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Local heights on hyperelliptic curves Jennifer Balakrishnan MIT, Department of Mathematics E ff ective Methods in p -adic Cohomology Mathematical Institute, University of Oxford Tuesday, March 16, 2010 Jennifer Balakrishnan (MIT) Local heights

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Local heights on hyperelliptic curves

Jennifer Balakrishnan

MIT, Department of Mathematics

Effective Methods in p-adic Cohomology Mathematical Institute, University of Oxford Tuesday, March 16, 2010

Jennifer Balakrishnan (MIT) Local heights on hyperelliptic curves Oxford, March 16, 2010 1 / 14

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Outline

1

Introduction Notation Coleman-Gross height pairing Local height: residue characteristic p

2

Differentials and cohomology Differentials Cohomology The map Ψ

3

Computing with Ψ Local and global symbols Splitting of H1

dR(C/k)

The normalized differential

4

Algorithm

Jennifer Balakrishnan (MIT) Local heights on hyperelliptic curves Oxford, March 16, 2010 2 / 14

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Introduction Notation

Notation

C: genus g hyperelliptic curve of the form y2 = f(x), with deg f(x) = 2g + 1 K: number field k: local field (char 0) with valuation ring O, uniformizer π F: residue field, O/πO, with |F| = q. J: Jacobian of C over k We’ll assume C has good ordinary reduction at π.

Jennifer Balakrishnan (MIT) Local heights on hyperelliptic curves Oxford, March 16, 2010 3 / 14

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Introduction Coleman-Gross height pairing

Definition The Coleman-Gross height pairing is a symmetric bilinear pairing h : Div0(C) × Div0(C) → Qp, which can be written as a sum of local height pairings h =

  • v

hv

  • ver all finite places v of K.

Techniques to compute hv depend on char(F): char(F) p: intersection theory (Prop 1.2 in C-G) char(F) = p: logarithms, normalized differentials, Coleman integration

Jennifer Balakrishnan (MIT) Local heights on hyperelliptic curves Oxford, March 16, 2010 4 / 14

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Introduction Local height: residue characteristic p

Local height: residue characteristic p

So for the rest of the talk, we’ll assume char(F) = p, and that k = Kv, the completion at v|p. Definition Let D1, D2 ∈ Div0(C) have disjoint support and ωD1 a normalized differential associated to D1. The local height pairing at v above p is given by hv(D1, D2) = trk/Qp

  • D2

ωD1

  • .

We will describe how to construct ωD1, using the analytic methods of Coleman and Gross.

Jennifer Balakrishnan (MIT) Local heights on hyperelliptic curves Oxford, March 16, 2010 5 / 14

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Differentials and cohomology Differentials

Differentials

Definition A differential on C over k is

  • f the first kind (denoted H1,0

dR(C/k)): regular everywhere

  • f the second kind: residue 0 everywhere
  • f the third kind (denoted T(k)): simple poles and integer residues

By the residual divisor homomorphism, T(k) fits into the following exact sequence:

H1,0

dR(C/k)

T(k)

Res Div0(C)

0.

Let Tl(k) ⊂ T(k) be the logarithmic differentials (df

f for f ∈ k(C)∗). Since

Tl(k) ∩ H1,0

dR(C/k) = {0} and Res( df f ) = (f),

0 −→ H1,0

dR(C/k) −→ T(k)/Tl(k) −→ J(k) −→ 0.

Jennifer Balakrishnan (MIT) Local heights on hyperelliptic curves Oxford, March 16, 2010 6 / 14

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Differentials and cohomology Cohomology

Cohomology

Recall the exact sequence 0 −→ H1,0

dR(C/k) −→ H1 dR(C/k) −→ H1 dR(C, OC/k) −→ 0,

(1) where H1,0

dR(C/k) has dimension g

H1

dR(C, OC/k) also has dimension g and may be canonically

identified with the tangent space at the origin of J. H1

dR(C/k) has a canonical non-degenerate alternating form given

by the algebraic cup product pairing H1

dR(C/k) × H1 dR(C/k) −→ k

([ν1], [ν2]) → [ν1] ∪ [ν2] =

  • x

Resx(ν2

  • ν1),

for νi differentials of the second kind.

Jennifer Balakrishnan (MIT) Local heights on hyperelliptic curves Oxford, March 16, 2010 7 / 14

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Differentials and cohomology The map Ψ

Theorem (Coleman-Gross) There is a canonical homomorphism Ψ : T(k)/Tl(k) −→ H1

dR(C/k)

which is the identity on differentials of the first kind and makes the following diagram commute:

H1,0(C/k) T(k)/Tl(k)

Ψ=logT

  • J(k)
  • logJ
  • H1,0

dR(C/k)

H1

dR(C/k)

H1

dR(C, OC/k)

0.

To compute with Ψ, we use the following Theorem (Besser) Let ω be a meromorphic form and ρ a form of the second kind. Then Ψ(ω) ∪ [ρ] = ω, ρ.

Jennifer Balakrishnan (MIT) Local heights on hyperelliptic curves Oxford, March 16, 2010 8 / 14

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Computing with Ψ

Local and global symbols

Definition For ω a meromorphic form and ρ a form of the second kind, we define the global symbol ω, ρ as a sum of local symbols ω, ρA. We have ω, ρ =

  • A

ω, ρA =

  • A

ResA

  • ω
  • ρ +

A

Z

ρ

  • ,

where A ∈ {Weierstrass points of C, poles of ω}, each ω, ρA is computed via local coordinates at A, the first integral is a formal antiderivative, and the second (Coleman) integral sets the constant of integration (for a fixed Z).

Jennifer Balakrishnan (MIT) Local heights on hyperelliptic curves Oxford, March 16, 2010 9 / 14

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Computing with Ψ

Global symbols and Ψ

We compute Ψ of a meromorphic differential via cup products and global symbols. Given a basis {ωi}2g−1

i=0 for H1 dR(C/k), we write

Ψ(ω) = c0ω0 + · · · + c2g−1ω2g−1. We solve for the coefficients ci by considering a linear system involving global symbols and cup products: ω, ωj = Ψ(ω) ∪ [ωj] =

2g−1

  • i=0

ci([ωi] ∪ [ωj]).

Jennifer Balakrishnan (MIT) Local heights on hyperelliptic curves Oxford, March 16, 2010 10 / 14

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Computing with Ψ

Splitting of H1

dR(C/k): getting ωD1

We fix a direct sum decomposition H1

dR(C/k) = H1,0 dR(C/k) ⊕ W,

where W is the unit root subspace for the action of Frobenius. Definition Let D1 ∈ Div0(C/k). We define ωD1 to be a differential of the third kind with residue divisor D1 such that Ψ(ωD1) ∈ W. Lemma ωD1 is unique.

Jennifer Balakrishnan (MIT) Local heights on hyperelliptic curves Oxford, March 16, 2010 11 / 14

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Computing with Ψ

The normalized differential ωD1

Thus choosing ω of the third kind with Res(ω) = D1, by the splitting H1

dR(C/k) = H1,0 dR(C/k) ⊕ W,

we have Ψ(ω) = η + Ψ(ωD1), for η holomorphic and some element Ψ(ωD1) ∈ W. Then taking ωD1 := ω − η, we have Ψ(ωD1) = Ψ(ω − η) = Ψ(ω) − Ψ(η) = Ψ(ω) − η.

Jennifer Balakrishnan (MIT) Local heights on hyperelliptic curves Oxford, March 16, 2010 12 / 14

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Algorithm

Algorithm: computing hp(D1, D2)

Input: C hyperelliptic curve over Qp with p a prime of good ordinary reduction, D1 = (P) − (Q), D2 = (R) − (S) ∈ Div0(C) with disjoint support Output: hp(D1, D2) =

  • D2 ωD1

(1) From D1 to ω. Choose ω a differential of the third kind with Res(ω) = D1. (2) The map Ψ. Compute log(ω) = Ψ(ω) for ω. (3) From ω to ωD1 and η. Via the decomposition H1

dR(C/k) ≃ H1,0 dR(C/k) ⊕ W,

write log(ω) = η + log(ωD1), where η is holomorphic, and log(ωD1) ∈ W. This gives ωD1 = ω − η.

Jennifer Balakrishnan (MIT) Local heights on hyperelliptic curves Oxford, March 16, 2010 13 / 14

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Algorithm

Algorithm, continued

(4) Coleman integration: holomorphic differential. Compute

  • D2 η.

(5) Coleman integration: meromorphic differential. Let φ be a p-power lift of Frobenius and set α := φ∗ω − pω. Then for β a differential with residue divisor D2 = (R) − (S), we compute

  • D2

ω = R

S

ω = 1 1 − p (Ψ(α) ∪ Ψ(β)) +

  • Res
  • α
  • β
  • −

1 1 − p S

φ(S)

ω + φ(R)

R

ω

  • .

(6) Height pairing. Subtract the integrals to recover the pairing: hp(D1, D2) = R

S

ω(P)−(Q) = R

S

ω − R

S

η.

Jennifer Balakrishnan (MIT) Local heights on hyperelliptic curves Oxford, March 16, 2010 14 / 14