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

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Local zeta functions

X algebraic variety {Fp, Nm :“ #XpFpmq m “ 1, 2, . . . ZXpTq :“ exp ˜ 8 ÿ

m“1

Nm T m m ¸ “ 1 ` N1T ` . . . P QrrTss Theorem (Dwork). ZXpTq P QpTq Example: a, b P Fp, 4a3 ` 27b2 ‰ 0 Ea,b : Y 2Z “ X 3 ` aXZ 2 ` bZ 3 Nm “ #Ea,bpFpmq “ 1 ` #tpx, yq P F2

pm : y2 “ x3 ` ax ` bu

ZEa,bpTq “ 1 ´ λT ` pT 2 p1 ´ Tqp1 ´ pTq “ 1 ` p1 ` p ´ λqT ` . . . λ “ p ` 1 ´ N1 determines all Nm : ˜ Nm`1 “ λ ˜ Nm ´ p ˜ Nm´1, where ˜ Nm :“ 1 ` pm ´ Nm

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Global (Hasse–Weil) zeta functions

X algebraic variety {Z ù ζXpsq :“ ź

p prime

ZX{Fppp´sq Example: X “ one point ZX{FppTq “ expp ÿ

mě1 T m m q “ 1 1´T

ζXpsq “ ź

p prime 1 1´p´s “

ź

p prime

´ 1 ` 1

ps ` 1 p2s ` . . .

¯ “ ÿ

ně1

1 ns

• Conjecture. ζXpsq can be analytically continued to a

meromorphic function of s in the whole C This is known only for very special classes of varieties.

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Hasse–Weil L-functions of elliptic curves

E : Y 2Z “ X 3 ` aXZ 2 ` bZ 3 a, b P Z ∆ “ ´4a3 ´ 27b2 ‰ 0 p ∤ ∆ : good reduction ZE{FppTq “

1´λpT`pT 2 p1´Tqp1´pTq,

λp “ p “ 1 ´ #EpFpq p|∆ : bad reduction ZE{FppTq “

1 p1´Tqp1´pTq or 1`T p1´Tqp1´pTq or 1 p1´pTq

• resp. additive / non-split / split multiplicative

ζEpsq “ ζpsqζps ´ 1q LEpsq , LEpsq “ ź

p∤∆

1 1 ´ λpp´s ` p1´2s ˆ ź

p|∆

...

• Proposition. LEpsq is convergent for Repsq ą 3

2.

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Compare: L-series of Hecke eigenforms

Recall: f “ ř

ně1 λnqn P Sk, a normalized Hecke eigenform

Lpf , sq “ ÿ

ně1

λn ns “ ź

p prime

1 1 ´ λpp´s ` pk´2s´1 (see Lecture 13, Proposition 5). For modular forms of higher level f P SkpΓ0pNqq the terms with p|N in the Euler product have different shape (there is an action of Hecke operators Tn 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: LEpsq “ ź

p∤∆

1 1 ´ λpp´s ` p1´2s ˆ ź

p|∆

...

• Definition. E is modular if there is N ą 1 and a normalized Hecke

eigenform f P S2pΓ0pNqq such that LEpsq “ Lpf , sq.

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Eichler–Shimura Theorem (1960’s)

• Theorem. Let N ą 1 and f P S2pΓ0pNqq be a normalized Hecke

eigenform with integral Fourier coefficients. Then there exists E{Z such that LEpsq “ Lpf , sq. Sketch of proof. f pzq “ ř8

n“1 λnqn “ ř8 n“1 λne2πinz

φpzq :“

8

ÿ

n“1

λn n e2πinz dφpzq dz “ 2πi f pzq ñ d dz pφpγzq ´ φpzqq “ 2πi ´

f pγzq pcz`dq2 ´ f pzq

¯ “ 0 φpγzq ´ φpzq ” Cpγq P C Λ :“ tCpγq : γ P Γ0pNqu Ă C lattice ñ φ : H Ñ C ù

Γ0pNqzH φ

Ñ C{Λ “: E Difficult part: E is defined over Q. There exist Γ0pNq-invariant functions xpzq, ypzq with Fourier coefficients in Q satisfying ypzq2 “ xpzq3 ` axpzq ` b and dxpzq

2ypzq “ 2πi f pzqdz.

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Is the converse true? Yes! (1990’s)

E elliptic curve / Q ù minimal Weierstrass equation {Z (minimzing |∆|)

• Definition. Conductor N “ NpEq “ ś

p|∆min pep,

ep “ 1 when E has multiplicative reduction at p ep “ 2 when E has additive reduction and p ‰ 2, 3 p2 ď e2 ď 8, 2 ď e3 ď 5 are special) Theorem (Wiles–Taylor, Breuil–Conrad–Diamond–Taylor) Let E{Q be an elliptic curve of conductor N. Then LEpsq “ Lpf , sq for a Hecke eigenform f P S2pΓ0pNqq.

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Example

E : y2 ´ y “ x3 ´ x2 N “ 11 p y “ 216Y ´ 108, x “ 36X ´ 12 Y 2 “ X 3 ´ 432X ` 8208, ∆ “ ´28 ¨ 312 ¨ 11 q

p “ 2 y 1 y 2 ´ y x 1 x3 ´ x #tpx, yq P F2 : y 2 ´ y “ x3 ´ xu “ 4 λ2 “ 2 ´ 4 “ ´2 p “ 3 y 1 2 y 2 ´ y 2 x 1 2 x3 ´ x 1 #tpx, yq P F3 : y 2 ´ y “ x3 ´ xu “ 4 λ3 “ 3 ´ 4 “ ´1 p “ 5 λ5 “ 5 ´ 4 “ 1 LEpsq “ 1 1 ` 2 ¨ 2´s ` 2 ¨ 2´2s ¨ 1 1 ` 3´s ` 3 ¨ 3´2s ¨ 1 1 ´ 5´s ` 5 ¨ 5´2s ¨ . . . “ 1 1s ´ 2 2s ´ 1 3s ` 2 4s ` 1 5s ` . . .

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Example (N “ 11)

E : y2 ´ y “ x3 ´ x2

LEpsq “ 1 1 ` 2 ¨ 2´s ` 2 ¨ 2´2s ¨ 1 1 ` 3´s ` 3 ¨ 3´2s ¨ 1 1 ´ 5´s ` 5 ¨ 5´2s ¨ . . . “ 1 1s ´ 2 2s ´ 1 3s ` 2 4s ` 1 5s ` . . . f pzq “ q

8

ź

n“1

p1 ´ qnq2p1 ´ q11nq2 “ q ´ 2q2 ´ q3 ` 2q4 ` q5 ` . . . P S2pΓ0p11qq dim S2pΓ0p11qq “ 1 “ genus of XpΓ0p11qq “: X0p11q X0p11q – E

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Fermat’s Last Theorem

For p ą 2 equation Ap ` Bp “ C p has no solutions pA, B, Cq P Z3 with A ¨ B ¨ C ‰ 0.

• Y. Hellegouarch (1970’s), G. Frey (1980’s) ù

E : y 2 “ xpx ´ Apqpx ` Bpq ∆ “ A2pB2pC 2p ‰ 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)

• ne of A, B, C is even ñ 2|N

suppose E is modular, f P S2pΓ0pNqq Ribet’s descent: f ù g P S2pΓ0p2qq, f ” g mod p but S2pΓ0p2qq “ t0u, a contradiction

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