The Distribution of 2-Selmer Ranks
of Quadratic Twists of Elliptic
Curves.
Zev Klagsbrun
partially joint with Karl Rubin and Barry Mazur
Department of Mathematics
University of Wisconsin - Madison
September 27, 2012
Theorem (Heath-Brown 1994, Swinnerton-Dyer 2008, Kane 2011)
Suppose E /Q is an elliptic curve with
E (Q)[2] ' Z/2Z × Z/2Z that does not have a cyclic
4-isogeny over Q. Then for r ≥ 2,
#{d squarefree, |d| < X : dimF2 Sel2 (E d /Q) = r }
= αr
X →∞
#{d squarefree, |d| < X }
lim
αr > 0 explicit constants with
P
αr = 1.
Theorem (Heath-Brown 1994, Swinnerton-Dyer 2008, Kane 2011)
Suppose E /Q is an elliptic curve with
E (Q)[2] ' Z/2Z × Z/2Z that does not have a cyclic
4-isogeny over Q. Then for r ≥ 2,
#{d squarefree, |d| < X : dimF2 Sel2 (E d /Q) = r }
= αr
X →∞
#{d squarefree, |d| < X }
lim
αr > 0 explicit constants with
α0 ≈ .21, α1 ≈ .42, α2 ≈ .21
P
αr = 1.
d2 (E /K ) := dimF2 Sel2 (E /K ) − dimF2 E (K )[2]
will be referred to as the 2-Selmer rank of E.
√
E := E , where F = K ( d)
F
d
Theorem (Dokchitser & Dokchitser)
Suppose K is a totally complex number field and
E /K has potential good reduction over an abelian
extension of K . Then
d2 (E F /K ) ≡ d2 (E /K )
for every quadratic twist E F of E .
(mod 2)
Theorem (Kramer, 1981)
We have a natural map
×
× 2
{F /K quadratic} = K /(K ) →
Y
v |2∆E ∞
Kv× /(Kv× )2
Theorem (Kramer, 1981)
We have a natural map
×
× 2
{F /K quadratic} = K /(K ) →
Y
Kv× /(Kv× )2
v |2∆E ∞
d2 (E F /K ) (mod 2) is determined entirely by
the image of F /K under this map.
Theorem (K., Mazur, Rubin)
Let
S(X ) := {F /K : NK /Q q < X ∀q ramified in F /K }.
Theorem (K., Mazur, Rubin)
Let
S(X ) := {F /K : NK /Q q < X ∀q ramified in F /K }.
Then for sufficiently large X ,
#{F /K ∈ S(X ) : d2 (E F /K ) is even } 1 + δ
=
#S(X )
2
δ ∈ [−1, 1] ∩ Z[ 12 ].
Y
δv , where the δv are explicit.
δ=
v |2∆E ∞
δv = 0 for all real places v , so δ = 0 if K has a
real embedding.
Example
Let E be the curve given by
y 2 + xy + y = x 3 + x 2 − 3x + 1,
√
K a finite extension of Q( −2) that is unramified at
primes above 5. Then
√
Y
δ = −1[K :Q( −2)] (1 − 2[Kv :Q2 ] )
v |2
δ is dense in [−1, 1] as K varies.
#{F /K ∈ S(X ) : d2 (E F /K ) is even }
is dense
#S(X )
in [0, 1] as K varies
Conjecture (Goldfeld’s Conjecture)
If E an elliptic curve over Q, then
50% of all quadratic twists of E have rank 0
50% have rank 1
0% have rank ≥ 2
Conjecture (Goldfeld’s Conjecture for Number Fields )
If E an elliptic curve over K , then there is a
computable factor δ ∈ [−1, 1] such that
The proportion of twists of E having rank 0 is
1+δ
2 .
The proportion of twists of E having rank 1 is
1−δ
2 .
Conjecture (Goldfeld’s Conjecture)
If E an elliptic curve over Q, then
50% of all quadratic twists of E have rank 0
50% have rank 1
0% have rank ≥ 2
Conjecture (Goldfeld’s Conjecture for Number Fields )
If E an elliptic curve over K , then there is a
computable factor δ ∈ [−1, 1] such that
The proportion of twists of E having rank 0 is
1+δ
2 .
The proportion of twists of E having rank 1 is
1−δ
2 .
δ may depend on how the twists are ordered.
Theorem (K., Mazur, Rubin)
Suppose E (K )[2] = 0 and Gal(K (E [2])/K ) ' S3 .
There exist “skew-boxes"
Bm (X ) for m ∈ N and
[
+
X ∈ R with
Bm (X ) = {[F : K ] = 2} such that
m,X
#{F /K ∈ Bm (X ) : d2 (E F /K ) = r }
lim lim
m→∞ X →∞
#Bm (X )
(1 + δ)αr if r is even
=
(1 − δ)αr if r is odd
• E [2] and E F [2] are GK -isomorphic so we can view H 1 (K , E F [2])
and sitting inside of H 1 (K , E [2]).
• For every F /K , the group Sel2 (E F /K ) is subgroup of
H 1 (K , E [2]) which is defined by local conditions in H 1 (Kv , E [2]) for
each v of K .
• The local conditions for E F and E vary in a controlled manner,
and we can understand how the subgroup of H 1 (K , E [2]) changes
as we change the local condition at a single prime from the local
condition for E to the local condition for E F .
• E [2] and E F [2] are GK -isomorphic so we can view H 1 (K , E F [2])
and sitting inside of H 1 (K , E [2]).
• For every F /K , the group Sel2 (E F /K ) is subgroup of
H 1 (K , E [2]) which is defined by local conditions in H 1 (Kv , E [2]) for
each v of K .
• The local conditions for E F and E vary in a controlled manner,
and we can understand how the subgroup of H 1 (K , E [2]) changes
as we change the local condition at a single prime from the local
condition for E to the local condition for E F .
• We work our way from Sel(E /K ) to Sel(E F /K ) by successively
changing the local conditions at each prime ramified in F /K . This
gives us a chain of intermediate subgroups Z1 , Z2 , . . . , Zm of
H 1 (K , E [2]) each defined by local conditions, where Zi and Zi+1
differ by a single local condition and Zm = Sel2 (E /K ).
• E [2] and E F [2] are GK -isomorphic so we can view H 1 (K , E F [2])
and sitting inside of H 1 (K , E [2]).
• For every F /K , the group Sel2 (E F /K ) is subgroup of
H 1 (K , E [2]) which is defined by local conditions in H 1 (Kv , E [2]) for
each v of K .
• The local conditions for E F and E vary in a controlled manner,
and we can understand how the subgroup of H 1 (K , E [2]) changes
as we change the local condition at a single prime from the local
condition for E to the local condition for E F .
• We work our way from Sel(E /K ) to Sel(E F /K ) by successively
changing the local conditions at each prime ramified in F /K . This
gives us a chain of intermediate subgroups Z1 , Z2 , . . . , Zm of
H 1 (K , E [2]) each defined by local conditions, where Zi and Zi+1
differ by a single local condition and Zm = Sel2 (E /K ).
• This process is a Markov process where the states are the ranks
of the intermediate groups Zi and the probabilities come from
understanding what happens when we change a single local
condition.
• The αr are the stable distribution of this Markov process.
Example (K., 2011)
Let K be a number field and let E be the elliptic
curve over K given by
E : y 2 = x 3 − 2(1 + 256n2 )x 2 + (1 + 256n2 )x
where n ∈ N with 1 + 256n2 6∈ (K × )2 . Then
d2 (E F /K ) ≥ r2 for every quadratic F /K where r2 is
the number of complex place of K .
Example (K., 2011)
Let K be a number field and let E be the elliptic
curve over K given by
E : y 2 = x 3 − 2(1 + 256n2 )x 2 + (1 + 256n2 )x
where n ∈ N with 1 + 256n2 6∈ (K × )2 . Then
d2 (E F /K ) ≥ r2 for every quadratic F /K where r2 is
the number of complex place of K .
E has a cyclic 4-isogeny defined over K (E [2])
Example (K., 2011)
Let K be a number field and let E be the elliptic
curve over K given by
E : y 2 = x 3 − 2(1 + 256n2 )x 2 + (1 + 256n2 )x
where n ∈ N with 1 + 256n2 6∈ (K × )2 . Then
d2 (E F /K ) ≥ r2 for every quadratic F /K where r2 is
the number of complex place of K .
E has a cyclic 4-isogeny defined over K (E [2])
Impossible when E does not have a cyclic
4-isogeny defined over K (E [2]).
Theorem (K., 2012)
Suppose E (Q)[2] ' Z/2Z and E does not have a
cyclic 4-isogeny defined over Q(E [2]). Then for any
fixed r ,
lim inf
X →∞
#{d squarefree, |d| < X : d2 (E d /Q) ≥ r }
#{d squarefree, |d| < X }
Theorem (K., 2012)
Suppose E (Q)[2] ' Z/2Z and E does not have a
cyclic 4-isogeny defined over Q(E [2]). Then for any
fixed r ,
lim inf
X →∞
#{d squarefree, |d| < X : d2 (E d /Q) ≥ r }
≥
#{d squarefree, |d| < X }
1
2
Theorem (K., 2012)
Suppose E (Q)[2] ' Z/2Z and E does not have a
cyclic 4-isogeny defined over Q(E [2]). Then for any
fixed r ,
lim inf
X →∞
#{d squarefree, |d| < X : d2 (E d /Q) ≥ r }
≥
#{d squarefree, |d| < X }
At least 12 of the twists of E have arbitrarily
large 2-Selmer rank
There is no distribution function on 2-Selmer
ranks within the twist family of E
1
2
We have a 2-isogeny φ : E → E 0 with C := ker φ = E (K )[2]
We have a 2-isogeny φ : E → E 0 with C := ker φ = E (K )[2]
E 0 (K )/φ(E (K ))
E 0 (Kv )/φ(E (Kv ))
κ
κv /
/
H 1 (K , C )
resv
H 1 (Kv , C )
We have a 2-isogeny φ : E → E 0 with C := ker φ = E (K )[2]
E 0 (K )/φ(E (K ))
E 0 (Kv )/φ(E (Kv ))
κ
κv /
/
H 1 (K , C )
resv
H 1 (Kv , C )
Hφ1 (Kv , C ) = image κv : E 0 (Kv )/φ(E (Kv )) → H 1 (Kv , C )
We have a 2-isogeny φ : E → E 0 with C := ker φ = E (K )[2]
E 0 (K )/φ(E (K ))
E 0 (Kv )/φ(E (Kv ))
κ
/
κv /
H 1 (K , C )
resv
H 1 (Kv , C )
Hφ1 (Kv , C ) = image κv : E 0 (Kv )/φ(E (Kv )) → H 1 (Kv , C )
n
Selφ (E /K ) = c ∈ H 1 (K , C ) :
resv (c)∈Hφ1 (Kv ,C )
for all v of K
o
• 0 → E 0 (K )[2] → Selφ (E /K ) → Sel2 (E /K ) → Selφ̂ (E 0 /K )
• 0 → E 0 (K )[2] → Selφ (E /K ) → Sel2 (E /K ) → Selφ̂ (E 0 /K )
• T (E /E 0 ) =
#Selφ (E /K )
#Selφ̂ (E 0 /K )
• 0 → E 0 (K )[2] → Selφ (E /K ) → Sel2 (E /K ) → Selφ̂ (E 0 /K )
• T (E /E 0 ) =
#Selφ (E /K )
#Selφ̂ (E 0 /K )
• d2 (E /K ) ≥ ord2 (T (E /E 0 ))
• 0 → E 0 (K )[2] → Selφ (E /K ) → Sel2 (E /K ) → Selφ̂ (E 0 /K )
• T (E /E 0 ) =
#Selφ (E /K )
#Selφ̂ (E 0 /K )
• d2 (E /K ) ≥ ord2 (T (E /E 0 ))
Y #Hφ1 (Kv , C )
• T (E /E ) =
2
0
v |2∆E ∞
Let ∆ be the discriminant of E and ∆0 be the discriminant of E 0 .
Let ∆ be the discriminant of E and ∆0 be the discriminant of E 0 .
• There exists a constant C = C (E ) such that
0 ∆
− ∆
X
p
p d
0d
≤C
ord2 T (E /E ) −
2
p|d
p -2∆
E
Let ∆ be the discriminant of E and ∆0 be the discriminant of E 0 .
• There exists a constant C = C (E ) such that
0 ∆
− ∆
X
p
p d
0d
≤C
ord2 T (E /E ) −
2
p|d
p -2∆
E
• The Erdös-Kac Theorem shows that
P
∆0
p
− ∆p
p|d
p -2∆E
p
2
log log d
has the standard normal distribution.
Let ∆ be the discriminant of E and ∆0 be the discriminant of E 0 .
• There exists a constant C = C (E ) such that
0 ∆
− ∆
X
p
p d
0d
≤C
ord2 T (E /E ) −
2
p|d
p -2∆
E
• The Erdös-Kac Theorem shows that
P
p|d
p -2∆E
p
∆0
p
− ∆p
2
log log d
has the standard normal distribution.
• For any fixed r ,
ord2 T (E d /E 0d ) ≥ r
for half of all squarefree d .