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 on hyperelliptic curves Oxford, March 16, 2010 1 / 14
Outline Introduction 1 Notation Coleman-Gross height pairing Local height: residue characteristic p Di ff erentials and cohomology 2 Di ff erentials Cohomology The map Ψ Computing with Ψ 3 Local and global symbols Splitting of H 1 dR ( C / k ) The normalized di ff erential Algorithm 4 Jennifer Balakrishnan (MIT) Local heights on hyperelliptic curves Oxford, March 16, 2010 2 / 14
Introduction Notation Notation C : genus g hyperelliptic curve of the form y 2 = f ( x ) , with deg f ( x ) = 2 g + 1 K : number field k : local field (char 0) with valuation ring O , uniformizer π F : residue field, O /π O , 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
Introduction Coleman-Gross height pairing Definition The Coleman-Gross height pairing is a symmetric bilinear pairing h : Div 0 ( C ) × Div 0 ( C ) → Q p , which can be written as a sum of local height pairings � h = h v v over all finite places v of K . Techniques to compute h v depend on char ( F ) : char ( F ) � p : intersection theory (Prop 1.2 in C-G) char ( F ) = p : logarithms, normalized di ff erentials, Coleman integration Jennifer Balakrishnan (MIT) Local heights on hyperelliptic curves Oxford, March 16, 2010 4 / 14
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 = K v , the completion at v | p . Definition Let D 1 , D 2 ∈ Div 0 ( C ) have disjoint support and ω D 1 a normalized di ff erential associated to D 1 . The local height pairing at v above p is given by � � � h v ( D 1 , D 2 ) = tr k / Q p ω D 1 . D 2 We will describe how to construct ω D 1 , using the analytic methods of Coleman and Gross. Jennifer Balakrishnan (MIT) Local heights on hyperelliptic curves Oxford, March 16, 2010 5 / 14
Di ff erentials and cohomology Di ff erentials Di ff erentials Definition A di ff erential on C over k is of the first kind (denoted H 1,0 dR ( C / k ) ): regular everywhere of the second kind: residue 0 everywhere of the third kind (denoted T ( k ) ): simple poles and integer residues By the residual divisor homomorphism, T ( k ) fits into the following exact sequence: Res �� Div 0 ( C ) � H 1,0 � T ( k ) � 0. dR ( C / k ) 0 Let T l ( k ) ⊂ T ( k ) be the logarithmic di ff erentials ( df f for f ∈ k ( C ) ∗ ). Since dR ( C / k ) = { 0 } and Res ( df T l ( k ) ∩ H 1,0 f ) = ( f ) , 0 −→ H 1,0 dR ( C / k ) −→ T ( k ) / T l ( k ) −→ J ( k ) −→ 0. Jennifer Balakrishnan (MIT) Local heights on hyperelliptic curves Oxford, March 16, 2010 6 / 14
Di ff erentials and cohomology Cohomology Cohomology Recall the exact sequence 0 −→ H 1,0 dR ( C / k ) −→ H 1 dR ( C / k ) −→ H 1 dR ( C , O C / k ) −→ 0, (1) where H 1,0 dR ( C / k ) has dimension g H 1 dR ( C , O C / k ) also has dimension g and may be canonically identified with the tangent space at the origin of J . H 1 dR ( C / k ) has a canonical non-degenerate alternating form given by the algebraic cup product pairing H 1 dR ( C / k ) × H 1 dR ( C / k ) −→ k � � ([ ν 1 ] , [ ν 2 ]) �→ [ ν 1 ] ∪ [ ν 2 ] = Res x ( ν 2 ν 1 ) , x for ν i di ff erentials of the second kind. Jennifer Balakrishnan (MIT) Local heights on hyperelliptic curves Oxford, March 16, 2010 7 / 14
� � � Di ff erentials and cohomology The map Ψ Theorem (Coleman-Gross) There is a canonical homomorphism Ψ : T ( k ) / T l ( k ) −→ H 1 dR ( C / k ) which is the identity on di ff erentials of the first kind and makes the following diagram commute: � H 1,0 ( C / k ) � T ( k ) / T l ( k ) � J ( k ) 0 0 log J Ψ = log T � H 1,0 � H 1 � H 1 � 0. dR ( C , O C / k ) 0 dR ( C / k ) dR ( C / k ) 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
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 , ρ � A . We have � A � � � �� � � � ω , ρ � = � ω , ρ � A = Res A ω ρ + ρ , Z A A where A ∈ { Weierstrass points of C , poles of ω } , each � ω , ρ � A 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
Computing with Ψ Global symbols and Ψ We compute Ψ of a meromorphic di ff erential via cup products and global symbols. Given a basis { ω i } 2 g − 1 i = 0 for H 1 dR ( C / k ) , we write Ψ ( ω ) = c 0 ω 0 + · · · + c 2 g − 1 ω 2 g − 1 . We solve for the coe ffi cients c i by considering a linear system involving global symbols and cup products: 2 g − 1 � � ω , ω j � = Ψ ( ω ) ∪ [ ω j ] = c i ([ ω i ] ∪ [ ω j ]) . i = 0 Jennifer Balakrishnan (MIT) Local heights on hyperelliptic curves Oxford, March 16, 2010 10 / 14
Computing with Ψ Splitting of H 1 dR ( C / k ) : getting ω D 1 We fix a direct sum decomposition dR ( C / k ) = H 1,0 H 1 dR ( C / k ) ⊕ W , where W is the unit root subspace for the action of Frobenius. Definition Let D 1 ∈ Div 0 ( C / k ) . We define ω D 1 to be a di ff erential of the third kind with residue divisor D 1 such that Ψ ( ω D 1 ) ∈ W . Lemma ω D 1 is unique. Jennifer Balakrishnan (MIT) Local heights on hyperelliptic curves Oxford, March 16, 2010 11 / 14
Computing with Ψ The normalized di ff erential ω D 1 Thus choosing ω of the third kind with Res ( ω ) = D 1 , by the splitting H 1 dR ( C / k ) = H 1,0 dR ( C / k ) ⊕ W , we have Ψ ( ω ) = η + Ψ ( ω D 1 ) , for η holomorphic and some element Ψ ( ω D 1 ) ∈ W . Then taking ω D 1 := ω − η , we have Ψ ( ω D 1 ) = Ψ ( ω − η ) = Ψ ( ω ) − Ψ ( η ) = Ψ ( ω ) − η . Jennifer Balakrishnan (MIT) Local heights on hyperelliptic curves Oxford, March 16, 2010 12 / 14
Algorithm Algorithm: computing h p ( D 1 , D 2 ) Input: C hyperelliptic curve over Q p with p a prime of good ordinary reduction, D 1 = ( P ) − ( Q ) , D 2 = ( R ) − ( S ) ∈ Div 0 ( C ) with disjoint support � Output: h p ( D 1 , D 2 ) = D 2 ω D 1 (1) From D 1 to ω . Choose ω a di ff erential of the third kind with Res ( ω ) = D 1 . (2) The map Ψ . Compute log ( ω ) = Ψ ( ω ) for ω . (3) From ω to ω D 1 and η . Via the decomposition H 1 dR ( C / k ) ≃ H 1,0 dR ( C / k ) ⊕ W , write log ( ω ) = η + log ( ω D 1 ) , where η is holomorphic, and log ( ω D 1 ) ∈ W . This gives ω D 1 = ω − η . Jennifer Balakrishnan (MIT) Local heights on hyperelliptic curves Oxford, March 16, 2010 13 / 14
Algorithm Algorithm, continued � (4) Coleman integration: holomorphic di ff erential. Compute D 2 η . (5) Coleman integration: meromorphic di ff erential. Let φ be a p -power lift of Frobenius and set α := φ ∗ ω − p ω . Then for β a di ff erential with residue divisor D 2 = ( R ) − ( S ) , we compute � R � ω = ω D 2 S � � 1 � � = 1 − p ( Ψ ( α ) ∪ Ψ ( β )) + Res α β � � S � φ ( R ) � 1 − ω + ω . 1 − p φ ( S ) R (6) Height pairing . Subtract the integrals to recover the pairing: � R � R � R h p ( D 1 , D 2 ) = ω ( P )−( Q ) = ω − η . S S S Jennifer Balakrishnan (MIT) Local heights on hyperelliptic curves Oxford, March 16, 2010 14 / 14
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