/ K . Then ` ≡ 3 mod 4 and
`
7 ≤ ` ≤ 6d+1.
Remark
This theorem implies the result obtained by Sutherland in the case
K = Q.
This theorem is proved studying the Galois representation ρE ,` .
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Let us assume that
q
−1
`
First case
Second case
` ∈ K.
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Let us assume that
q
−1
`
First case
Second case
` ∈ K.
Lemma (A.)
Let G be a subgroup of GL2 (F` ) whose image H in PGL2 (F` ) is
contained in SL2 (F` )/ {±1}. Suppose |P1 (F` )g | > 0 for all g ∈ G but
|P1 (F` )G | = 0. Then ` ≡ 1 mod 4 and one of the followings holds:
(1) H is dihedral of order 2n, where n ∈ Z >1 is a divisor of `−1;
(2) H is isomorphic to one of the following exceptional groups: A4 , S4 or
A5 .
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
First case
Second case
Proposition (A.)
Let E be an elliptic curve defined over a number q
field K of degree d over
−1
` ∈ K . Suppose
Q and let ` be a prime number. Let us suppose
`
E /K admits an `-isogeny locally at a set of primes with density one.
Then:
(1) if `≡3 mod 4 the elliptic curve E admits a global `-isogeny over K ;
(2) if `≡1 mod 4 the elliptic curve E admits an `-isogeny over L, finite
extension of K , which can ramify only at primes dividing the
conductor of E and `.
Moreover, if `≡−1 mod 3 or if ` ≥ 60d+1, then E admits an
`-isogeny over a quadratic extension L of K .
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
First case
Second case
Proposition (A.)
Let (`, j(E )) be an exceptional pair for the number field K of degree d
over Q and discriminant ∆. Then
` ≤ max {∆, 6d+1} .
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
1
Preliminaries and introduction
2
Exceptional pairs
3
New results
4
Finiteness result
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Question
Let K be a number field and let ` be a prime number, how many
exceptional pairs (`, j(E )) do exist over K ?
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Proposition (A.)
Given a number field K :
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Proposition (A.)
Given a number field K :
if ` = 2, 3 then there exists no exceptional pair;
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Proposition (A.)
Given a number field K :
if ` = 2, 3 then there exists no exceptional pair;
there exist infinitely many
√ exceptional pairs (5, j(E )) for the number
field K if and only if 5 belongs to K ;
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Proposition (A.)
Given a number field K :
if ` = 2, 3 then there exists no exceptional pair;
there exist infinitely many
√ exceptional pairs (5, j(E )) for the number
field K if and only if 5 belongs to K ;
if ` > 7, then the number of exceptional pairs (`, j(E )) is finite.
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Sketch of the Proof
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Sketch of the Proof
If ` = 2, 3 then the result follows from the previous combinatorics.
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Sketch of the Proof
If ` = 2, 3 then the result follows from the previous combinatorics.
For ` ≥ 5 the result follows from counting rational points on
modular curves.
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Sketch of the Proof
If ` = 2, 3 then the result follows from the previous combinatorics.
For ` ≥ 5 the result follows from counting rational points on
modular curves.
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Modular curves
Let ` ≥ 5 be a prime. The modular curve X (`) is the compactified fine
moduli space which classify, up to isomorphism, pairs (E , α), where E is
a generalized elliptic curve over a scheme S over Spec(Z [1/`, ζ` ]) and
∼
α : (Z /`Z )2S → E [`] is an isomorphism of group schemes over S which is
a full level `-structure.
A full level `-structure on a generalized elliptic curve E over S is a pair of
points (P1 , P2 ), satisfying P1 , P2 ∈ E [`] and e` (P1 , P2 ) = ζ` where e` is
the Weil pairing on E [`].
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
In the case ` = 5 the modular curve considered is
XV4 (5) := G \X (5),
where G ⊂ GL2 (Z /`Z ) is the inverse image of V4 ⊂ PGL2 (Z /`Z ).
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
In the case ` = 5 the modular curve considered is
XV4 (5) := G \X (5),
where G ⊂ GL2 (Z /`Z ) is the inverse image of V4 ⊂ PGL2 (Z /`Z ).
Proposition
√
Over Spec(Q ( 5)) the modular curve XV4 (5) is isomorphic to P1 .
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
For ` > 7 the result follows applying Faltings’ Theorem.
In this case, an exceptional pair essentially (we are not considering
exceptional subgroups) corresponds to a rational point of
Xsplit (`) := G \X (`),
where G is the normalizer of a split Cartan subgroup of GL2 (F` ). The
curve Xsplit (`) parametrizes elliptic curves endowed with a pair of
independent cyclic `-isogenies.
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
The local-global principle for 7-isogenies leads us to a dichotomy between
a finite and an infinite number of counterexamples according to the rank
of a specific elliptic curve that we call the Elkies-Sutherland curve:
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
The local-global principle for 7-isogenies leads us to a dichotomy between
a finite and an infinite number of counterexamples according to the rank
of a specific elliptic curve that we call the Elkies-Sutherland curve:
Proposition (A.)
If ` = 7 then the number of exceptional pairs (7, j(E )) for a number field
K , is finite or infinite, depending on the rank of the elliptic curve
E 0 : y 2 = x 3 − 1715x + 33614
being respectively 0 or positive.
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Examples for 7-isogenies:
√
For Q ( −23) and Q (i) there are infinitely many counterexamples to
the local-global principle about 7-isogenies.
For any field in the following table there are finitely many
counterexamples to the local-global principle about 7-isogenies:
√
Q (√−14)
Q (√−21)
Q (√−35)
Q (√−42)
Q (√ −91)
Q ( −105)
√
Q (√−119)
Q (√−133)
Q (√−154)
Q (√−161)
Q (√−182)
Q ( −203)
Samuele Anni
√
Q (√−210)
Q (√−217)
Q (√−231)
Q (√−238)
Q (√−259)
Q ( −287)
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
Further directions
Generalization for simple abelian varieties of dimension d over Q
which are principally polarized i.e. study of the subgroups of
P GSp2d (F` )...
Generalization for abelian varieties of GL2 -type;
Generalization for isogenies of prime power degree;
Generalization for isogenies of degree given by products of primes;
...
Samuele Anni
A local-global principle for isogenies over number fields
Preliminaries and introduction
Exceptional pairs
New results
Finiteness result
“Think Globally, Act Locally”
Patrick Geddes
Thanks!
Samuele Anni
Samuele Anni
A local-global principle for isogenies over number fields