NUMBERS. Outline Ching-Li Chai

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1 Institute of Mathematics Academia Sinica and Deartment of Mathematics University of Pennsylvania National Chiao Tung University, July 6, 2012 Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication Outline 1 Samle arithmetic Diohantine equations diohantine equation rime numbers 2 Samle of geometric Ellitic curve basics Comlex multilication Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication

2 The general theme Geometry and symmetry influences arithmetic through zeta functions and modular forms Remark. (i) zeta functions = L-functions; modular forms = automorhic reresentations. (ii) There are two kinds of L-functions, from harmonic analysis and arithmetic resectively. Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication Fermat s infinite descent I. Samle arithmetic questions and results 1. Diohantine equations Examle. Fermat roved (by his infinite descent) that the diohantine equation x 4 y 4 = z 2 does not have any non-trivial integer solution. Remark. (i) The above equation can be rojectivized to x 4 y 4 = x 2 z 2, which gives an ellitic curve E with comlex multilication by Z[ 1]. Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication

Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication Figure: Fermat Fermat s infinite descent continued (ii) Idea: Show

3 Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication Figure: Fermat Fermat s infinite descent continued (ii) Idea: Show that every non-trivial rational oint P E(Q) is the image [2] E of another smaller rational oint. (Construct another rational variety X and mas f : E X and g : X E such that g f = [2] E and descent in two stages. Here X is a twist of E, and f,g corresonds to [1 + 1] and [1 1] resectively.) Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication

4 Interlude: Euler s addition formula In 1751, Fagnano s collection of aers Produzioni Mathematiche reached the Berlin Academy. Euler was asked to examine the book and draft a letter to thank Count Fagnano. Soon Euler discovered the addition formula r dρ u = dη v + dψ, 0 1 ρ η ψ 4 where r = u 1 v 4 + v 1 u u 2 v 2. Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication Figure: Euler

5 Counting sums of squares 2. diohantine equation Examle. Counting sums of squares. For n,k N, let r k(n) := #{(x 1,...,x k) Z n : x x 2 k = n} be the number of ways to reresent n as a sum of k squares. { (i) r 2(n) = 4 ( 1) (d 1)/2 0 if n2 = d n, n odd d n1 1 if n 2 = where n = 2 f n 1 n 2, and every rime divisor of n 1 (res. n 2) is 1 (mod 4) (res. 3 (mod 4)). Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication { 8 d n d if n is odd (ii) r 4(n) = 24 d n,d odd d if n is even How to count number of sum of squares Method. Exlicitly identify the theta series θ k (τ) = ( q m2) k m N where q = e 2π 1τ with modular forms obtained in a different way, such as Eisenstein series. Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication

6 3. and L-functions (a) Count the number of congruence solutions of a given diohantine equation modulo a (fixed) rime number (b) Identify the L-function for a given diohantine equation (basically the generating function for the number of congruence solutions modulo as varies) with an L-function coming from harmonic analysis. (The latter is associated to a modular form). Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication Remark. (b) is an essential asect of the Langlands rogram. The Riemann zeta function 4. L-functions and the the distribution of rime numbers for a given diohantine roblem Examles. (i) The Riemann zeta function ζ (s) is a meromorhic function on C with only a simle ole at s = 0, ζ (s) = n s = (1 s ) 1 for Re(s) > 1, n 1 such that the function ξ (s) = π s/2 Γ(s/2) ζ (s) satisfies ξ (1 s) = ξ (s). Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication

7 Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication Figure: Riemann Dirichlet L-functions (ii) Similar roerties hold for the Dirichlet L-function L(χ,s) = χ(n) n s Re(s) > 1 n N,(n,N)=1 for a rimitive Dirichlet character χ : (Z/NZ) C. Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication

8 Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication Figure: Dirichlet rime numbers (a) Dirichlet s theorem for rimes in arithmetic rogression L(χ,1) 0 Dirichlet character χ. (b) The rime number theorem zero free region of ζ (s) near {Re(s) = 1}. (c) Riemann s hyothesis the first term after the main term in the asymtotic exansion of ζ (s). Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication

9 Bernoulli numbers and zeta values 5. Secial values of L-functions Examles. (a) zeta and L-values for Q. Recall that the Bernoulli numbers B n are defined by x e x 1 = B n n N n! xn B 0 = 1, B 1 = 1/2, B 2 = 1/6, B 4 = 1/30, B 6 = 1/42, B 8 = 1/30, B 10 = 5/66, B 12 = 691/2730. (i) (Euler) ζ (1 k) = B k/k even integer k > 0. Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication (ii) (Leibniz s formula, 1678; Madhava, 1400) = π 4 Content of L-values (b) L-values often contain dee arithmetic/geometric information. (i) Leibniz s formula: Z[ 1] is a PID (because the formula imlies that the class number h(q( 1)) is 1). (ii) B k/k aears in the formula for the number of (isomorhism classes of) exotic (4k 1)-sheres. Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication

Kummer congruence (c) (Kummer congruence) (i) ζ (m) Z for m 0 with m 1 (mod 1) (ii) ζ (m) ζ (m ) (mod ) for all m,m 0 with m m 1 (mod 1). Examles. ζ ( 1) = 1 2 2 ; 1 1 (mod 1) only for = 2,3.

10 Kummer congruence (c) (Kummer congruence) (i) ζ (m) Z for m 0 with m 1 (mod 1) (ii) ζ (m) ζ (m ) (mod ) for all m,m 0 with m m 1 (mod 1). Examles. ζ ( 1) = ; 1 1 (mod 1) only for = 2, ζ ( 11) = ; 11 1 (mod 1) holds only for = 2,3,5,7,13. 1 ζ ( 5) = ζ ( 1) (mod 5). 7 Note that (mod 5). Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication Figure: Kummer

11 Ellitic curves basics II. Samle of geometric 1. Review of ellitic curves Equivalent definitions of an ellitic curve E : a rojective curve with an algebraic grou law; a rojective curve of genus one together with a rational oint (= the origin); over C: a comlex torus of the form E τ = C/Zτ + Z, where τ H := uer-half lane; over a field F with 6 F : given by an affine equation Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication y 2 = 4x 3 g 2 x g 3, g 2,g 3 F. Weistrass theory For E τ = C/Zτ + Z, let x τ(z) = (τ,z) y τ(z) = d dz (τ,z) = 1 z 2 + (m,n) (0,0) Then E τ satisfies the Weistrass equation ( 1 (z mτ n) 2 1 (mτ + n) 2 y 2 τ = 4x 3 τ g 2(τ)x τ g 3(τ) with 1 g 2(τ) = 60 (mτ + n) (0,0) (m,n) Z g 3(τ) = 140 (mτ + n) (0,0) (m,n) Z 6 2 ) Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication

12 The j-invariant Ellitic curves are classified by their j-invariant g 3 2 j = 1728 g g2 3 Over C, j(e τ) deends only on the lattice Zτ + Z of E τ. So j(τ) is a modular function for SL 2(Z): ( ) aτ + b j = j(τ) cτ + d for all a,b,c,d Z with ad bc = 1. We have a Fourier exansion Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication j(τ) = 1 q q q2 +, where q = q τ = e 2π 1τ. curves and Hecke symmetry 2. Modular forms and modular curves Let Γ SL 2(Z) be a congruence subgrou of SL 2(Z), i.e. Γ contains all elements which are I 2 (mod N) for some N. (a) A holomorhic function f (τ) on the uer half lane H is said to be a modular form of weight k and level Γ if f ((aτ + b)(cτ + d) 1 ) = (cτ + d) k ( ) a b f (τ) γ = Γ c d and has moderate growth at all cuss. (b) The quotient Y Γ := Γ\H has a natural structure as an (oen) algebraic curve, definable over a natural number field; it arametrizes ellitic curves with suitable level structure. Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication

$det>0 on H survives on the modular curve Y Γ = Γ\H and takes a reincarnated form as a family of algebraic corresondences.$

13 Modular (c) Modular forms of weight k for Γ = H 0 (X Γ,ω k ), where X Γ is the natural comactification of Y Γ, and ω is the Hodge line bundle on X Γ ω [E] = Lie(E) [E] X Γ (d) The action of GL 2(Q) det>0 on H survives on the modular curve Y Γ = Γ\H and takes a reincarnated form as a family of algebraic corresondences. The L-function attached to a cus form which is a common eigenvector of all Hecke corresondences admits an Euler roduct. Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication Figure: Hecke

14 The Ramanujan τ function Examle. Weight 12 cus forms for SL 2(Z) are constant multiles of and = q (1 q m ) 24 = τ(n)q n m 1 n T ( ) = τ(), where T is the Hecke oerator reresented by Let L(,s) = n 1 a n n s. We have L(,s) = (1 τ() s s ) 1. ( ) Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication CM ellitic curves 3. Comlex multilication An ellitic E over C is said to have comlex multilication if its endomorhism algebra End 0 (E) is an imaginary quadratic field. Examle. Consequences of j(c/o K) is an algebraic integer K j(c/o K) = the Hilbert class field of K. e π 67 = ( ) j = = Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication e π 163 = ( ) j = =

15 Mod oints for a CM curve A tyical feature of CM ellitic curves is that there are exlicit formulas: Let E be the ellitic curve y 2 = x 3 + x, which has CM by Z[ 1]. We have #E(F ) = 1 + a and for odd we have ( u a = 3 ) + u u F { 0 if 3 (mod 4) = 2a if = a 2 + 4b 2 with a 1 (mod 4) Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication A CM curve and its associated modular form, continued The L-function L(E,s) attached to E with (1 a s + 1 2s ) 1 = a n n s odd n is equal to a Hecke L-function L(ψ,s), where the Hecke character ψ is the given by { 0 if 2 N(a) ψ(a) = λ if a = (λ), λ 1 + 4Z + 2Z 1 The function f E(τ) = n a n q n is a modular form of weight 2 and level 4, and f E(τ) = a ψ(a) q N(a) = a 1 (mod 4) b 0 (mod 2) a q a2 +b 2 Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication

16 4. Every algebraic variety X over a finite field F q has a ma Fr q : X X, induced by the ring endomorhism f f q of the function field of X. Deligne s roof of Weil s conjecture imlies that τ() 2 11/2 Idea: Ste 1. Use Hecke symmetry to cut out a 2-dimensional Galois reresentation inside H 1 et ( X,Sym 10 (H(E /X)) ), which contains the cus form via the Eichler-Shimura integral. Ste 2. Aly the Eichler-Shimura congruence relation, which relates Fr and the Hecke corresondence T ; invoke the Weil bound. Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication A hyergeometric differential equation 5. (a) The hyergeometric differential equation has a classical solution 4x(1 x) d2 y dy + 4(1 2x) dx2 dx y = 0 F(1/2,1/2,1,x) = ( 1/2 n )x n n 0 The global monodromy grou of the above differential is the rincial congruence subgrou Γ(2). Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication

17 Historic origin Remark. The word monodromy means run around singly ; it was (?first) used by Riemann in Beiträge zur Theorie der durch die Gauss sche Reihe F(α,β,γ,x) darstellbaren Functionen, ; für einen Werth in welchem keine Verzweigung statfindet, heist die Function einändrig order monodrom... Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication The Legendre family of ellitic curves The family of equations y 2 = x(x 1)(x λ) 0,1, λ P 1 defines a family π : E S = P 1 {0,1, } of ellitic curves, with j(e λ ) = 28 [1 λ(1 λ)] 3 λ 2 (1 λ) 2 This formula exhibits the λ-line as an S 3-cover of the j-line, such that the 6 conjugates of λ are λ, 1 λ, 1 λ, 1 1 λ, λ λ 1, λ 1 λ. Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication

18 The Legendre family, continued The formula [ 4λ (1 λ) d dλ 2 + 4(1 2λ) d dλ 1 ] ( dx y ) ( = d y (x λ) 2 means that the global section [dx/y] of H 1 dr (E /S) satisfies the above hyergeometric ODE. ) Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication and symmetry 1. can be regarded as attainable among otential. 2. To say that the monodromy is as large as ossible is an irreducibility statement. 3. Maximality of monodromy has imortant consequences. E.g. the key geometric inut in Deligne-Ribet s roof of -adic interolation for secial values of Hecke L-functions attached to totally real fields. Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication

19 Suersingular ellitic curves 6. Fine structure in char. > 0 Examle. (ordinary/suersingular dichotomy) Ellitic curves over an algebraically closed field k F come in two flavors. Those with E(k) (0) are called suersingular. There is only a finite number of suersingular j-values. An ellitic curve E over a finite field F q is suersingular if and only if E(F q) 0 (mod ). Those with E[](k) Z/Z are said to be ordinary. An ellitic curve E over a finite field F q is suersingular if and only if E(F q) 0 (mod ) Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication The Hasse invariant For the Legendre family, the suersingular locus (for > 2) is the zero locus of A(λ) = ( 1) ( 1)/2 ( 1)/2 j=0 where (c) m := c(c + 1) (c + m 1). ( ) 2 (1/2)j λ j Remark. The above formula for the coefficients a j satisfy and a 1,...,a ( 1)/2 Z () a (+1)/2 a 1 0 (mod ). j! Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication

20 Counting suersingular j-values Theorem. (Eichler 1938) The number h of suersingular j-values is /12 if 1 (mod 12) h = /12 if 5 or 7 (mod 12) /12 +1 if 11 (mod 12) Remark. (i) It is known that h is the class number for the quaternion division algebra over Q ramified (exactly) at and. (ii) Deuring thought that it is nicht leicht that the above class number formula can be obtained by counting suersingular j-invariants directly. Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication Igusa s roof From the hyergeometric equation for F(1/2,1/2,1,x) we conclude that ] [4λ (1 λ) d2 d + 4(1 2λ) dλ 2 dλ 1 A(λ) 0 (mod ) for all > 3. It follows immediately that A(λ) has simle zeroes. The formula for h is now an easy consequence. (Hint: Use the formula 6-to-1 cover of the j-line by the λ-line.) Q.E.D. Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication

21 -adic monodromy for modular curves For the ordinary locus of the Legendre family π : E ord S ord the monodromy reresentation ( ρ : π 1 S ord ) Aut ( E ord [ ](F ) ) = Z (defined by Galois theory) is surjective. Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication -adic monodromy for the modular curve Sketch of a roof: Given any n > 0 and any ū (Z/ n Z), ick a reresentative u N of ū with 0 < u < n and let let ι : Q[T]/(T 2 u T + 4n ) Q be the embedding such that ι(t) Z. Then ι(t) u (mod 2n ). By a result of Deuring, there exists an ellitic curve E over F 2n whose Frobenius is the Weil number ι(t). So the image of the monodromy reresentation contains ι(t), which is congruent to the given element ū Z/ n Z. Q.E.D. Samle arithmetic Diohantine equations diohantine equation rime numbers Samle of geometric Ellitic curve basics Comlex multilication

FERMAT S WORLD A TOUR OF. Outline. Ching-Li Chai. Philadelphia, March, Samples of numbers. 2 More samples in arithemetic. 3 Congruent numbers

FERMAT S WORLD A TOUR OF. Outline. Ching-Li Chai. Philadelphia, March, Samples of numbers. 2 More samples in arithemetic. 3 Congruent numbers Department of Mathematics University of Pennsylvania Philadelphia, March, 2016 Outline 1 2 3 4 5 6 7 8 9 Some familiar whole numbers 1. Examples of numbers 2, the only even prime number. 30, the largest