Numerical computation of Coleman integrals Kiran S. Kedlaya Department of Mathematics, Massachusetts Institute of Technology p -adic Methods and Rational Points R´ enyi Institute (Budapest), May 20, 2007 Kiran S. Kedlaya (MIT, Dept. of Mathematics) Numerical computation of Coleman integrals R´ enyi Institute, May 20, 2007 1 / 26
Contents Contents Introduction and motivation 1 A framework for computing Coleman integrals 2 Example: hyperelliptic curves 3 Implementation and demonstration 4 What to do next? 5 Kiran S. Kedlaya (MIT, Dept. of Mathematics) Numerical computation of Coleman integrals R´ enyi Institute, May 20, 2007 2 / 26
Introduction and motivation Contents Introduction and motivation 1 A framework for computing Coleman integrals 2 Example: hyperelliptic curves 3 Implementation and demonstration 4 What to do next? 5 Kiran S. Kedlaya (MIT, Dept. of Mathematics) Numerical computation of Coleman integrals R´ enyi Institute, May 20, 2007 3 / 26
Introduction and motivation Overview of Coleman integration Let C be a smooth proper curve over Z q = W ( F q ) . Coleman described a � Q P ω whenever ω is a meromorphic 1-form on C Q q , and canonical integral P , Q ∈ C ( Q q ) are points where ω is holomorphic. Properties include: � Q � Q � Q P ( αω 1 + βω 2 ) = α P ω 1 + β P ω 2 . Linearity: � R � Q � R P ω = P ω + Q ω . Additivity: Change of variables: if C ′ is another such curve, and f : U → U ′ is a rigid � Q � f ( Q ) P f ∗ ω = f ( P ) ω . analytic map between wide opens, then � Q Fundamental theorem of calculus: P df = f ( Q ) − f ( P ) . Kiran S. Kedlaya (MIT, Dept. of Mathematics) Numerical computation of Coleman integrals R´ enyi Institute, May 20, 2007 4 / 26
Introduction and motivation Application: Chabauty-Coleman method Let C be a smooth curve over Z [ 1 / N ] admitting a compactification C which is smooth proper over Z [ 1 / N ] , with C − C a relative normal crossings divisor. (E.g., a smooth proper curve, or P 1 −{ 0 , 1 , ∞ } .) Assume p � | N . The Chabauty condition is rank J ( C )( Z [ 1 / N ]) < dim J ( C ) . When this is satisfied, J ( C )( Z [ 1 / N ]) lies in a closed analytic subspace of J ( C ) an , which meets C an in finitely many points. Equivalently, there exists a � P 1-form ω on J ( C ) an with O ω = 0 for P ∈ J ( C )( Z [ 1 / N ]) . � P If we can find all points P ∈ C an where O ω = 0, we may be able to determine C ( Z [ 1 / N ]) . Kiran S. Kedlaya (MIT, Dept. of Mathematics) Numerical computation of Coleman integrals R´ enyi Institute, May 20, 2007 5 / 26
Introduction and motivation Application: Kim’s nonabelian Chabauty method What if the Chabauty condition fails? Instead of J ( C ) , one can work with a Selmer variety corresponding to a unipotent quotient of π geom ( C ) . Kim 1 conjectures that a suitable analogue of the Chabauty condition holds for a sufficiently large quotient (true for P 1 −{ 0 , 1 , ∞ } ). If one can describe the Selmer variety, one can proceed as the original Chabauty method, but replacing the Coleman integral by an iterated version. (Cf. talk of Wewers.) Kiran S. Kedlaya (MIT, Dept. of Mathematics) Numerical computation of Coleman integrals R´ enyi Institute, May 20, 2007 6 / 26
Introduction and motivation Application: p -adic heights One can use Coleman integrals to compute p -adic heights on Jacobians of curves over number fields (Coleman-Gross, Besser). These heights appear in analogues of the Birch-Swinnerton-Dyer conjecture for p -adic L -functions (Mazur-Tate-Teitelbaum). (Cf. talk of Besser.) Kiran S. Kedlaya (MIT, Dept. of Mathematics) Numerical computation of Coleman integrals R´ enyi Institute, May 20, 2007 7 / 26
A framework for computing Coleman integrals Contents Introduction and motivation 1 A framework for computing Coleman integrals 2 Example: hyperelliptic curves 3 Implementation and demonstration 4 What to do next? 5 Kiran S. Kedlaya (MIT, Dept. of Mathematics) Numerical computation of Coleman integrals R´ enyi Institute, May 20, 2007 8 / 26
A framework for computing Coleman integrals The fundamental linear system Fix an open dense subscheme U of C F q , and let φ : V 1 → V 2 be a q -power Frobenius lift between two strict neighborhoods of the tube ] U [ of U in C Q q . Let ω 1 ,..., ω n be 1-forms forming a basis for H 1 dR ( V ) for V a wide open strict neighborhood of ] U [ . We can then write n ∑ φ ∗ ω i = df i + A ij ω j j = 1 for some functions f i and some n × n matrix A over Q q . Kiran S. Kedlaya (MIT, Dept. of Mathematics) Numerical computation of Coleman integrals R´ enyi Institute, May 20, 2007 9 / 26
A framework for computing Coleman integrals Using the fundamental linear system � Q P ω , for ω a meromorphic 1-form on C Q q and Say we want to compute P , Q ∈ ] U [ . We can write ω = df + c 1 ω 1 + ··· + c n ω n � Q P ω i . for some function f and some c i ∈ Q q , so it suffices to compute the Using the fundamental linear system, we write � φ ( Q ) � Q � Q n ∑ φ ( P ) ω i = P φ ∗ ω i = f i ( Q ) − f i ( P )+ P ω j . A ij j = 1 In other words, � φ ( P ) � Q � Q � Q n ∑ P ω i = ω i + φ ( Q ) ω i + f i ( Q ) − f i ( P )+ P ω j . A ij P j = 1 Kiran S. Kedlaya (MIT, Dept. of Mathematics) Numerical computation of Coleman integrals R´ enyi Institute, May 20, 2007 10 / 26
A framework for computing Coleman integrals Using the fundamental linear system (contd.) The last equation from the previous slide is equivalent to � φ ( P ) � Q � Q n ∑ P ω j = f i ( P ) − f i ( Q ) − ω i − φ ( Q ) ω i . ( A − I ) ij P j = 1 The integrals on the right side are within a single residue disc, where the formal antiderivative of ω i converges. So we can numerically approximate the right side of the equation. The matrix A − I is invertible because the eigenvalues of A have C -norm q 1 / 2 or q , so we can solve the linear system. (This is almost Coleman’s original construction.) If q � = p , it is easier to write down an analogous semilinear system for a p -power Frobenius lift φ p , then derive the linear system for the appropriate power of φ p . Kiran S. Kedlaya (MIT, Dept. of Mathematics) Numerical computation of Coleman integrals R´ enyi Institute, May 20, 2007 11 / 26
A framework for computing Coleman integrals Teichm¨ uller points In each residue disc, there is a unique point P with φ ( P ) = P ; this is a uller point for the map φ . Teichm¨ When computing Coleman integrals, it may be convenient to first compute the � φ ( P ) ω i = 0) integral between Teichm¨ uller points in the right discs (for which P and then correct the endpoints afterward. Kiran S. Kedlaya (MIT, Dept. of Mathematics) Numerical computation of Coleman integrals R´ enyi Institute, May 20, 2007 12 / 26
Example: hyperelliptic curves Contents Introduction and motivation 1 A framework for computing Coleman integrals 2 Example: hyperelliptic curves 3 Implementation and demonstration 4 What to do next? 5 Kiran S. Kedlaya (MIT, Dept. of Mathematics) Numerical computation of Coleman integrals R´ enyi Institute, May 20, 2007 13 / 26
Example: hyperelliptic curves Hyperelliptic curves Assume p � = 2. Let C / Z q be a hyperelliptic curve of genus g with a rational Weierstrass point; we can then write C as y 2 = P ( x ) for P ( x ) monic of degree 2 g + 1. We will take our subscheme U of C F q to be the complement of the Weierstrass points; i.e., U = Spec F q [ x , y , z ] / ( y 2 − P ( x ) , yz − 1 ) . Kiran S. Kedlaya (MIT, Dept. of Mathematics) Numerical computation of Coleman integrals R´ enyi Institute, May 20, 2007 14 / 26
Example: hyperelliptic curves Cohomology of hyperelliptic curves The first de Rham cohomology of C minus its Weierstrass point is generated by x i dx x i dx ( i = 0 ,..., 2 g − 1 ) , ( i = 0 ,..., 2 g ) . y 2 y Moreover, there is a simple procedure to express any 1-form as an exact 1-form plus a linear combination of these, using relations such as: A ( s − 2 ) P ′ dy ≡ 2 A ′ dx ( s � = 2 ) . y s − 2 y s This extends to a wide open V in which one removes a closed disc of radius < 1 around each Weierstrass point. Kiran S. Kedlaya (MIT, Dept. of Mathematics) Numerical computation of Coleman integrals R´ enyi Institute, May 20, 2007 15 / 26
Example: hyperelliptic curves Computing the Frobenius action We use the Frobenius lift x �→ x q � 1 / 2 � P ( x q ) − P ( x ) q y �→ y q P ( x ) q truncated to some appropriate p -adic precision. Again, if q � = p , it is easier to work with a p -power Frobenius instead. Kiran S. Kedlaya (MIT, Dept. of Mathematics) Numerical computation of Coleman integrals R´ enyi Institute, May 20, 2007 16 / 26
Implementation and demonstration Contents Introduction and motivation 1 A framework for computing Coleman integrals 2 Example: hyperelliptic curves 3 Implementation and demonstration 4 What to do next? 5 Kiran S. Kedlaya (MIT, Dept. of Mathematics) Numerical computation of Coleman integrals R´ enyi Institute, May 20, 2007 17 / 26
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