2-Selmer groups and Heegner points
Chao Li
Department of Mathematics
Harvard University
May 24, 2014 / FRG Workshop in Cambridge
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
BSD Conjecture
Let E/Q be an elliptic curve.
I
(Rank Conjecture)
?
rank E(Q) = ords=1 L(E/Q, s).
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
BSD Conjecture
Let E/Q be an elliptic curve.
I
(Rank Conjecture)
?
rank E(Q) = ords=1 L(E/Q, s).
I
Known when ords=1 L(E/Q, s) ≤ 1 (Gross-Zagier, Kolyvagin, ...)
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
BSD Conjecture
I
(Refined BSD formula)
L(r ) (E/Q, 1) ?
=
r!
Chao Li (Harvard)
Z
ω·
E(R)
Y
cp · |Ш(E/Q)| ·
p
2-Selmer groups and Heegner points
det(hPi , Pj iri,j=1 )
|E(Q)tor |2
.
FRG 2014
BSD Conjecture
I
(Refined BSD formula)
L(r ) (E/Q, 1) ?
=
r!
Z
ω·
E(R)
Y
cp · |Ш(E/Q)| ·
p
Or,
L(r ) (E/Q, 1)
?
=
r !Ω(E/Q)R(E/Q)
Chao Li (Harvard)
det(hPi , Pj iri,j=1 )
Q
p
|E(Q)tor |2
cp · |Ш(E/Q)|
|E(Q)tor |2
2-Selmer groups and Heegner points
.
.
FRG 2014
BSD Conjecture
I
(Refined BSD formula)
L(r ) (E/Q, 1) ?
=
r!
Z
ω·
E(R)
Y
cp · |Ш(E/Q)| ·
p
Or,
L(r ) (E/Q, 1)
?
=
r !Ω(E/Q)R(E/Q)
I
det(hPi , Pj iri,j=1 )
Q
p
|E(Q)tor |2
cp · |Ш(E/Q)|
|E(Q)tor |2
.
.
Known cases of p-part of BSD formula (under mild assumptions)
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
BSD Conjecture
I
(Refined BSD formula)
L(r ) (E/Q, 1) ?
=
r!
Z
ω·
E(R)
Y
cp · |Ш(E/Q)| ·
p
Or,
L(r ) (E/Q, 1)
?
=
r !Ω(E/Q)R(E/Q)
I
det(hPi , Pj iri,j=1 )
Q
p
|E(Q)tor |2
cp · |Ш(E/Q)|
|E(Q)tor |2
.
.
Known cases of p-part of BSD formula (under mild assumptions)
I
p ≥ 3 when ords=1 L(E/Q, s) = 0 (Skinner-Urban, Kato, ...)
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
BSD Conjecture
I
(Refined BSD formula)
L(r ) (E/Q, 1) ?
=
r!
Z
ω·
E(R)
Y
cp · |Ш(E/Q)| ·
p
Or,
L(r ) (E/Q, 1)
?
=
r !Ω(E/Q)R(E/Q)
I
det(hPi , Pj iri,j=1 )
Q
p
|E(Q)tor |2
cp · |Ш(E/Q)|
|E(Q)tor |2
.
.
Known cases of p-part of BSD formula (under mild assumptions)
I
I
p ≥ 3 when ords=1 L(E/Q, s) = 0 (Skinner-Urban, Kato, ...)
p ≥ 5 when ords=1 L(E/Q, s) = 1 (Wei Zhang, ...)
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
BSD Conjecture
I
(Refined BSD formula)
L(r ) (E/Q, 1) ?
=
r!
Z
ω·
E(R)
Y
cp · |Ш(E/Q)| ·
p
Or,
L(r ) (E/Q, 1)
?
=
r !Ω(E/Q)R(E/Q)
I
Q
p
|E(Q)tor |2
cp · |Ш(E/Q)|
|E(Q)tor |2
.
.
Known cases of p-part of BSD formula (under mild assumptions)
I
I
I
det(hPi , Pj iri,j=1 )
p ≥ 3 when ords=1 L(E/Q, s) = 0 (Skinner-Urban, Kato, ...)
p ≥ 5 when ords=1 L(E/Q, s) = 1 (Wei Zhang, ...)
p=2?
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
BSD Conjecture
I
(Refined BSD formula)
L(r ) (E/Q, 1) ?
=
r!
Z
ω·
E(R)
Y
cp · |Ш(E/Q)| ·
p
Or,
L(r ) (E/Q, 1)
?
=
r !Ω(E/Q)R(E/Q)
I
Q
p
|E(Q)tor |2
cp · |Ш(E/Q)|
|E(Q)tor |2
.
.
Known cases of p-part of BSD formula (under mild assumptions)
I
I
I
det(hPi , Pj iri,j=1 )
p ≥ 3 when ords=1 L(E/Q, s) = 0 (Skinner-Urban, Kato, ...)
p ≥ 5 when ords=1 L(E/Q, s) = 1 (Wei Zhang, ...)
p = 2 ? Why p = 2?
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
BSD formula
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
I
Our strategy: study the 2-Selmer group
Sel2 (E/Q) ⊆ H 1 (Q, E[2])
via level raising of modular forms (mod 2).
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
I
Our strategy: study the 2-Selmer group
Sel2 (E/Q) ⊆ H 1 (Q, E[2])
via level raising of modular forms (mod 2).
I
Inspired by W. Zhang’s recent work.
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
I
E/Q: elliptic curve of odd conductor N.
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
I
I
E/Q: elliptic curve of odd conductor N.
∼ GL2 (F2 )
ρ̄ = ρ̄E,2 : GQ → Aut(E[2]) =
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
I
I
E/Q: elliptic curve of odd conductor N.
∼ GL2 (F2 )
ρ̄ = ρ̄E,2 : GQ → Aut(E[2]) =
Assume (throughout this talk) the following mild hypothesis:
1. (surj) ρ̄ is surjective
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
I
I
E/Q: elliptic curve of odd conductor N.
∼ GL2 (F2 )
ρ̄ = ρ̄E,2 : GQ → Aut(E[2]) =
Assume (throughout this talk) the following mild hypothesis:
1. (surj) ρ̄ is surjective
2. (ram) Serre conductor N(ρ̄) = N
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
I
I
E/Q: elliptic curve of odd conductor N.
∼ GL2 (F2 )
ρ̄ = ρ̄E,2 : GQ → Aut(E[2]) =
Assume (throughout this talk) the following mild hypothesis:
1. (surj) ρ̄ is surjective
2. (ram) Serre conductor N(ρ̄) = N
3. (nontrivial at 2) ρ̄|GQ is nontrivial
2
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
I
I
E/Q: elliptic curve of odd conductor N.
∼ GL2 (F2 )
ρ̄ = ρ̄E,2 : GQ → Aut(E[2]) =
Assume (throughout this talk) the following mild hypothesis:
1. (surj) Q(E[2])/Q is a GL2 (F2 ) ∼
= S3 -extension (=⇒ E[2](Q) = 0).
2. (ram) Serre conductor N(ρ̄) = N
3. (nontrivial at 2) ρ̄|GQ is nontrivial
2
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
I
I
E/Q: elliptic curve of odd conductor N.
∼ GL2 (F2 )
ρ̄ = ρ̄E,2 : GQ → Aut(E[2]) =
Assume (throughout this talk) the following mild hypothesis:
1. (surj) Q(E[2])/Q is a GL2 (F2 ) ∼
= S3 -extension (=⇒ E[2](Q) = 0).
2. (ram) the order of component
group of the Neron model of E is
Q
odd at any p | N (=⇒ 2 - p cp ).
3. (nontrivial at 2) ρ̄|GQ is nontrivial
2
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
I
I
E/Q: elliptic curve of odd conductor N.
∼ GL2 (F2 )
ρ̄ = ρ̄E,2 : GQ → Aut(E[2]) =
Assume (throughout this talk) the following mild hypothesis:
1. (surj) Q(E[2])/Q is a GL2 (F2 ) ∼
= S3 -extension (=⇒ E[2](Q) = 0).
2. (ram) the order of component
group of the Neron model of E is
Q
odd at any p | N (=⇒ 2 - p cp ).
3. (nontrivial at 2) 2 does not split in Q(E[2])/Q
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
I
Under these assumptions,
E[2] (as GQ -module) + knowledge of reduction type at p
pins down the local condition for Sel2 (E/Q) at p
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
I
Under these assumptions,
E[2] (as GQ -module) + knowledge of reduction type at p
pins down the local condition for Sel2 (E/Q) at p
I
Keep E[2], but at a prime q - 2N of choice,
good reduction at q
Chao Li (Harvard)
multiplicative reduction at q
2-Selmer groups and Heegner points
FRG 2014
I
Under these assumptions,
E[2] (as GQ -module) + knowledge of reduction type at p
pins down the local condition for Sel2 (E/Q) at p
I
Keep E[2], but at a prime q - 2N of choice,
good reduction at q
I
multiplicative reduction at q
Necessarily:
ρ̄(Frobq ) =
Chao Li (Harvard)
q ∗
0 1
(mod 2)
2-Selmer groups and Heegner points
FRG 2014
I
Under these assumptions,
E[2] (as GQ -module) + knowledge of reduction type at p
pins down the local condition for Sel2 (E/Q) at p
I
Keep E[2], but at a prime q - 2N of choice,
good reduction at q
I
Necessarily:
ρ̄(Frobq ) =
I
multiplicative reduction at q
q ∗
0 1
(mod 2)
In GL2 (F2 ) ∼
= S3 ,
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
I
Under these assumptions,
E[2] (as GQ -module) + knowledge of reduction type at p
pins down the local condition for Sel2 (E/Q) at p
I
Keep E[2], but at a prime q - 2N of choice,
good reduction at q
I
Necessarily:
ρ̄(Frobq ) =
I
multiplicative reduction at q
q ∗
0 1
(mod 2)
In GL2 (F2 ) ∼
= S3 ,
I
10
01
(trivial)
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
I
Under these assumptions,
E[2] (as GQ -module) + knowledge of reduction type at p
pins down the local condition for Sel2 (E/Q) at p
I
Keep E[2], but at a prime q - 2N of choice,
good reduction at q
I
Necessarily:
ρ̄(Frobq ) =
I
multiplicative reduction at q
q ∗
0 1
(mod 2)
In GL2 (F2 ) ∼
= S3 ,
I
I
1
0
1
0
0
1
1
1
(trivial)
(order 2)
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
I
Under these assumptions,
E[2] (as GQ -module) + knowledge of reduction type at p
pins down the local condition for Sel2 (E/Q) at p
I
Keep E[2], but at a prime q - 2N of choice,
good reduction at q
I
Necessarily:
ρ̄(Frobq ) =
I
multiplicative reduction at q
q ∗
0 1
(mod 2)
In GL2 (F2 ) ∼
= S3 ,
I
I
I
1
0
1
0
1
1
0
1
1
1
1
0
(trivial)
(order 2)
(order 3)
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
I
Under these assumptions,
E[2] (as GQ -module) + knowledge of reduction type at p
pins down the local condition for Sel2 (E/Q) at p
I
Keep E[2], but at a prime q - 2N of choice,
good reduction at q
I
Necessarily:
ρ̄(Frobq ) =
I
multiplicative reduction at q
q ∗
0 1
(mod 2)
In GL2 (F2 ) ∼
= S3 ,
I
I
I
1
0
1
0
1
1
0
1
1
1
1
0
(trivial)
(order 2)
(order 3)
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
I
Under these assumptions,
E[2] (as GQ -module) + knowledge of reduction type at p
pins down the local condition for Sel2 (E/Q) at p
I
Keep E[2], but at a prime q - 2N of choice,
good reduction at q
I
Necessarily:
ρ̄(Frobq ) =
I
multiplicative reduction at q
q ∗
0 1
(mod 2)
In GL2 (F2 ) ∼
= S3 ,
I
I
I
1
0
1
0
1
1
0
1
1
1
1
0
(trivial)
(order 2)
(order 3)
Theorem (Ribet’s level raising)
Let q - 2N be a prime. Suppose ρ̄(Frobq ) =
comes from a weight 2 newform of level Nq.
Chao Li (Harvard)
10
01
2-Selmer groups and Heegner points
or
11
01
, then ρ̄
FRG 2014
I
E/Q
f =
Chao Li (Harvard)
P
n≥1 an q
n
∈ S2 (N)new .
2-Selmer groups and Heegner points
FRG 2014
I
E/Q
f =
I
ρ̄(Frobq ) =
Chao Li (Harvard)
n
new .
n≥1 an q ∈ S2 (N)
1 0 or 1 1 ⇐⇒ 2 | a .
q
01
01
P
2-Selmer groups and Heegner points
FRG 2014
n
new .
n≥1 an q ∈ S2 (N)
I ρ̄(Frobq ) = 1 0 or 1 1 ⇐⇒ 2 | aq .
01
01
P
I 2 | aq
g = n≥1 bn q n ∈ S2 (Nq)new
I
E/Q
f =
P
f ≡g
Chao Li (Harvard)
such that
(mod 2).
2-Selmer groups and Heegner points
FRG 2014
n
new .
n≥1 an q ∈ S2 (N)
I ρ̄(Frobq ) = 1 0 or 1 1 ⇐⇒ 2 | aq .
01
01
P
I 2 | aq
g = n≥1 bn q n ∈ S2 (Nq)new
I
E/Q
f =
P
f ≡g
I
such that
(mod 2).
More precisely,
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
n
new .
n≥1 an q ∈ S2 (N)
I ρ̄(Frobq ) = 1 0 or 1 1 ⇐⇒ 2 | aq .
01
01
P
I 2 | aq
g = n≥1 bn q n ∈ S2 (Nq)new
I
E/Q
f =
P
f ≡g
I
such that
(mod 2).
More precisely,
I
F = Q(g) (totally real) with ring of integers O = OF .
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
n
new .
n≥1 an q ∈ S2 (N)
I ρ̄(Frobq ) = 1 0 or 1 1 ⇐⇒ 2 | aq .
01
01
P
I 2 | aq
g = n≥1 bn q n ∈ S2 (Nq)new
I
E/Q
f =
P
f ≡g
I
such that
(mod 2).
More precisely,
I
I
F = Q(g) (totally real) with ring of integers O = OF .
∃λ | 2 a prime of O such that for any p 6= q,
ap ≡ bp
Chao Li (Harvard)
(mod λ)
2-Selmer groups and Heegner points
FRG 2014
n
new .
n≥1 an q ∈ S2 (N)
I ρ̄(Frobq ) = 1 0 or 1 1 ⇐⇒ 2 | aq .
01
01
P
I 2 | aq
g = n≥1 bn q n ∈ S2 (Nq)new
I
E/Q
f =
P
f ≡g
I
(mod 2).
More precisely,
I
I
F = Q(g) (totally real) with ring of integers O = OF .
∃λ | 2 a prime of O such that for any p 6= q,
ap ≡ bp
I
such that
(mod λ)
g
A/Q an abelian variety up to isogeny, of dimension [F : Q]
with real multiplication by F .
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
n
new .
n≥1 an q ∈ S2 (N)
I ρ̄(Frobq ) = 1 0 or 1 1 ⇐⇒ 2 | aq .
01
01
P
I 2 | aq
g = n≥1 bn q n ∈ S2 (Nq)new
I
E/Q
f =
P
f ≡g
I
such that
(mod 2).
More precisely,
I
I
F = Q(g) (totally real) with ring of integers O = OF .
∃λ | 2 a prime of O such that for any p 6= q,
ap ≡ bp
(mod λ)
I
g
A/Q an abelian variety up to isogeny, of dimension [F : Q]
with real multiplication by F .
I
Choose A in the isogeny class so that O ,→ End(A). It is unique
up to prime-to-λ isogenies.
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Definition
Such q is called a level raising prime for E. We say that A is obtained
from E via level raising at q; A and E are congruent mod 2.
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Definition
Such q is called a level raising prime for E. We say that A is obtained
from E via level raising at q; A and E are congruent mod 2.
Remark
There are a lot of level raising primes!
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Definition
Such q is called a level raising prime for E. We say that A is obtained
from E via level raising at q; A and E are congruent mod 2.
Remark
There are a lot of level raising primes!
Example
E = X0 (11) : y 2 + y = x 3 − x 2 − 10x − 20.
ap
11a
Chao Li (Harvard)
2
-2
3
-1
5
1
7
-2
11
1
13
4
2-Selmer groups and Heegner points
17
2
19
0
FRG 2014
Definition
Such q is called a level raising prime for E. We say that A is obtained
from E via level raising at q; A and E are congruent mod 2.
Remark
There are a lot of level raising primes!
Example
E = X0 (11) : y 2 + y = x 3 − x 2 − 10x − 20. q = 7 is a level raising
prime. There are three (isogeny classes of) elliptic curves of conductor
77.
ap
11a
77a
77b
77c
Chao Li (Harvard)
2
-2
0
0
1
3
-1
-3
1
2
5
1
-1
3
-2
7
-2
-1
1
-1
11
1
-1
-1
1
13
4
-4
-4
4
2-Selmer groups and Heegner points
17
2
2
-6
4
19
0
-6
2
0
FRG 2014
Definition
Such q is called a level raising prime for E. We say that A is obtained
from E via level raising at q; A and E are congruent mod 2.
Remark
There are a lot of level raising primes!
Example
E = X0 (11) : y 2 + y = x 3 − x 2 − 10x − 20. q = 7 is a level raising
prime. There are three (isogeny classes of) elliptic curves of conductor
77. Two of them are congruent to E mod 2.
ap
11a
77a
77b
77c
Chao Li (Harvard)
2
-2
0
0
1
3
-1
-3
1
2
5
1
-1
3
-2
7
-2
-1
1
-1
11
1
-1
-1
1
13
4
-4
-4
4
2-Selmer groups and Heegner points
17
2
2
-6
4
19
0
-6
2
0
FRG 2014
I
Let k = O/λ. Then
E[2] ⊗F2 k ∼
= A[λ].
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
I
Let k = O/λ. Then
E[2] ⊗F2 k ∼
= A[λ].
I
2-descent:
0 → E(Q) ⊗Z F2 → Sel2 (E/Q) → Ш(E/Q)[2] → 0,
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
I
Let k = O/λ. Then
E[2] ⊗F2 k ∼
= A[λ].
I
2-descent:
0 → E(Q) ⊗Z F2 → Sel2 (E/Q) → Ш(E/Q)[2] → 0,
I
λ-descent:
0 → A(Q) ⊗O k → Selλ (A/Q) → Ш(A/Q)[λ] → 0.
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
I
Let k = O/λ. Then
E[2] ⊗F2 k ∼
= A[λ].
I
2-descent:
0 → E(Q) ⊗Z F2 → Sel2 (E/Q) → Ш(E/Q)[2] → 0,
I
λ-descent:
0 → A(Q) ⊗O k → Selλ (A/Q) → Ш(A/Q)[λ] → 0.
I
Can compare the Selmer groups of E and A,
Sel2 (E/Q)
⊗k

_
H 1 (Q, E[2]) ⊗ k
Chao Li (Harvard)
Selλ (A/Q)

_
H 1 (Q, A[λ])
2-Selmer groups and Heegner points
FRG 2014
I
Let k = O/λ. Then
E[2] ⊗F2 k ∼
= A[λ].
I
2-descent:
0 → E(Q) ⊗Z F2 → Sel2 (E/Q) → Ш(E/Q)[2] → 0,
I
λ-descent:
0 → A(Q) ⊗O k → Selλ (A/Q) → Ш(A/Q)[λ] → 0.
I
Can compare the Selmer groups of E and A,
Sel2 (E/Q)
⊗k

_
H 1 (Q, E[2]) ⊗ k
Chao Li (Harvard)
Selλ (A/Q)

_
H 1 (Q, A[λ])
2-Selmer groups and Heegner points
FRG 2014
Parity conjecture
I
Assume K is an imaginary quadratic field such that dK 6= −4, −3
satisfying the Heegner hypothesis:
p | N =⇒ p is split in K .
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Parity conjecture
I
Assume K is an imaginary quadratic field such that dK 6= −4, −3
satisfying the Heegner hypothesis:
p | N =⇒ p is split in K .
I
Then
Q K /Q is disjoint with Q(E[2])/Q (=⇒ E[2](K ) = 0 and
2 - v |N cv ).
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Parity conjecture
I
Assume K is an imaginary quadratic field such that dK 6= −4, −3
satisfying the Heegner hypothesis:
p | N =⇒ p is split in K .
I
Then
Q K /Q is disjoint with Q(E[2])/Q (=⇒ E[2](K ) = 0 and
2 - v |N cv ).
I
Half of the level raising primes q are inert in K . Choose such q
henceforth, then
ε(f /K ) = −ε(g/K ).
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Parity conjecture
I
Assume K is an imaginary quadratic field such that dK 6= −4, −3
satisfying the Heegner hypothesis:
p | N =⇒ p is split in K .
I
Then
Q K /Q is disjoint with Q(E[2])/Q (=⇒ E[2](K ) = 0 and
2 - v |N cv ).
I
Half of the level raising primes q are inert in K . Choose such q
henceforth, then
ε(f /K ) = −ε(g/K ).
I
BSD =⇒ rank E(K ) and rankF A(K ) have different parity.
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Parity conjecture
I
Assume K is an imaginary quadratic field such that dK 6= −4, −3
satisfying the Heegner hypothesis:
p | N =⇒ p is split in K .
I
Then
Q K /Q is disjoint with Q(E[2])/Q (=⇒ E[2](K ) = 0 and
2 - v |N cv ).
I
Half of the level raising primes q are inert in K . Choose such q
henceforth, then
ε(f /K ) = −ε(g/K ).
I
BSD =⇒ rank E(K ) and rankF A(K ) have different parity.
I
dimF2 Ш(E/K )[2] and dimk Ш(A/K )[λ] should be even.
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
BSD predicts that
Parity Conjecture
dimF2 Sel2 (E/K ) and dimk Selλ (A/K ) have different parity.
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
BSD predicts that
Parity Conjecture
dimF2 Sel2 (E/K ) and dimk Selλ (A/K ) have different parity.
Theorem A (–)
When ρ̄(Frobq ) =
11
01
,
dimF2 Sel2 (E/K ) = dimk Selλ (A/K ) ± 1.
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
BSD predicts that
Parity Conjecture
dimF2 Sel2 (E/K ) and dimk Selλ (A/K ) have different parity.
Theorem A (–)
When ρ̄(Frobq ) =
11
01
,
dimF2 Sel2 (E/K ) = dimk Selλ (A/K ) ± 1.
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Remark
The conclusion of the theorem may fail when (nontrivial at 2) is not
satisfied due to the uncertainty of the local condition at 2.
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Remark
The conclusion of the theorem may fail when (nontrivial at 2) is not
satisfied due to the uncertainty of the local condition at 2.
Example
The elliptic curve
E = 2351a1 : y 2 + xy + y = x 3 − 5x − 5
has trivial ρ̄|GQ .
2
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Remark
The conclusion of the theorem may fail when (nontrivial at 2) is not
satisfied due to the uncertainty of the local condition at 2.
Example
The elliptic curve
E = 2351a1 : y 2 + xy + y = x 3 − 5x − 5
has trivial ρ̄|GQ . The elliptic curve
2
A = 25861i1 : y 2 + xy + y = x 3 + x 2 − 17x + 30
is obtained from E via level raising at q = 11.
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Remark
The conclusion of the theorem may fail when (nontrivial at 2) is not
satisfied due to the uncertainty of the local condition at 2.
Example
The elliptic curve
E = 2351a1 : y 2 + xy + y = x 3 − 5x − 5
has trivial ρ̄|GQ . The elliptic curve
2
A = 25861i1 : y 2 + xy + y = x 3 + x 2 − 17x + 30
√
is obtained from E via level raising at q = 11. For K = Q( −111),
rank E(K ) = dimF2 Sel2 (E/K ) = 1,
rank A(K ) = dimF2 Sel2 (A/K ) = 4.
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Remark
dimk Ш(A/Q)[λ] is not necessarily even. No parity prediction for
2-Selmer groups over Q!
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Remark
dimk Ш(A/Q)[λ] is not necessarily even. No parity prediction for
2-Selmer groups over Q!
Remark
When A has an odd degree polarization, Poonen-Stoll constructed
c ∈ Ш(A/Q)[2] with Cassels-Tate pairing hc, ci = 0 or 1/2 ∈ Q/Z so
that h, i on Ш(A/Q)[2] is alternating if and only if hc, ci = 0. A
quadratic base change K /Q kills this obstruction.
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Heegner points
Corollary
If dim Sel2 (E/K ) = 1, then dimk Selλ (A/K ) = 0 or 2.
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Heegner points
Corollary
If dim Sel2 (E/K ) = 1, then dimk Selλ (A/K ) = 0 or 2.
Theorem B (–)
Suppose dim Sel2 (E/K ) =
1. Let yK ∈ E(K ) be a Heegner point.
Suppose ρ̄(Frobq ) = 10 11 and # ker(J0 (N) → J1 (N)) is odd. For
simplicity assume dim A = 1.
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Heegner points
Corollary
If dim Sel2 (E/K ) = 1, then dimk Selλ (A/K ) = 0 or 2.
Theorem B (–)
Suppose dim Sel2 (E/K ) =
1. Let yK ∈ E(K ) be a Heegner point.
Suppose ρ̄(Frobq ) = 10 11 and # ker(J0 (N) → J1 (N)) is odd. For
simplicity assume dim A = 1.
If the 2-part of BSD formula is true for A/K . Then
Sel2 (A/K ) = 0 ⇐⇒ yK mod q 6∈ 2E(Fq 2 ) + E(Fq ).
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Heegner points
Corollary
If dim Sel2 (E/K ) = 1, then dimk Selλ (A/K ) = 0 or 2.
Theorem B (–)
Suppose dim Sel2 (E/K ) =
1. Let yK ∈ E(K ) be a Heegner point.
Suppose ρ̄(Frobq ) = 10 11 and # ker(J0 (N) → J1 (N)) is odd. For
simplicity assume dim A = 1.
If the 2-part of BSD formula is true for A/K . Then
Sel2 (A/K ) = 0 ⇐⇒ yK mod q 6∈ 2E(Fq 2 ) + E(Fq ).
(The latter is an index 2 subgroup of E(Fq 2 )).
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Heegner points
Corollary
If dim Sel2 (E/K ) = 1, then dimk Selλ (A/K ) = 0 or 2.
Theorem B (–)
Suppose dim Sel2 (E/K ) =
1. Let yK ∈ E(K ) be a Heegner point.
Suppose ρ̄(Frobq ) = 10 11 and # ker(J0 (N) → J1 (N)) is odd. For
simplicity assume dim A = 1.
If the 2-part of BSD formula is true for A/K . Then
Sel2 (A/K ) = 0 ⇐⇒ yK mod q 6∈ 2E(Fq 2 ) + E(Fq ).
(The latter is an index 2 subgroup of E(Fq 2 )).
Remark
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Heegner points
Corollary
If dim Sel2 (E/K ) = 1, then dimk Selλ (A/K ) = 0 or 2.
Theorem B (–)
Suppose dim Sel2 (E/K ) =
1. Let yK ∈ E(K ) be a Heegner point.
Suppose ρ̄(Frobq ) = 10 11 and # ker(J0 (N) → J1 (N)) is odd. For
simplicity assume dim A = 1.
If the 2-part of BSD formula is true for A/K . Then
Sel2 (A/K ) = 0 ⇐⇒ yK mod q 6∈ 2E(Fq 2 ) + E(Fq ).
(The latter is an index 2 subgroup of E(Fq 2 )).
Remark
Establish an instance of “Jochnowitz congruences” in the terminology
of Bertolini-Darmon: f ≡ g
L0 (f /K , 1)“ ≡ ”L(g/K , 1).
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Heegner points
Corollary
If dim Sel2 (E/K ) = 1, then dimk Selλ (A/K ) = 0 or 2.
Theorem B (–)
Suppose dim Sel2 (E/K ) =
1. Let yK ∈ E(K ) be a Heegner point.
Suppose ρ̄(Frobq ) = 10 11 and # ker(J0 (N) → J1 (N)) is odd. For
simplicity assume dim A = 1.
If the 2-part of BSD formula is true for A/K . Then
Sel2 (A/K ) = 0 ⇐⇒ yK mod q 6∈ 2E(Fq 2 ) + E(Fq ).
(The latter is an index 2 subgroup of E(Fq 2 )).
Remark
Establish an instance of “Jochnowitz congruences” in the terminology
of Bertolini-Darmon: f ≡ g
L0 (f /K , 1)“ ≡ ”L(g/K , 1). This level
raising process tells us about Heegner points!
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
I
When dim Sel2 (E/K ) = 1, BSD predicts that
rank(E(K )) = 1,
Chao Li (Harvard)
Ш(E/K )[2] = 0.
2-Selmer groups and Heegner points
FRG 2014
I
When dim Sel2 (E/K ) = 1, BSD predicts that
rank(E(K )) = 1,
I
Ш(E/K )[2] = 0.
By Gross-Zagier formula,
BSD formula for E/K ⇐⇒ #Ш(E/K ) =
Chao Li (Harvard)
2-Selmer groups and Heegner points
[E(K ) : ZyK ]
Q
v |N cv
!2
.
FRG 2014
I
When dim Sel2 (E/K ) = 1, BSD predicts that
rank(E(K )) = 1,
I
Ш(E/K )[2] = 0.
By Gross-Zagier formula,
BSD formula for E/K ⇐⇒ #Ш(E/K ) =
[E(K ) : ZyK ]
Q
v |N cv
!2
.
Conjecture (2-part of BSD formula)
yK 6∈ 2E(K ).
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
I
When dim Sel2 (E/K ) = 1, BSD predicts that
rank(E(K )) = 1,
I
Ш(E/K )[2] = 0.
By Gross-Zagier formula,
BSD formula for E/K ⇐⇒ #Ш(E/K ) =
[E(K ) : ZyK ]
Q
v |N cv
!2
.
Conjecture (2-part of BSD formula)
yK 6∈ 2E(K ).
Corollary
If Sel2 (A/K ) = 0, then
2-part of BSD for A/K =⇒ 2-part of BSD for E/K .
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Question
yK mod q 6∈ 2E(Fq 2 ) + E(Fq ) for some q?
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Question
yK mod q 6∈ 2E(Fq 2 ) + E(Fq ) for some q?
Answer
No.
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Question
yK mod q 6∈ 2E(Fq 2 ) + E(Fq ) for some q?
Answer
No.
E(K )/2E(K ) ∼
= Z/2Z.
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Question
yK mod q 6∈ 2E(Fq 2 ) + E(Fq ) for some q?
Answer
No.
E(K )/2E(K ) ∼
= Z/2Z. Gal(K /Q) always acts trivially on Z/2Z.
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Question
yK mod q 6∈ 2E(Fq 2 ) + E(Fq ) for some q?
Answer
No.
E(K )/2E(K ) ∼
= Z/2Z. Gal(K /Q) always acts trivially on Z/2Z. So
yK mod q ∈ E(Fq 2 )/2E(Fq 2 )
is invariant under Frobq .
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Question
yK mod q 6∈ 2E(Fq 2 ) + E(Fq ) for some q?
Answer
No.
E(K )/2E(K ) ∼
= Z/2Z. Gal(K /Q) always acts trivially on Z/2Z. So
yK mod q ∈ E(Fq 2 )/2E(Fq 2 )
is invariant under Frobq . When ρ̄(Frobq ) =
subspace is exactly the image of
11
01
, the invariant
Z/2Z ∼
= E(Fq )/2E(Fq ) ,→ E(Fq 2 )/2E(Fq 2 ) ∼
= (Z/2Z)2 .
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Question
yK mod q 6∈ 2E(Fq 2 ) + E(Fq ) for some q?
Answer
No.
E(K )/2E(K ) ∼
= Z/2Z. Gal(K /Q) always acts trivially on Z/2Z. So
yK mod q ∈ E(Fq 2 )/2E(Fq 2 )
is invariant under Frobq . When ρ̄(Frobq ) =
subspace is exactly the image of
11
01
, the invariant
Z/2Z ∼
= E(Fq )/2E(Fq ) ,→ E(Fq 2 )/2E(Fq 2 ) ∼
= (Z/2Z)2 .
Thus always
yK mod q ∈ 2E(Fq 2 ) + E(Fq ).
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Obstruction for rank lowering
Theorem C (–)
Suppose ρ̄(Frobq ) =
11
01
. If dim Sel2 (E/K ) = 1, then
dimk Selλ (A/K ) = 2.
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Obstruction for rank lowering
Theorem C (–)
Suppose ρ̄(Frobq ) =
11
01
. If dim Sel2 (E/K ) = 1, then
dimk Selλ (A/K ) = 2.
Over K , there is an intrinsic reason forcing the 2-Selmer rank to go up.
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Obstruction for rank lowering
Theorem C (–)
Suppose ρ̄(Frobq ) =
11
01
. If dim Sel2 (E/K ) = 1, then
dimk Selλ (A/K ) = 2.
Over K , there is an intrinsic reason forcing the 2-Selmer rank to go up.
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
Example
E = X0 (11).
q
7
7
13
13
17
17
19
19
29
29
Chao Li (Harvard)
A
77a
77b
143a
143a
187a
187a
209a
209a
319a
319a
dK
-8
-8
-7
-8
-7
-24
-7
-19
-8
-19
rank A(K )
2
0
2
2
2
0
2
2
2
0
dim Ш(A/K )[2]
0
2
0
0
0
2
0
0
0
2
2-Selmer groups and Heegner points
FRG 2014
Example
E = X0 (11).
q
7
7
13
13
17
17
19
19
29
29
A
77a
77b
143a
143a
187a
187a
209a
209a
319a
319a
dK
-8
-8
-7
-8
-7
-24
-7
-19
-8
-19
rank A(K )
2
0
2
2
2
0
2
2
2
0
dim Ш(A/K )[2]
0
2
0
0
0
2
0
0
0
2
This never happens for Ш(A/K )[p] when p is odd!
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
In contrast to the situation over Q
In the joint work (in progress) with Bao Le Hung, we enhance Ribet’s
level raising theorem (mod 2). One curious consequence:
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
In contrast to the situation over Q
In the joint work (in progress) with Bao Le Hung, we enhance Ribet’s
level raising theorem (mod 2). One curious consequence:
Theorem D (Le Hung, –)
Suppose E/Q satisfies our assumptions in the beginning and that E
has negative discriminant. Then for any given r ≥ 0, there exists an
abelian variety A/Q obtained from E/Q via a sequence of level raising,
such that
dimk Selλ (A/Q) = r .
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014
In contrast to the situation over Q
In the joint work (in progress) with Bao Le Hung, we enhance Ribet’s
level raising theorem (mod 2). One curious consequence:
Theorem D (Le Hung, –)
Suppose E/Q satisfies our assumptions in the beginning and that E
has negative discriminant. Then for any given r ≥ 0, there exists an
abelian variety A/Q obtained from E/Q via a sequence of level raising,
such that
dimk Selλ (A/Q) = r .
Thanks!
Chao Li (Harvard)
2-Selmer groups and Heegner points
FRG 2014