Rational points on elliptic curves and cycles on Shimura varieties - PDF document

Rational points on elliptic curves and cycles on Shimura varieties Harvard-MIT-Brandeis-Northeastern Joint Colloquium Henri Darmon McGill University February 28, 2008 http://www.math.mcgill.ca/darmon /slides/slides.html

Diophantine equations f 1 , . . . , f m ∈ Z [ x 1 , . . . , x n ] ,  f 1 ( x 1 , . . . , x n ) = 0  . . .  . . . X : . . . f m ( x 1 , . . . , x n ) = 0 .   Question : What is an interesting Diophantine equation? A “working definition” . A Diophantine equa- tion is interesting if it reveals or suggests a rich underlying mathematical structure. (In other words, a Diophantine question is in- teresting if it has an interesting answer...!) 1

Some examples Fermat, 1635 : Pell’s equation x 2 − ny 2 = 1 has infinitely many solutions because the class group of binary quadratic forms of discriminant 4 n is finite. Kummer, 1847 : Fermat’s equation x n + y n = z n has no non-zero solution for 2 < n < 37 because all primes p < 37 are regular . Mazur, Frey, Serre, Ribet, Wiles, Taylor, Fermat’s equation x n + y n = z n has 1994 : no non-zero solution for all n > 2 because all elliptic curves are modular . 2

Elliptic Curves An elliptic curve is an equation of the form E : y 2 = x 3 + ax + b, with ∆ := 4 a 3 − 27 b 2 � = 0 . If F is a field, E ( F ) := Mordell-Weil group of E over F . Why elliptic curves? 3

The addition law Elliptic curves are algebraic groups . y 2 3 y = x + a x + b R Q x P P+Q The addition law on an elliptic curve 4

Modularity Let N = conductor of E . � p + 1 − # E ( Z /p Z ) if p � | N ; a ( p ) := 0 , ± 1 if p | N. a ( mn ) = a ( m ) a ( n ) if gcd( m, n ) = 1 , a ( p n ) = a ( p ) a ( p n − 1 ) − pa ( p n − 2 ) , if p � | N. Generating series : ∞ a ( n ) e 2 πinz , � f E ( z ) = z ∈ H , n =1 H := Poincar´ e upper half-plane 5

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 ). � az + b � = ( cz + d ) 2 f E ( z ) . f E cz + d In particular, the differential form ω f := f E ( z ) dz is defined on the quotient X := Γ 0 ( N ) \H . 6

Cycles and modularity The Riemann surface X contains many natural cycles , which convey a tremendous amount of arithmetic information about E . These cycles are indexed by the commutative subrings of M 0 ( N ): orders in Q [ ǫ ], Q × Q , or in a quadratic field. 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 . 7

The special cycles γ R ⊂ X Case 1 . Disc( R ) > 0. Then ( R ⊗ Q ) × 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 } 8

An (idealised) picture 5 −3 −4 8 −7 13 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 arith- metic of E over quadratic fields. 9

Periods of ω f : the case D > 0 Theorem (Eichler, Shimura) The set �� � Λ := ω f , R ∈ Σ > 0 ⊂ C γ R 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 , parametris- ing 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 � � ω f = Φ( γ R ) ω E f ∈ Λ E f . γ R 10

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 ≥ 1 . 3. Isogeny conjecture for curves (Faltings): E f is isogenous to E over Q . 11

Arithmetic information Conjecture (BSD) Let D > 0 be a fundamen- tal discriminant. Then √ � J D := ω f � = 0 iff # E ( Q ( D )) < ∞ . γ D “The position of γ D in the homology H 1 ( X, Z ) encodes an obstruction to the presence of ra- √ tional points on E ( Q ( D )). ” Gross-Zagier, Kolyvagin . If J D � = 0, then √ E ( Q ( D )) is finite. 12

Periods of ω f : 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 triv- ial 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. γ # ∆ := any smooth one-chain on X having γ ∆ as boundary, � P ∆ := ω f ∈ C / Λ f ≃ E ( C ) . γ ♯ ∆ 13

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 . 14

Generalisations? Principle of functoriality : modularity admits many incarnations. Simple example: quadratic base change. Choose a fixed real quadratic field F , and consider E as an elliptic curve over this field. 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 . Problem : To package these coefficients into a modular generating series. 15

Modularity Generating series � n 1 d 1 z 1 + n 2 � 2 πi d 2 z 2 � G ( z 1 , z 2 ) := a (( n )) e , n>> 0 where d := totally positive generator of the different of F . Theorem : (Doi-Naganuma, Shintani). 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 = ∈ SL 2 ( O F ) . c d 16

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 ( ǫ 1 z 1 , ǫ 2 ¯ z 2 ) dz 1 d ¯ z 2 , where ǫ ∈ O × F satisfies ǫ 1 > 0, ǫ 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 ? 17

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 ), i.e., by O F -orders in quadratic ex- tensions 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. 18

The special cycles γ R ⊂ X F Case 1 . Disc( R ) > > 0. Then, for j = 1 , 2, ( R ⊗ v j R ) × 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, for j = 1 , 2, ( R ⊗ v j R ) × has a single fixed point τ j ∈ H . γ R := { ( τ 1 , τ 2 ) } 19

The ATR case Case 3 . D 1 < 0 , D 2 > 0. Then ( R ⊗ v 1 R ) × has a unique fixed point τ 1 ∈ H . ( R ⊗ v 2 R ) × 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 . 20

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

Periods of ω G : the case D > > 0 Conjecture (Oda) The set �� � Λ G := R ∈ Σ > ω G , ⊂ C > 0 γ R 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 := ω G � = 0 iff # E ( K ) < ∞ . γ D “The position of γ D in H 2 ( X F , Z ) encodes an obstruction to the presence of rational points √ on E ( F ( D )). ” 22

Periods of ω G : 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 := ω G ∈ C / Λ G ≃ E ( C ) . γ ♯ R 23

The conjecture on ATR points Assume still that D 1 < 0, D 2 > 0. ATR points conjecture . If R ∈ Σ D , then the point P R belongs to E ( H D ) ⊗ Q , 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 algebraic points, and Euler systems attached to elliptic curves. b) “Explicit” construction of class fields. 24