Elliptic Curves and Modularity

Elliptic curves and modularity

Manami Roy

Fordham University

July 30, 2021

PRiME (Pomona Research in Mathematics Experience)

Manami Roy Elliptic curves and modularity Outline

elliptic curves

reduction of elliptic curves over ﬁnite ﬁelds

the modularity theorem with an explicit example

some applications of the modularity theorem

generalization the modularity theorem Elliptic curves Elliptic curves

Over Q, we can write an elliptic curve E as

2 3 2 E : y + a1xy + a3y = x + a2x + a4x + a6

or more commonly E : y 2 = x3 + Ax + B

3 2 where ai , A, B ∈ Q and ∆(E) = −16(4A + 27B ) 6= 0.

∆(E) is called the discriminant of E.

The conductor NE of a rational elliptic curve is a product of the form

Y fp NE = p . p|∆ Example

The elliptic curve E : y 2 = x3 − 432x + 8208

12 12 has discriminant ∆ = −2 · 3 · 11 and conductor NE = 11.

A minimal model of E is

2 3 2 Emin : y + y = x − x

with discriminant ∆min = −11. Example Elliptic curves of ﬁnite ﬁelds

2 3 2 E : y + y = x − x over F113 Elliptic curves of ﬁnite ﬁelds

Let us consider E˜ : y 2 + y = x3 − x2 ¯ ¯ ¯ over the ﬁnite ﬁeld of p elements Fp = Z/pZ = {0, 1, 2 ··· , p − 1}.

˜ Speciﬁcally, we consider the solution of E over Fp. Let

˜ 2 2 3 2 #E(Fp) = 1 + #{(x, y) ∈ Fp : y + y ≡ x − x (mod p)} and ˜ ap(E) = p + 1 − #E(Fp). Elliptic curves of ﬁnite ﬁelds

E˜ : y 2 − y = x3 − x2

˜ ˜ p #E(Fp) ap(E) = p + 1 − #E(Fp) 2 5 −2 3 5 −1 5 5 1 7 10 −2 13 10 4 ...... 47 40 8 Elliptic curves of ﬁnite ﬁelds

˜ Note that E/F11 is not an elliptic curve anymore since 11 | ∆. In this case, we get a curve of the shape:

2 3 2 For E : y − y = x − x , a11(E) = 1.

For the “bad” primes that divide ∆, one deﬁnes ap to be 0, 1, or −1, depending on the type of singularity E has when reduced mod p Some “nice” functions

∞ ∞ Y n 2 11n 2 X n f (q) = q (1 − q ) (1 − q ) = an(f )q . n=1 n=1

q − 2q2 − q3 + 2q4 + q5 + 2q6 − 2q7 − 2q9 − 2q10 + q11 − 2q12 + 4q13 + 4q14 −q15 −4q16 −2q17 +4q18 +2q20 +2q21 −2q22 −q23 −4q25 −8q26 + 5q27 −4q28 +2q30 +7q31 +8q32 −q33 +4q34 −2q35 −4q36 +3q37 −4q39 − 8q41 − 4q42 − 6q43 + 2q44 − 2q45 + 2q46 + 8q47 + 4q48 − 3q49 + O(q50).

ap(E) = ap(f ) for all primes p Some “nice” functions

2πiz Now assume q = e where z = x + iy ∈ C with y > 0. Then ∞ ∞ Y n 2 11n 2 X 2πinz f (z) = q (1 − q ) (1 − q ) = an(f )e . n=1 n=1

az + b f = (cz + d)2 f (z), a b ∈ Γ (11) = ZZ ∩ SL(2, ) cz + d c d 0 11ZZ Z This function is known as a modular form. Modularity theorem

Elliptic curve E/Q Andrew Wiles Modular form f of of conductor N Breuil, Conrad, weight 2 and level N Diamond,Taylor

ap(f ) = ap(E) for all p.

∞ P an(f ) Q −s 1−2s L(s, f ) = ns = (1 − ap(f )p + p ). n=1 p ∞ P an(E) Q −s 1−2s L(s, E) = ns = (1 − ap(E)p + p ). n=1 p

L(s, f ) = L(s, E) Little bit of history of the modularity theorem

Taniyama-Shimura-Weil conjecture

Yutaka Taniyama Goro Shimura Andr´eWeil Andrew Wiles Richard Taylor and Andrew Wiles (1995) -proof for all semistable elliptic curves. Christophe Breuil, Brian Conrad, Fred Diamond, Richard Taylor (2001) - proof for the full conjecture.

The conjecture became known as the modularity theorem. Fermat’s Last Theorem

Theorem xn + y n = zn has no positive integer solutions for n > 2.

Suppose an + bn = cn with a, b, c > 0 and n > 3 (n = 3: Euler).

Consider the elliptic curve E/Q deﬁned as y 2 = x(x − an)(x − bn)

J.P. Serre and Ken Ribet proved that E is not modular.

J.P. Serre Ken Ribet E is semistable, hence modular by Wiles (with Taylor). Other forms of modularity theorem Recall Γ (N) = ZZ ∩ SL(2, ). 0 NZZ Z

X0(N): the modular curve such that the set of its complex points is nat- ∗ urally isomorphic to the quotient Γ0(N)\H as a Riemann surface.

Theorem Let E be an elliptic curve over C with j(E) ∈ Q. Then for some positive integer N there exists a surjective holomorphic function of compact Riemann surfaces X0(N) −→ E.

Theorem Let E be an elliptic curve over Q. Then for some positive integer N there exists a surjective morphism over Q

X0(N)alg −→ E. Coming back to elliptic curves over Fp ˜ Recall that given an elliptic curve E/Q we can consider E over Fp. What ˜ can we say about #E(Fp)? The following theorem of Hasse was originally conjectured by Emil Artin. Theorem (Hesse, 1933) ˜ 1/2 |ap(E)| = |p + 1 − #E(Fp)| ≤ 2p

Emil Artin Helmut Hasse

a (E) p√ Then xp = 2 p is a real number in the interval [ − 1, 1].

Question: What is the distribution of xp as p varies? Is it uniform? Sato-Tate Conjecture

Let E/Q be an elliptic curve without complex multiplication. Then the a (E) p√ xp = 2 p have a semi-circular distribution in [−1, 1].

Mikio Sato John Tate

For −1 ≤ α < β ≤ 1,

a (E) p√ β #{p ≤ T : α ≤ 2 p ≤ β} 2 Z p lim = 1 − x2dx. T →∞ #{p ≤ T } π α Sato-Tate Conjecture

−1/2 In fact, we can write ap(E)p = 2 cos(θp), where θp ∈ [0, π].

Then θp are equidistributed (as p varies) in [0, π] according to the function 2 2 π sin (θ).

For 0 ≤ α < β ≤ π,

#{p ≤ T : α ≤ θ ≤ β} 2 Z β lim p = sin2 θ dθ. T →∞ #{p ≤ T } π α

Theorem (Taylor et al., 2008 and 2011) Sato-Tate conjecture is true if any elliptic curve without complex multiplication. Example 1

Consider the curve E : y 2 + y = x3 − x2 of conductor 11. Example 1

Figure: E : y 2 + y = x 3 − x 2 Example 2

Consider the curve E : y 2 = x3 − x of conductor 32. Another application of the modularity theorem

∞ ∞ Y n 2 11n 2 X 2πinz f (z) = q (1 − q ) (1 − q ) = an(f )e . n=1 n=1

It follows from the modularity Theorem: ap(f ) = ap(E), where E : y 2 + y = x3 − x2

ap(f ) Sato-Tate Conjecture: √ are equidistributed in [−1, 1]. 2 p Sato-Tate conjecture for modular forms

∞ ∞ Y X ∆(z) = q (1−qn)24 = q−24q2+252q3−1472q4+··· = τ(n)e2πinz . n=1 n=1

11 11 Ramanujan 1917, Deligne 1971: |τ(p)| ≤ 2p 2 ⇒ τ(p) = 2p 2 cos(θp).

Srinivasa Ramanujan Pierre Deligne

Sato-Tate Conjecture: θp are equidistributed (as p varies) in [0, π] according to the function τ(p) 2 sin2(θ). In other words are equidistributed in [−2, 2]. π p11/2 Equidistribution of eigenvalues of modular forms

P∞ 2πinz f : a modular form with level N and a weight k; f (z) = n=1 an(f )e .

Sk (N): the vector space of these functions.

Consider some linear operators Tp on Sk (N): Tpf = ap(f )f . a (f ) The values p (as the prime p varies) are equidistributed on [−2, 2]. p(k−1)/2 Automorphic forms

Recall that a modular form f : H → C satisﬁes az + b f = (cz + d)k f (z), a b ∈ Γ (N) = ZZ ∩ SL(2, ). cz + d c d 0 NZZ Z

Consider a function Φf : SL(2, R) → C such that

−k a b Φf (g) = (ci + d) f (g · i), g = c d ∈ SL(2, R).

Then for any γ ∈ Γ0(N),

Φf (γg) = Φf (g).

This is called an automorphic form. Question

Elliptic curves Automorphic forms modular forms on groups like SL(2)

Equidistribution results Are there equidistribution respect to results for certain measure automorphic forms? ∞ P 1 L-functions ζ(s) = ns n=1

Langlands Automorphic world Geometric world program

Galois modular forms, elliptic representations automorphic forms curves ∞ P 1 L-functions ζ(s) = ns n=1

Langlands Automorphic world Geometric world program

Galois modular forms, elliptic representations automorphic forms curves L-functions

Langlands Automorphic world Geometric world program modular forms, Galois elliptic automorphic representations curves forms

Thank you for your attention!