Rational points on elliptic curves. cycles on modular varieties

Transcription

1 Rational points on elliptic curves and cycles on modular varieties Mathematics Colloquium January 2009 TIFR, Mumbai Henri Darmon McGill University /slides/slides.html

2 Elliptic Curves An elliptic curve is an equation of the form E : y 2 = x 3 + ax + b, with := 4a 3 27b 2 0. Why elliptic curves? 1

3 The addition law Elliptic curves are algebraic groups. y R 2 3 y = x + a x + b Q P x P+Q The addition law on an elliptic curve 2

4 The main question E(F ) := group of rational points of E over F. Mordell-Weil Theorem: If F is a number field, then E(F ) is finitely generated. r(e, F ) := rank(e(f )). Question: Given a fixed curve E, understand the function F r(e, F ) := rank(e(f )). This function is erratic, poorly understood... But it can also display surprising regularities. Conjecture (Coates, Sujatha). elliptic curve E = y 2 y = x 3 x 2 Consider the of conductor 11. Then r(e, Q(11 1/3n )) = n. Question: What are the arithmetic/geometric structures that might account for patterns of this kind? 3

5 Modularity N = conductor of E. a(p) := { p + 1 #E(Z/pZ) if p N; 0, or ± 1 if p N. a(p n ) = { a(p)a(p n 1 ) pa(p n 2 ), if p N; a(p) n if p N. a(mn) = a(m)a(n) if gcd(m, n) = 1. Generating series: f E (z) = n=1 a(n)e 2πinz, z H, H := Poincaré upper half-plane 4

6 Modularity Modularity: the series f E (z) satisfies a deep symmetry property. M 0 (N) := ring of 2 2 integer matrices which are upper triangular modulo N. Γ 0 (N) := M 0 (N) 1 = units of determinant 1. Theorem: The series f E is a modular form of weight two on Γ 0 (N). f E ( az + b cz + d ) = (cz + d) 2 f E (z). In particular, the differential form ω f := f E (z)dz is defined on the quotient X := Γ 0 (N)\H. 5

7 Cycles and modularity The Riemann surface X contains many natural cycles, which convey a tremendous amount of arithmetic information about E( Q). These cycles are indexed by the commutative subrings of M 0 (N): e.g., orders in quadratic fields. Disc(R) := discriminant of R. Σ D = Γ 0 (N)\{R M 0 (N) with Disc(R) = D}. G D := Equivalence classes of binary quadratic forms of discriminant D. The set Σ D, if non-empty, is equipped with an action of the class group G D. 6

8 The special cycles γ R X Case 1. Disc(R) > 0. Then R has two real fixed points τ R, τ R R. Υ R := geodesic from τ R to τ R ; γ R := R 1 \Υ R Case 2. Disc(R) < 0. Then (R Q) has a single fixed point τ R H. γ R := {τ R } 7

9 An (idealised) picture For each discriminant D, define: γ D = γ R, the sum being taken over a G D -orbit in Σ D. Convention: γ D = 0 if Σ D is empty. Fact: The periods of ω f against γ R and γ D convey alot of information about the arithmetic of E over quadratic fields. 8

10 The case D > 0 Theorem (Eichler, Shimura) The set Λ := γ R ω f, R Σ >0 C is a lattice in C, which is commensurable with the Weierstrass lattice of E. Proof (Sketch) 1. Modular curves: X = Y 0 (N)(C), where Y 0 (N) is an algebraic curve over Q, parametrising elliptic curves over Q. 2. Eichler-Shimura: There exists an elliptic curve E f and a quotient map Φ f : Y 0 (N) E f such that γ R ω f = Φ(γ R ) ω E f Λ Ef. 9

11 Hence, γ R ω f is a period of E f. The curves E f and E are related by: a n (E f ) = a n (E) for all n Isogeny conjecture for curves (Faltings): E f is isogenous to E over Q. 10

12 Arithmetic information Conjecture (BSD) Let D > 0 be a fundamental discriminant. Then J D := γ D ω f 0 iff #E(Q( D)) <. The position of γ D in the homology H 1 (X, Z) represents an obstruction to the presence of rational points on E(Q( D)). One implication is known: Gross-Zagier, Kolyvagin. E(Q( D)) is finite. If J D 0, then 11

13 The case D < 0 The γ R are 0-cycles, and their image in H 0 (X, Z) is constant (independent of R). Hence we can produce many homologically trivial 0-cycles suppported on Σ D : Σ 0 D := ker(div(σ D) H 0 (X, Z)). Extend R γ R to Σ 0 D by linearity. Let γ # := any smooth one-chain on X having γ as boundary, P := γ ω f C/Λ f E(C). 12

14 CM points CM point Theorem For all Σ 0 D, the point P belongs to E(H D ) Q, where H D is the Hilbert class field of Q( D). Proof (Sketch) 1. Complex multiplication: If R Σ D, the 0- cycle γ R is a point of Y 0 (N)(C) corresponding to an elliptic curve with complex multiplication by Q( D). Hence it is defined over H D. 2. Explicit formula for Φ: Φ(γ ) = P. The systematic supply of algebraic points on E given by the CM point theorem is an essential tool in studying the arithmetic of E over K. 13

15 Generalisations? Principle of functoriality: modularity admits many incarnations. Simple example: quadratic base change. Choose a fixed real quadratic field F, and let E be an elliptic curve over F. Notation: (v 1, v 2 ) : F R R, x (x 1, x 2 ). Assumptions: h + (F ) = 1, N = 1. Counting points mod p yields n a(n) Z, on the integral ideals of O F. We can package these coefficients into a modular generating series. 14

16 Modularity Generating series G(z 1, z 2 ) := n>>0 a((n))e 2πi ( n1 d1 z 1 + n 2 d2 z 2 ), where d := totally positive generator of the different of F. Modularity Conjecture/Theorem: G(γ 1 z 1, γ 2 z 2 ) = (c 1 z 1 +d 2 ) 2 (c 2 z 2 +d 2 ) 2 G(z 1, z 2 ), for all ( a b ) γ = c d SL 2 (O F ). 15

17 Geometric formulation The differential form α G := G(z 1, z 2 )dz 1 dz 2 is a holomorphic (hence closed) 2-form defined on the quotient X F := SL 2 (O F )\(H H). It is better to work with the harmonic form ω G := G(z 1, z 2 )dz 1 dz 2 + G(ɛ 1 z 1, ɛ 2 z 2 )dz 1 d z 2, where ɛ O F satisfies ɛ 1 > 0, ɛ 2 < 0. ω G is a closed two-form on the four-dimensional manifold X F. Question: What do the periods of ω G, against various natural cycles on X F, know about the arithmetic of E over F? 16

18 Cycles on the four-manifold X F The natural cycles on the four-manifold X F are now indexed by commutative O F -subalgebras of M 2 (O F ); e.g., by O F -orders in quadratic extensions of F. D := Disc(R) := relative discriminant of R over F. There are now three cases to consider. 1. D 1, D 2 > 0: the totally real case. 2. D 1, D 2 < 0: the complex multiplication (CM) case. 3. D 1 < 0, D 2 > 0: the almost totally real (ATR) case. 17

19 The special cycles γ R X F Case 1. Disc(R) >> 0. Then, for j = 1, 2, R acting on H j has two fixed points τ j, τ j R. Let Υ j := geodesic from τ j to τ j ; γ R := R 1 \(Υ 1 Υ 2 ) Case 2. Disc(R) << 0. Then, (R Q) has a single fixed point τ R H H. γ R := {τ R } 18

20 The ATR case Case 3. D 1 < 0, D 2 > 0. Then R has a unique fixed point τ 1 H 1. R acting on H 2 has two fixed points τ 2, τ 2 R. Let Υ 2 := geodesic from τ 2 to τ 2 ; γ R := R 1 \({τ 1} Υ 2 ) The cycle γ R is a closed one-cycle in X F. It is called an ATR cycle. 19

21 R 1 An (idealised) picture R 8 R R 12 R 10 R 6 R 9 R R 2 7 R 3 R 4 R 5 R 11 Cycles on the four-manifold X F 20

22 The case D >> 0 Conjecture (Oda) The set Λ G := γ R ω G, R Σ >>0 C is a lattice in C which is commensurable with the Weierstrass lattice of E. Conjecture (BSD) Let D := Disc(K/F ) >> 0. Then J D := γ D ω G 0 iff #E(K) <. The position of γ D in H 2 (X F, Z) encodes an obstruction to the presence of rational points on E(F ( D)). 21

23 The ATR case Theorem: The cycles γ R are homologically trivial (after tensoring with Q). This is because H 1 (X F, Q) = 0. Given R Σ D, let γ # R := any smooth two-chain on X F having γ R as boundary. P R := γ R ω G C/Λ G E(C). 22

24 The conjecture on ATR points Assume still that D 1 < 0, D 2 > 0. ATR points conjecture. (D., Logan, Charollois...) If R Σ D, then the point P R belongs to E(H D ), where H D is the Hilbert class field of F ( D). Question: Understand the process whereby the one-dimensional ATR cycles γ R on X F lead to the construction of algebraic points on E. Several potential applications: a) Construction of natural collections of algebraic points on E; b) Explicit construction of class fields. 23

25 Algebraic cycles Basic set-up (Bertolini, Prasanna, D). K=quadratic imaginary field, (h(k) = 1). E = C/O K (elliptic curve with CM). L(E, s) = L(θ ψ, s), where θ ψ = a O K ψ(a)q aā. Idea: Replace the weight 2 form θ ψ by the form θ ψ r+1 of weight r + 2 on Γ SL 2 (Z). W r = r-fold product of the universal elliptic curve over X Γ (Kuga-Sato variety). Modular curves are replaced by X r = W r E r (a variety of dimension 1 + 2r). CH r+1 (X r ) 0 = null-homologous, codimension r + 1 algebraic cycles on X r /. 24

26 Exotic modular parametrisations Conjecture There is an algebraic correspondence Π X r E, of complex dimension r + 1, which induces a non constant exotic modular parametrisation : Φ : CH r+1 (X r ) 0 E, by the rule Φ( ) = π E (π 1 ( ) Π). π : X r E X r, π E : X r E E. Explanation: The l-adic Galois representation Het 1 (Ē, Q l )(1) occurs as a constituent of Het 2r+1 ( X r, Q l )(r+1). The Tate conjecture asserts that such a coincidence cannot be fortuitous, but must be accounted for by the existence of a suitable algebraic correspondence in X r E. 25

27 Exotic modular parametrisations, cont d Geometric examples: When r = 1 and K = Q(i), Q( 3), Q( 2), or Q( 7), the surface W 1 is a singular K3 surface, and X 1 is a Calabi-Yau threefold. The correspondence Π can then be exhibited explicitly using Shioda-Inose structures. Why is Φ useful? CH r+1 (X r ) 0 contains a large supply of interesting algebraic cycles, indexed by isogenies ϕ : E E. ϕ = graph(ϕ) r (E E ) r = (E ) r E r W r E r 26

28 Analytic calculations Key remark: Even though the map Φ cannot be shown to exist, it can still be computed, analytically. Complex formulae. (Well-suited for computer calculations) Write E (C) = C/ 1, τ, Φ( ϕ ) = τ i (z τ)r θ ψ r+1(z)dz C/Λ = E (C). p-adic formulae (Well-suited for theoretical calculations) For F any finite extension of Q p, we have: Φ p : CH r+1 (X r ) 0 (F ) E(F ). Key Ingredients: Hodge theory. p-adic integration, p-adic 27

29 Theorem (Bertolini, Prasanna, D). Let H be the field of definition of ϕ, and let p be a prime which splits in K/Q. There exists a global point P ϕ E(H) such that for all p p. Φ p ( ϕ ) = P ϕ in E(H p ), Addendum: The points P ϕ can be related to special values of Hecke L-series, as in the Gross-Zagier formula. This gives support (however fragmentary) for the general program of constructing algebraic points on elliptic curves using (algebraic) cycles on modular varieties. 28

30 A final question. Vague Definition: A point P E( Q) is said to be modular if there exists: a modular variety X, an exotic modular parametrisation Φ : CH r (X) 0 E, and a modular cycle CH r (X), such that P = λφ( ), for some λ Q. Question. Given E, what points in E( Q) are modular? Very optimistic: All algebraic points on E are modular. Optimistic: All algebraic points on E satisfying a suitable rank one hypothesis are modular. Legitimate question: Find a simple characterisation of the modular points. 29