L OCAL ARITHMETIC FOR HYPERELLIPTIC CURVES OF GENUS 2
C ÉLINE M AISTRET
C . MAISTRET @ WARWICK . AC . UK
2- PARITY CONJECTURE
R ICHELOT ISOGENY
Let K be a finite extension of Q and C/K a hyperelliptic curve of genus 2 admitting a Richelot
isogeny. Consider J(C) the Jacobian of C.
2
Let C : y = F (x) be a hyperelliptic curve of
genus 2 defined over K such that K[x]:
Y
Y
gi (x − αi )(x − βi )
Gi (x) =
F (x) =
2-parity conjecture
1≤i≤3
1≤i≤3
=
rk2 (J(C)(K))
(−1)
= w(J(C)/K),
Y
gi2 x2 + gi1 x + gi0 .
1≤i≤3
• rk2 (J(C)(K)) is the 2∞ -Selmer rank of J(C)/K
i.e.
rk2 (J(C)/K) = rk(J(C)/K) + δ2 ,
• X(J(C)/K)[2∞ ] ' (Q2 /Z2 )δ2 × X0 [2∞ ]
• w(J(C)/K) global root number of J(C)/K.
ˆ s.t.
Let ∆ = det(gij )1≤i,j≤3 then φ : J(C) → J(C)
Y
2
0
0
ˆ
C : ∆y =
Gi+1 (x)Gi+2 (x) − Gi+1 (x)Gi+2 (x)
1≤i≤3
Let K be a finite extension of Qp , p 6= 2, and C/K a hyperelliptic curve of genus 2
Q
2
C : y = f (x) = c r∈S (x − r) , r ∈ K.
Definition 0.1. Let S denote the set of roots of f (x)
in K.
A cluster s ⊂ S is a non-empty subset of S cut out
n
nr
by a disk D = z + p OK , with z ∈ K . Its depth
ds := n = min{k : |r − z| ≤ k, ∀r ∈ s}.
Theorem 0.2. [1] Let C/K be semistable, then the special fibre Ck of the minimal regular model of C over k̄
can be derived from its cluster picture.
ker(φ) := {O, {(αi , 0), (βi , 0)} | 1 ≤ i ≤ 3}.
Proposition 0.3. C(Kv ) = ∅ if and only if the order
of the Gal(k/k)-orbit of each irreducible component of
Ck is even (follows from remark 9.2.1. in [3]).
R ESULTS
The parity of the 2∞ -Selmer rank can be expressed as a sum of local terms:
Q
c (J(C))Ω
∞ X
|X
(J(C))[2
]|
v
J(C)
0
≡
λv mod 2
rk2 (J(C)/K) ≡ ord2 Q
∞ ]|
ˆ
ˆ
cv (J(C))Ω
|X
(J(
C))[2
0
J(Ĉ)
v∈MK
with trivial contribution when Kv = C and
1 for Kv = R,
λv ≡ ord2
2 F INITE CONTRIBUTION : λv . J OIN WORK WITH T., V. D OKCHITSER AND A. M ORGAN [1]
n(J(C))Mv (C)
ˆ
ˆ
| ker(α)|n(J(C))M
(
C)
v
Definition 0.4. We define the Tamagawa graph J
with an automorphism FJC associated to a cluster
picture, Z[FJC ]-lattices ΛJC ≤ ΛoJC and a pairing
<, >JC .
Example : K = Q17 , C : y 2 = f (x)
f (x)=(x−172 )(x+172 )(x−1−173 )(x+1+173 )(x−5)(x−7)
• Cluster Picture:
•r2
•r1
•r3
s1
S
•r4
•r5 •r6
s2
where dS = 0, ds1 = 2, ds2 = 3.
• Theorem 0.2 gives CF17 :
ΓS
•
•
Γs1
•
•
•
•
•
•
•
Γs2
•
• By proposition 0.3 C(Q17 ) 6= ∅.
• Tamagawa Tree :
mod 2,
2 for a finite place v,
c M (C) v
v
λv ≡ ord2
mod 2.
ˆ
ĉv Mv (C)
Main result
2
Theorem 0.7. Let C : y = F (x) be a hyperelliptic curve of genus 2 admitting a Richelot isogeny with F (x) a monic
integral polynomial of degree 6 defined over a number field K.
Assume K totally complex, all primes above 2 in K have even residue degree, J(C) has good reduction at all primes
ˆ v ) 6= ∅ , all double roots of F (x) have split reduction at every prime, then
above 2, for all v ∈ MK C(K
rk2 (J(C))
(−1)
= w(J(C)).
1 R EAL CONTRIBUTION : λR
Proposition 0.8. [2]
Let n(C(R)) denote the number of connected components on C(R) and g the genus of C then:
n(C(R))−1
2
if n(C) > 0
n(J(C)(R)) =
1
if n(C) = 0 and g ≡ 0 mod 2.
Theorem 0.5. [1] For a semistable hyperelliptic curve
C over a local field of odd residue characteristic
ΛJ(C) ' ΛJC .
where ΛJ(C) is the dual character group lattice of J(C)
together with a Frobenius action and the monodromy
pairing.
•vS,s1
•vS,s2
•
•
•
•
•
•
•vs1
•
•
•vs2
FJ
Lemma 0.6. cv = |(Λ∨
/Λ
)
|.
JC
JC
Moreover, there exists an algorithm that computes cv
from JC .
• From Lemma 0.6, c17 = 24.
R EFERENCES
[1] T. Dokchitser, V. Dokchitser, C. Maistret, and A. Morgan. Arithmetic of semistable hyperelliptic curves over
local fields. In preparation.
[2] Harris Gross. Real algebraic curves. Annales scientifiques de ENS, 1981.
[3] B. Poonen and M. Stoll. The Cassels-Tate pairing on polarized abelian varieties. Annals of Mathematics, 150,
1999.