Lecture 13, part II: Modularity of elliptic curves and Fermats Last - PowerPoint PPT Presentation

Lecture 13, part II: Modularity of elliptic curves and Fermat’s Last Theorem (an overview) June 9, 2020 1 / 10

Local zeta functions X algebraic variety { F p , N m : “ # X p F p m q m “ 1 , 2 , . . . ˜ 8 ¸ T m ÿ Z X p T q : “ exp “ 1 ` N 1 T ` . . . P Q rr T ss N m m m “ 1 Theorem (Dwork). Z X p T q P Q p T q 4 a 3 ` 27 b 2 ‰ 0 Example: a , b P F p , E a , b : Y 2 Z “ X 3 ` aXZ 2 ` bZ 3 p m : y 2 “ x 3 ` ax ` b u N m “ # E a , b p F p m q “ 1 ` # tp x , y q P F 2 1 ´ λ T ` pT 2 Z E a , b p T q “ p 1 ´ T qp 1 ´ pT q “ 1 ` p 1 ` p ´ λ q T ` . . . λ “ p ` 1 ´ N 1 determines all N m : N m : “ 1 ` p m ´ N m N m ` 1 “ λ ˜ ˜ N m ´ p ˜ where ˜ N m ´ 1 , 2 / 10

Global (Hasse–Weil) zeta functions ź Z X { F p p p ´ s q X algebraic variety { Z ζ X p s q : “ ù p prime Example: X “ one point T m ÿ 1 Z X { F p p T q “ exp p m q “ 1 ´ T m ě 1 1 ´ ¯ ź 1 ź 1 ` 1 1 ÿ ζ X p s q “ 1 ´ p ´ s “ p s ` p 2 s ` . . . “ n s p prime p prime n ě 1 Conjecture. ζ X p s q can be analytically continued to a meromorphic function of s in the whole C This is known only for very special classes of varieties. 3 / 10

Hasse–Weil L-functions of elliptic curves E : Y 2 Z “ X 3 ` aXZ 2 ` bZ 3 a , b P Z ∆ “ ´ 4 a 3 ´ 27 b 2 ‰ 0 p ∤ ∆ : good reduction 1 ´ λ p T ` pT 2 Z E { F p p T q “ p 1 ´ T qp 1 ´ pT q , λ p “ p “ 1 ´ # E p F p q p | ∆ : bad reduction 1 1 ` T 1 Z E { F p p T q “ p 1 ´ T qp 1 ´ pT q or p 1 ´ T qp 1 ´ pT q or p 1 ´ pT q resp. additive / non-split / split multiplicative ζ E p s q “ ζ p s q ζ p s ´ 1 q 1 ź ź , L E p s q “ 1 ´ λ p p ´ s ` p 1 ´ 2 s ˆ ... L E p s q p ∤ ∆ p | ∆ Proposition. L E p s q is convergent for Re p s q ą 3 2 . 4 / 10

Compare: L-series of Hecke eigenforms n ě 1 λ n q n P S k , a normalized Hecke eigenform Recall: f “ ř λ n 1 ÿ ź L p f , s q “ n s “ 1 ´ λ p p ´ s ` p k ´ 2 s ´ 1 n ě 1 p prime (see Lecture 13, Proposition 5). For modular forms of higher level f P S k p Γ 0 p N qq the terms with p | N in the Euler product have different shape (there is an action of Hecke operators T n with n ∤ N ). In 1950’s Y. Taniyama notices similiarity of the above (with k “ 2) and the Hasse–Weil L-functions of elliptic curves: 1 ź ź L E p s q “ 1 ´ λ p p ´ s ` p 1 ´ 2 s ˆ ... p ∤ ∆ p | ∆ Definition. E is modular if there is N ą 1 and a normalized Hecke eigenform f P S 2 p Γ 0 p N qq such that L E p s q “ L p f , s q . 5 / 10

Eichler–Shimura Theorem (1960’s) Theorem. Let N ą 1 and f P S 2 p Γ 0 p N qq be a normalized Hecke eigenform with integral Fourier coefficients. Then there exists E { Z such that L E p s q “ L p f , s q . n “ 1 λ n q n “ ř 8 Sketch of proof. f p z q “ ř 8 n “ 1 λ n e 2 π inz 8 d φ p z q λ n ÿ n e 2 π inz φ p z q : “ “ 2 π i f p z q dz n “ 1 ñ d ´ ¯ f p γ z q dz p φ p γ z q ´ φ p z qq “ 2 π i p cz ` d q 2 ´ f p z q “ 0 φ p γ z q ´ φ p z q ” C p γ q P C Λ : “ t C p γ q : γ P Γ 0 p N qu Ă C lattice φ ñ φ : H Ñ C Γ 0 p N q z H Ñ C { Λ “ : E ù Difficult part: E is defined over Q . There exist Γ 0 p N q -invariant functions x p z q , y p z q with Fourier coefficients in Q satisfying y p z q 2 “ x p z q 3 ` ax p z q ` b and dx p z q 2 y p z q “ 2 π i f p z q dz . 6 / 10

Is the converse true? Yes! (1990’s) E elliptic curve / Q ù minimal Weierstrass equation { Z (minimzing | ∆ | ) p | ∆ min p e p , Definition. Conductor N “ N p E q “ ś e p “ 1 when E has multiplicative reduction at p e p “ 2 when E has additive reduction and p ‰ 2 , 3 p 2 ď e 2 ď 8, 2 ď e 3 ď 5 are special) Theorem (Wiles–Taylor, Breuil–Conrad–Diamond–Taylor) Let E { Q be an elliptic curve of conductor N . Then L E p s q “ L p f , s q for a Hecke eigenform f P S 2 p Γ 0 p N qq . 7 / 10

Example E : y 2 ´ y “ x 3 ´ x 2 N “ 11 p y “ 216 Y ´ 108 , x “ 36 X ´ 12 Y 2 “ X 3 ´ 432 X ` 8208 , ∆ “ ´ 2 8 ¨ 3 12 ¨ 11 q y 0 1 x 0 1 p “ 2 y 2 ´ y x 3 ´ x 0 0 0 0 # tp x , y q P F 2 : y 2 ´ y “ x 3 ´ x u “ 4 λ 2 “ 2 ´ 4 “ ´ 2 y 0 1 2 x 0 1 2 p “ 3 y 2 ´ y x 3 ´ x 0 0 2 0 0 1 # tp x , y q P F 3 : y 2 ´ y “ x 3 ´ x u “ 4 λ 3 “ 3 ´ 4 “ ´ 1 p “ 5 λ 5 “ 5 ´ 4 “ 1 1 1 1 L E p s q “ 1 ` 2 ¨ 2 ´ s ` 2 ¨ 2 ´ 2 s ¨ 1 ` 3 ´ s ` 3 ¨ 3 ´ 2 s ¨ 1 ´ 5 ´ s ` 5 ¨ 5 ´ 2 s ¨ . . . 1 s ´ 2 1 2 s ´ 1 3 s ` 2 4 s ` 1 “ 5 s ` . . . 8 / 10

Example ( N “ 11) E : y 2 ´ y “ x 3 ´ x 2 1 1 1 L E p s q “ 1 ` 2 ¨ 2 ´ s ` 2 ¨ 2 ´ 2 s ¨ 1 ` 3 ´ s ` 3 ¨ 3 ´ 2 s ¨ 1 ´ 5 ´ s ` 5 ¨ 5 ´ 2 s ¨ . . . 1 s ´ 2 1 2 s ´ 1 3 s ` 2 4 s ` 1 “ 5 s ` . . . 8 p 1 ´ q n q 2 p 1 ´ q 11 n q 2 “ q ´ 2 q 2 ´ q 3 ` 2 q 4 ` q 5 ` . . . ź f p z q “ q n “ 1 P S 2 p Γ 0 p 11 qq dim S 2 p Γ 0 p 11 qq “ 1 “ genus of X p Γ 0 p 11 qq “ : X 0 p 11 q X 0 p 11 q – E 9 / 10

Fermat’s Last Theorem For p ą 2 equation A p ` B p “ C p has no solutions p A , B , C q P Z 3 with A ¨ B ¨ C ‰ 0. Y. Hellegouarch (1970’s), G. Frey (1980’s) ù E : y 2 “ x p x ´ A p qp x ` B p q ∆ “ A 2 p B 2 p C 2 p ‰ 0 1985: J.-P. Serre (almost) shows that Taniyama–Shimura–Weil (modularity) conjecture implies FLT 1986: K. Ribet fills the missing part in Serre’s proof ( ε -conjecture) one of A , B , C is even ñ 2 | N suppose E is modular, f P S 2 p Γ 0 p N qq Ribet’s descent: f ù g P S 2 p Γ 0 p 2 qq , f ” g mod p but S 2 p Γ 0 p 2 qq “ t 0 u , a contradiction 10 / 10