Explicit Coleman integration for hyperelliptic curves Jennifer Balakrishnan 1 Robert Bradshaw 2 Kiran Kedlaya 1 1 Massachusetts Institute of Technology 2 University of Washington ANTS-IX INRIA Nancy, France Thursday, July 22, 2010 Balakrishnan, Bradshaw, Kedlaya (MIT) Coleman integration for hyperelliptic curves ANTS-IX 1 / 21
Introduction Introduction: making sense of p -adic integrals Let C be the hyperelliptic curve y 2 = x 5 − x 4 + x 3 + x 2 − 2 x + 1 over Q 7 and let P 1 = ( 0, 1 ) , P 2 = ( 1, − 1 ) . Two questions: How do we compute things like 1 � P 2 dx 2 y ? P 1 What do these (Coleman) integrals tell us? 2 Balakrishnan, Bradshaw, Kedlaya (MIT) Coleman integration for hyperelliptic curves ANTS-IX 2 / 21
Introduction Introduction: making sense of p -adic integrals Let C be the hyperelliptic curve y 2 = x 5 − x 4 + x 3 + x 2 − 2 x + 1 over Q 7 and let P 1 = ( 0, 1 ) , P 2 = ( 1, − 1 ) . Two questions: How do we compute things like 1 � P 2 dx 2 y ? P 1 What do these (Coleman) integrals tell us? 2 Balakrishnan, Bradshaw, Kedlaya (MIT) Coleman integration for hyperelliptic curves ANTS-IX 2 / 21
Integrals for hyperelliptic curves Notation and setup X : genus g hyperelliptic curve (of the form y 2 = f ( x ) with deg f ( x ) = 2 g + 1) over K = Q p p : prime of good reduction X : special fibre of X X Q : generic fibre of X (as a rigid analytic space) Balakrishnan, Bradshaw, Kedlaya (MIT) Coleman integration for hyperelliptic curves ANTS-IX 3 / 21
Integrals for hyperelliptic curves Notation and setup, in pictures There is a natural reduction map from X Q to X ; the inverse image of any point of X is a subspace of X Q X Q -1 red (P) -1 red (R) isomorphic to an open unit disc. We -1 red (S) call such a disc a residue disc of X . A wide open subspace of X Q is the complement in X Q of the union of a red finite collection of disjoint closed discs of radius λ i < 1: P R X S λ 1 λ 2 1 1 Balakrishnan, Bradshaw, Kedlaya (MIT) Coleman integration for hyperelliptic curves ANTS-IX 4 / 21
Integrals for hyperelliptic curves Tiny integrals Computing tiny integrals � Q We refer to any Coleman integral of the form P ω in which P , Q lie in the same residue disc as a tiny integral . To compute such an integral: Construct a linear interpolation from P to Q . For instance, in a non-Weierstrass residue disc, we may take x ( t ) = ( 1 − t ) x ( P ) + tx ( Q ) � y ( t ) = f ( x ( t )) , where y ( t ) is expanded as a formal power series in t . Formally integrate the power series in t : � Q � 1 ω = ω ( x ( t ) , y ( t )) . Q P P 0 Balakrishnan, Bradshaw, Kedlaya (MIT) Coleman integration for hyperelliptic curves ANTS-IX 5 / 21
Integrals for hyperelliptic curves Example Tiny integral: example Let X be the hyperelliptic curve y 2 = f ( x ) = x 5 − x 4 + x 3 + x 2 − 2 x + 1 over Q 7 , ω = dx 2 y , and P = ( 1, − 1 ) = ( 1 + O ( 7 5 ) , 6 + 6 · 7 + 6 · 7 2 + 6 · 7 3 + 6 · 7 4 + O ( 7 5 )) , Q = ( 1 + 7 + O ( 7 5 ) , 6 + 4 · 7 + 4 · 7 2 + 3 · 7 3 + 2 · 7 4 + O ( 7 5 )) . � Q We compute P ω . Balakrishnan, Bradshaw, Kedlaya (MIT) Coleman integration for hyperelliptic curves ANTS-IX 6 / 21
Integrals for hyperelliptic curves Example Tiny integral: example, continued � Q Computing P ω : Interpolate: we have 1 x ( t ) = ( 1 − t ) x ( P ) + tx ( Q ) = 1 + O ( 7 5 ) + 7 + O ( 7 5 ) � � t � f ( x ( t )) = 6 + 6 · 7 + 6 · 7 2 + 6 · 7 3 + 6 · 7 4 + O ( 7 5 )+ y ( t ) = � � 5 · 7 + 6 · 7 2 + 6 · 7 3 + 6 · 7 4 + O ( 7 5 ) t + · · · . Integrate: 2 � Q � 1 7 + O ( 7 5 ) dx 2 y = ( 5 + 6 · 7 + · · · ) + ( 3 · 7 + 6 · 7 2 + · · · ) t + · · · dt P 0 = 3 · 7 + 2 · 7 3 + 5 · 7 4 + O ( 7 5 ) . Balakrishnan, Bradshaw, Kedlaya (MIT) Coleman integration for hyperelliptic curves ANTS-IX 7 / 21
Integrals for hyperelliptic curves Example Tiny integral: example, continued � Q Computing P ω : Interpolate: we have 1 x ( t ) = ( 1 − t ) x ( P ) + tx ( Q ) = 1 + O ( 7 5 ) + 7 + O ( 7 5 ) � � t � f ( x ( t )) = 6 + 6 · 7 + 6 · 7 2 + 6 · 7 3 + 6 · 7 4 + O ( 7 5 )+ y ( t ) = � � 5 · 7 + 6 · 7 2 + 6 · 7 3 + 6 · 7 4 + O ( 7 5 ) t + · · · . Integrate: 2 � Q � 1 7 + O ( 7 5 ) dx 2 y = ( 5 + 6 · 7 + · · · ) + ( 3 · 7 + 6 · 7 2 + · · · ) t + · · · dt P 0 = 3 · 7 + 2 · 7 3 + 5 · 7 4 + O ( 7 5 ) . Balakrishnan, Bradshaw, Kedlaya (MIT) Coleman integration for hyperelliptic curves ANTS-IX 7 / 21
Integrals for hyperelliptic curves Properties of the Coleman integral Properties of the Coleman integral Coleman formulated an integration theory on wide open subspaces of curves over O , exhibiting no phenomena of path dependence. � Q This allows us to define P ω whenever ω is a meromorphic 1-form on X , and P , Q ∈ X ( Q p ) are points where ω is holomorphic. Properties of the Coleman integral include: Theorem (Coleman) � Q � Q � Q P ( αω 1 + βω 2 ) = α P ω 1 + β Linearity: P ω 2 . � R � Q � R Additivity: P ω = P ω + Q ω . Change of variables: if X ′ is another such curve, and f : U → U ′ is a � Q � f ( Q ) P f ∗ ω = rigid analytic map between wide opens, then f ( P ) ω . � Q Fundamental theorem of calculus: P df = f ( Q ) − f ( P ) . Balakrishnan, Bradshaw, Kedlaya (MIT) Coleman integration for hyperelliptic curves ANTS-IX 8 / 21
Integrals for hyperelliptic curves Properties of the Coleman integral Properties of the Coleman integral Coleman formulated an integration theory on wide open subspaces of curves over O , exhibiting no phenomena of path dependence. � Q This allows us to define P ω whenever ω is a meromorphic 1-form on X , and P , Q ∈ X ( Q p ) are points where ω is holomorphic. Properties of the Coleman integral include: Theorem (Coleman) � Q � Q � Q P ( αω 1 + βω 2 ) = α P ω 1 + β Linearity: P ω 2 . � R � Q � R Additivity: P ω = P ω + Q ω . Change of variables: if X ′ is another such curve, and f : U → U ′ is a � Q � f ( Q ) P f ∗ ω = rigid analytic map between wide opens, then f ( P ) ω . � Q Fundamental theorem of calculus: P df = f ( Q ) − f ( P ) . Balakrishnan, Bradshaw, Kedlaya (MIT) Coleman integration for hyperelliptic curves ANTS-IX 8 / 21
Integrals for hyperelliptic curves Properties of the Coleman integral Properties of the Coleman integral Coleman formulated an integration theory on wide open subspaces of curves over O , exhibiting no phenomena of path dependence. � Q This allows us to define P ω whenever ω is a meromorphic 1-form on X , and P , Q ∈ X ( Q p ) are points where ω is holomorphic. Properties of the Coleman integral include: Theorem (Coleman) � Q � Q � Q P ( αω 1 + βω 2 ) = α P ω 1 + β Linearity: P ω 2 . � R � Q � R Additivity: P ω = P ω + Q ω . Change of variables: if X ′ is another such curve, and f : U → U ′ is a � Q � f ( Q ) P f ∗ ω = rigid analytic map between wide opens, then f ( P ) ω . � Q Fundamental theorem of calculus: P df = f ( Q ) − f ( P ) . Balakrishnan, Bradshaw, Kedlaya (MIT) Coleman integration for hyperelliptic curves ANTS-IX 8 / 21
Integrals for hyperelliptic curves Properties of the Coleman integral Properties of the Coleman integral Coleman formulated an integration theory on wide open subspaces of curves over O , exhibiting no phenomena of path dependence. � Q This allows us to define P ω whenever ω is a meromorphic 1-form on X , and P , Q ∈ X ( Q p ) are points where ω is holomorphic. Properties of the Coleman integral include: Theorem (Coleman) � Q � Q � Q P ( αω 1 + βω 2 ) = α P ω 1 + β Linearity: P ω 2 . � R � Q � R Additivity: P ω = P ω + Q ω . Change of variables: if X ′ is another such curve, and f : U → U ′ is a � Q � f ( Q ) P f ∗ ω = rigid analytic map between wide opens, then f ( P ) ω . � Q Fundamental theorem of calculus: P df = f ( Q ) − f ( P ) . Balakrishnan, Bradshaw, Kedlaya (MIT) Coleman integration for hyperelliptic curves ANTS-IX 8 / 21
Integrals for hyperelliptic curves Frobenius Coleman’s construction How do we integrate if P , Q aren’t in the same residue disc? Coleman’s key idea: use Frobenius to move between di ff erent residue discs (Dwork’s “analytic continuation along Frobenius”) Frobenius “Tiny” integral P’ Q’ P Q So we need to calculate the action of Frobenius on di ff erentials. Balakrishnan, Bradshaw, Kedlaya (MIT) Coleman integration for hyperelliptic curves ANTS-IX 9 / 21
Integrals for hyperelliptic curves Frobenius Frobenius, MW-cohomology X ′ : a ffi ne curve ( X − { Weierstrass points of X } ) A : coordinate ring of X ′ To discuss the di ff erentials we will be integrating, we recall: The Monsky-Washnitzer (MW) weak completion of A is the ring A † consisting of infinite sums of the form � � ∞ � B i ( x ) , B i ( x ) ∈ K [ x ] , deg B i � 2 g , y i i =− ∞ further subject to the condition that v p ( B i ( x )) grows faster than a linear function of i as i → ± ∞ . We make a ring out of these using the relation y 2 = f ( x ) . These functions are holomorphic on wide opens, so we will integrate 1-forms ω = g ( x , y ) dx g ( x , y ) ∈ A † . 2 y , Balakrishnan, Bradshaw, Kedlaya (MIT) Coleman integration for hyperelliptic curves ANTS-IX 10 / 21
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