Iterated integrals, diagonal cycles and rational points on elliptic curves November 2, 2011
Classical Heegner points Let E /Q be an elliptic curve and f = n 1 a n q n S 2 (N) with L(E, s) = L(f, s). The modular parametrization is ϕ : X 0 (N)(C) = Γ 0 (N)\H E(C) τ P τ := 2πi τ f (z)dz = n 1 an n e2πin τ If τ P 1 (Q) is a cusp: P τ E(Q) tors. If τ H K, where K is imaginary quadratic: P τ E(K ab ).
Stark-Heegner points Bertolini, Darmon, Greenberg replaced H by the p-adic upper half-plane, using Coleman p-adic path integrals. For E /Q, K real quadratic where p is inert and H/K ring class field, Darmon constructs points on E(C p ) which should be H-rational. Bertolini, Dasgupta, Greenberg, Longo, R., Seveso, Vigni complete the conjectural picture. For E /F modular over a totally real F, Darmon and Logan use a similar cohomological formalism to construct points on ring class fields H/K of ATR quadratic extensions K /F. Gartner generalizes to any K /F provided the signs of the functional equations match, but is not effective.
The modular parametrization revisited The universal covering of X 0 (N) is P(X 0 (N); ) = {γ : [0, 1] X 0 (N), γ(0) = }/homotopy. The modular parametrization factors through ϕ : X 0 (N) = π 1 (X 0 (N))\P(X 0 (N)) J 0 (N) E γ : τ P τ := γ ω f, as π 1 (X 0 (N)) C, γ γ ω f factors through H 1 (X 0 (N), Z). Chen s iterated integrals may give rise to anabelian modular parametrizations of points in E(C).
Chen s iterated path integrals Y smooth quasi-projective curve, o Y base point, Ỹ universal covering. The iterated integral attached to a tuple of smooth 1-forms (ω 1,..., ω n ) on Y is the functional γ ω 1 ω 2... ω n := (γ ω 1 )(t 1 )(γ ω 2 )(t 2 ) (γ ω n )(t n ), γ where = {0 t n t n 1 t 1 1}. When n = 2 : γ ω η = γ ωf η, for F η primitive of η on Ỹ. A linear combination of iterated integrals which is homotopy invariant yields J : P(Y ; o) C.
Iterated integrals of modular forms X = X 0 (N), Y = X \ { }, cusp 0 as base point. Let ω Ω 1 (X) and η Ω 1 (Y ), with a pole at. Let α = α ω,η Ω 1 (Y ) such that ωf η α ω,η on Ỹ has log poles over. J ω,η := ω η α ω,η is homotopy-invariant.
Iterated integrals of modular forms Let E /Q be an elliptic curve and f = f E S 2 (N E ). Let g S 2 (M) be a newform of some level M, with [Q({a n (g)}) : Q] = t 1. Put N = lcm(m, N E ). γ f H 1 (X, C) Poincaré dual of ω f. Let {ω g,i, η g,i } i=1,...,t be a symplectic basis of H 1 (X)[g]. Define P g,f := t i=1 γ ω f g,i η g,i η g,i ω g,i 2α i E(C). The point is independent of the choice of base point 0, path γ f or basis of H 1 (X)[g].
Numerical computation With Michael Daub, Henri Darmon and Sam Lichtenstein we have an algorithm to compute P g,f : Given N 1, define c N the smallest integer for which there are γ j = ( ) a b Γ0 (N), c c N cn d such that H 1 (X, Z) =..., [γ j ],... Z. The number n D of Fourier coefficients required to compute P g,f to a given number D of digits of accuracy is n D = O(max{N c N (D + N 11σ 0(N)+2 ), c 2 N N2σ 0(N)+2 }). We represent the 1-forms η g,i as differentials of the 2nd kind: u i ω g,i where u i are modular units given as eta products.
Some points on curves of rank 1 and conductor < 100 E P gen g n P g,f,n 37a1 (0, 1) 1 1 6P 43a1 (0, 1) 1 1 4P 53a1 (0, 1) 1 1 2P 4 57a1 (2, 1) 1 1 3 P 2 1 16 3 P 3 1 4P 58a1 (0, 1) 1 1 4P 2 1 0 2 4P 12 77a1 (2, 3) 1 1 5 P 2 1 4 3 P 4 3 1 3 P 4 1 12 5 P 79a1 (0, 0) 1 1 4P 82a1 (0, 0) 1 1 0 3 2P 2 1 2P 83a1 (0, 0) 1 1 0 2 2P 88a1 (2, 2) 1 1 0 2 1 0 2 8P 3 1 0 2 8P 91a1 (0, 0) 1 1 2P 2 1 2P 3 1 4P 91b1 ( 1, 3) 0 1 0 2 1 0 3 1 0 92b1 (1, 1) 1 1 0 2 1 0 99a1 (2, 0) 1 1 2 3 P 2 1 0 3 1 2 3 P
Connection with diagonal cycles Theorem 1 (Darmon-R.-Sols) The points P f,g are Q-rational. P g,f is the Chow-Heegner point associated with the [g, g, f ]-isotypical component of Gross-Kudla-Schoen s diagonal cycle = {(x, x, x), x X} {(x, x, 0)} {(x, 0, x)} {(0, x, x)}+ +{(0, 0, x)} + {(0, x, 0)} + {(x, 0, 0)} X 3, a null-homologous cycle of codimension two in X 3 : Putting Π = {(x, x, y, y)} X 4, P g,f = π f, (Π π123 ( [g, g, f ])).
Connection with diagonal cycles In fact, for any divisor T we obtain a point Pic(X X) π 1 Pic(X) π 2 Pic(X) End(J 0(N)) P T = Π π 123 ( T ) for a suitable T CH 2 (X 3 ) 0. This gives rise to a new modular parametrization of points End(J 0 (N)) Hodge(X 0 (N) 2 ) J 0 (N)(Q) π f, E(Q), T P T, which is Gal( Q/Q)-equivariant for its natural extension to Q.
Connection with L-functions The triple L-function of f S k (N f ), g S l (N g ), h S m (N h ) is L(f, g, h; s) = L(V f V g V h ; s) = p L (p) (f, g, h; p s ) 1, For p N = lcm(n f, N g, N h ), the Euler factor L (p) (f, g, h; T ) is (1 α f α g α h T ) (1 α f α g β h T )... (1 β f β g β h T ). The completed L-function satisfies Λ(f, g, h; s) = ε p (f, g, h) Λ(f, g, h; k + l + m 2 s). ε (f, g, h) = p N { 1 if (k, l, m) are balanced. +1 if (k, l, m) are unbalanced.
Combining our result with Yuan-Zhang-Zhang Theorem 2. Let E /Q be an elliptic curve of conductor N E and g a newform of level M. Assume ε p (g, g, f ) = +1 at the primes p N = lcm(m, N E ). Then the module of points P g,f := π f (d){p T, T End 0 (J 0 (N))[g]} E(Q) d N N E is nonzero if and only if: i. L(f, 1) = 0, ii. L (f, 1) 0, and iii. L(f Sym 2 (g σ ), 2) 0 for all σ : K g C.
Examples E = 37a, g = 37b. P g,f = P g,f is not torsion. ε 37 (g, g, f ) = +1 and L(f Sym 2 (g), 2) 0. E = 58a and g = 29a. ε 2 (g, g, f ) = ε 29 (g, g, f ) = +1 and L(f Sym 2 (g), 2) 0. But P g,f is torsion. P g,f contains the non-torsion point P g,f,2 := π f (P Tg T 2 ). E = 91b, g = 91a. P g,f = P g,f is torsion, because ε 7 (g, g, f ) = ε 13 (g, g, f ) = 1. Wants a Shimura curve. E = 158b, g = 158d. While ε 2 (g, g, f ) = ε 79 (g, g, f ) = +1, P g,f = P g,f is torsion, because L(f Sym 2 (g), 2) = 0.
Motivic explanation of the rationality of P g,f Set Y = X 0 (N) \ { }, Γ = π 1 (Y ; 0). I = γ 1 augmentation ideal of Z[Γ]: Z[Γ]/I = Z. I/I 2 = Γ ab = H 1 (Y, Z) = H 1 (X, Z). I 2 /I 3 = (Γ ab Γ ab ), γ 1 γ 2 (γ 1 1)(γ 2 1). {P(Y ; 0) C of length n} Hom(I/I n+1, C).
An extension of mixed motives The exact sequence 0 I 2 /I 3 I/I 3 I/I 2 0 becomes 0 H 1 B (Y ) M B H 1 B (Y ) 2 0. The first and third groups are the Betti realizations of a pure motive defined over Q. The complexification of both is equipped with a Hodge filtration: both are pure Hodge structures. M B = Hom(I/I 3, C) = {J : P(Y ; 0) C of length 2} underlies a mixed Hodge structure and should arise from a mixed motive over Q.
Motivic explanation of the rationality of P g,f M B yields an extension class κ Ext 1 MHS (H1 B (X) 2, H 1 B (X)). Any ξ : Z( 1) H 1 B (X) 2 yields 0 H 1 B (X) M B (ξ) Z( 1) 0 0 H 1 B (X) M B H 1 B (X) 2 0. ξ ϕ : Ext 1 MHS (Z( 1), H1 B (X)) = ξ g : 1 cl(t g ) H 2 B (X 2 ) Kunneth H 1 B (X) 2, P g := ϕ(ξ g ) J 0 (N), P g,f := π f (P g ). H 1 dr (X/C) Ω 1 (X(C))+H 1 B (X) J 0(N)(C).
P g,f as a Chow-Heegner point The complex Abel-Jacobi map for curves AJ C : CH 1 (X) 0 Ω 1, X /H 1(X, Z), D generalizes to varieties V of higher dimension d and null-homologous cycles of codimension c: AJ C : CH c (V ) 0 J c (V ) = Fild c+1 H 2d 2c+1 dr (V C ) H 2d 2c+1 (V,Z),, where is a 2(d c) + 1-differentiable chain on the real manifold V (C) with boundary. D
P g,f as a complex Chow-Heegner point CH 2 (X 3 ) 0 AJ C J 2 (X 3 ) = Fil2 H 3 dr (X 3 ) H 3 (X 3,Z) Π Π C E AJ C C/ΛE, CH 2 (X 3 ) 0 π123 π 123 Π P := π E, (π123 Π) E. Theorem. (Darmon-R.-Sols) In E(Q) Z Q: AJ C ( GKS )(cl(t g ) ω f ) = γ f ( t ) ω g,i η g,i η g,i ω g,i 2α i. i=1
P g,f as a p-adic Chow-Heegner point via Coleman integration The p-adic Abel-Jacobi map at a prime p N is AJ p : CH r+2 (X 3 ) 0 (Q p ) Fil 2 H 3 dr (X 3 /Q p ). log ωf (P g,f ) = 2AJ p ( GKS )(η g ω g ω f ). Theorem. (Darmon-R.) Let (W, Φ) be a wide open of X 0 (N)(C p ) \ red 1 (X( F p ) ss ) and a lift of Frobenius. Let ρ Ω 1 (W W) such that dρ = P(Φ)(ω g ω f ) for a suitable polynomial P. Then AJ p ( )(η g ω g ω f ) = η, P(Φ) 1 ɛ ρ X where ɛ = ɛ 12 ɛ 1 ɛ 2, for ɛ 12, ɛ 1, ɛ 2 : X X 2.