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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 2 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 : pj∆ 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 = f0; 1; 2 ··· ; p − 1g. ~ Specifically, we consider the solution of E over Fp. Let ~ 2 2 3 2 #E(Fp) = 1 + #f(x; y) 2 Fp : y + y ≡ x − x (mod p)g 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 j ∆. 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 1 1 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 2 C with y > 0. Then 1 1 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 2 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: 1 P an(f ) Q −s 1−2s L(s; f ) = ns = (1 − ap(f )p + p ). n=1 p 1 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)nH as a Riemann surface. Theorem Let E be an elliptic curve over C with j(E) 2 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 jap(E)j = jp + 1 − #E(Fp)j ≤ 2p Emil Artin Helmut Hasse a (E) pp 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) pp xp = 2 p have a semi-circular distribution in [−1; 1]. Mikio Sato John Tate For −1 ≤ α < β ≤ 1, a (E) pp β #fp ≤ T : α ≤ 2 p ≤ βg 2 Z p lim = 1 − x2dx: T !1 #fp ≤ T g π α Sato-Tate Conjecture −1=2 In fact, we can write ap(E)p = 2 cos(θp), where θp 2 [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, #fp ≤ T : α ≤ θ ≤ βg 2 Z β lim p = sin2 θ dθ: T !1 #fp ≤ T g 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 1 1 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: p are equidistributed in [−1; 1]. 2 p Sato-Tate conjecture for modular forms 1 1 Y X ∆(z) = q (1−qn)24 = q−24q2+252q3−1472q4+··· = τ(n)e2πinz : n=1 n=1 11 11 Ramanujan 1917, Deligne 1971: jτ(p)j ≤ 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 P1 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 2 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 2 SL(2; R): Then for any γ 2 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? 1 P 1 L-functions ζ(s) = ns n=1 Langlands Automorphic world Geometric world program Galois modular forms, elliptic representations automorphic forms curves 1 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!.
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