GEOMETRY AND NUMBERS Ching-Li Chai Sample arithmetic statements Diophantine equations Counting solutions of a GEOMETRY AND NUMBERS diophantine equation Counting congruence solutions L-functions and distribution of prime numbers Ching-Li Chai Zeta and L-values Sample of geometric structures and symmetries Institute of Mathematics Elliptic curve basics Academia Sinica Modular forms, modular and curves and Hecke symmetry Complex multiplication Department of Mathematics Frobenius symmetry University of Pennsylvania Monodromy Fine structure in characteristic p National Chiao Tung University, July 6, 2012 GEOMETRY AND Outline NUMBERS Ching-Li Chai Sample arithmetic 1 Sample arithmetic statements statements Diophantine equations Diophantine equations Counting solutions of a diophantine equation Counting solutions of a diophantine equation Counting congruence solutions L-functions and distribution of Counting congruence solutions prime numbers L-functions and distribution of prime numbers Zeta and L-values Sample of geometric Zeta and L-values structures and symmetries Elliptic curve basics Modular forms, modular 2 Sample of geometric structures and symmetries curves and Hecke symmetry Complex multiplication Elliptic curve basics Frobenius symmetry Monodromy Modular forms, modular curves and Hecke symmetry Fine structure in characteristic p Complex multiplication Frobenius symmetry Monodromy Fine structure in characteristic p GEOMETRY AND The general theme NUMBERS Ching-Li Chai Sample arithmetic statements Diophantine equations Counting solutions of a Geometry and symmetry influences diophantine equation Counting congruence solutions L-functions and distribution of arithmetic through zeta functions and prime numbers Zeta and L-values modular forms Sample of geometric structures and symmetries Elliptic curve basics Modular forms, modular Remark. (i) zeta functions = L-functions; curves and Hecke symmetry Complex multiplication modular forms = automorphic representations. Frobenius symmetry Monodromy Fine structure in characteristic (ii) There are two kinds of L-functions, from harmonic analysis p and arithmetic respectively. GEOMETRY AND Fermat’s infinite descent NUMBERS Ching-Li Chai Sample arithmetic statements I. Sample arithmetic questions and results Diophantine equations Counting solutions of a diophantine equation Counting congruence solutions 1. Diophantine equations L-functions and distribution of prime numbers Example. Fermat proved (by his infinite descent) that the Zeta and L-values Sample of geometric diophantine equation structures and symmetries Elliptic curve basics 4 4 2 Modular forms, modular x − y = z curves and Hecke symmetry Complex multiplication Frobenius symmetry does not have any non-trivial integer solution. Monodromy Fine structure in characteristic p Remark. (i) The above equation can be “projectivized” to 4 4 2 2 x − y = x z , which gives anp elliptic curve E with complex multiplication by Z[ −1]. GEOMETRY AND NUMBERS Ching-Li Chai Sample arithmetic statements Diophantine equations Counting solutions of a diophantine equation Counting congruence solutions L-functions and distribution of prime numbers Zeta and L-values Sample of geometric structures and symmetries Elliptic curve basics Modular forms, modular curves and Hecke symmetry Complex multiplication Frobenius symmetry Monodromy Fine structure in characteristic p Figure: Fermat GEOMETRY AND Fermat’s infinite descent continued NUMBERS Ching-Li Chai Sample arithmetic statements Diophantine equations Counting solutions of a diophantine equation Counting congruence solutions L-functions and distribution of (ii) Idea: Show that every non-trivial rational point P 2 E( ) is prime numbers Q Zeta and L-values [ ] the image 2 E of another “smaller” rational point. Sample of geometric structures and (Construct another rational variety X and maps f : E ! X and symmetries Elliptic curve basics g : X ! E such that g ◦ f = [2]E and descent in two stages. Here Modular forms, modular p curves and Hecke symmetry X is a twist of E, and f ;g corresponds to [1 + −1] and Complex multiplication p Frobenius symmetry [1 − −1] respectively.) Monodromy Fine structure in characteristic p GEOMETRY AND Interlude: Euler’s addition formula NUMBERS Ching-Li Chai Sample arithmetic statements Diophantine equations Counting solutions of a In 1751, Fagnano’s collection of papers Produzioni diophantine equation Counting congruence solutions Mathematiche reached the Berlin Academy. Euler was asked to L-functions and distribution of prime numbers examine the book and draft a letter to thank Count Fagnano. Zeta and L-values Soon Euler discovered the addition formula Sample of geometric structures and Z r dr Z u dh Z v dy symmetries = + ; Elliptic curve basics p p p Modular forms, modular 0 1 − r4 0 1 − h4 0 1 − y4 curves and Hecke symmetry Complex multiplication Frobenius symmetry where Monodromy p p Fine structure in characteristic u 1 − v4 + v 1 − u4 p r = : 1 + u2v2 GEOMETRY AND NUMBERS Ching-Li Chai Sample arithmetic statements Diophantine equations Counting solutions of a diophantine equation Counting congruence solutions L-functions and distribution of prime numbers Zeta and L-values Sample of geometric structures and symmetries Elliptic curve basics Modular forms, modular curves and Hecke symmetry Complex multiplication Frobenius symmetry Monodromy Fine structure in characteristic p Figure: Euler GEOMETRY AND Counting sums of squares NUMBERS Ching-Li Chai 2. Counting solutions of a diophantine equation Sample arithmetic statements Diophantine equations Example. Counting sums of squares. Counting solutions of a diophantine equation For n;k 2 N, let Counting congruence solutions L-functions and distribution of prime numbers n 2 2 Zeta and L-values rk(n) := #f(x1;:::;xk) 2 Z : x1 + ··· + xk = ng Sample of geometric structures and be the number of ways to represent n as a sum of k squares. symmetries Elliptic curve basics Modular forms, modular curves and Hecke symmetry (d−1)=2 0 if n2 6= Complex multiplication (i) r2(n) = 4 · ∑ (−1) = Frobenius symmetry 1 if n = Monodromy djn; n odd ∑djn1 2 Fine structure in characteristic f p where n = 2 · n1 · n2, and every prime divisor of n1 (resp. n2) is ≡ 1 (mod 4) (resp. ≡ 3 (mod 4)). 8 · ∑djn d if n is odd (ii) r4(n) = 24 · ∑djn;d odd d if n is even GEOMETRY AND How to count number of sum of squares NUMBERS Ching-Li Chai Sample arithmetic statements Diophantine equations Counting solutions of a diophantine equation Counting congruence solutions L-functions and distribution of Method. Explicitly identify the theta series prime numbers Zeta and L-values p 2 k Sample of geometric k( ) = qm where q = e2p −1t structures and q t ∑ symmetries m2N Elliptic curve basics Modular forms, modular curves and Hecke symmetry with modular forms obtained in a different way, such as Complex multiplication Frobenius symmetry Eisenstein series. Monodromy Fine structure in characteristic p GEOMETRY AND Counting congruence solutions NUMBERS Ching-Li Chai Sample arithmetic statements 3. Counting congruence solutions and L-functions Diophantine equations Counting solutions of a diophantine equation (a) Count the number of congruence solutions of a given Counting congruence solutions L-functions and distribution of diophantine equation modulo a (fixed) prime number p prime numbers Zeta and L-values Sample of geometric structures and (b) Identify the L-function for a given diophantine equation symmetries (basically the generating function for the number of Elliptic curve basics Modular forms, modular congruence solutions modulo p as p varies) curves and Hecke symmetry Complex multiplication with Frobenius symmetry Monodromy an L-function coming from harmonic analysis. (The latter is Fine structure in characteristic p associated to a modular form). Remark. (b) is an essential aspect of the Langlands program. GEOMETRY AND The Riemann zeta function NUMBERS Ching-Li Chai Sample arithmetic statements 4. L-functions and the the distribution of prime Diophantine equations Counting solutions of a numbers for a given diophantine problem diophantine equation Counting congruence solutions L-functions and distribution of Examples. (i) The Riemann zeta function z(s) is a prime numbers Zeta and L-values meromorphic function on C with only a simple pole at s = 0, Sample of geometric structures and −s −s −1 symmetries z(s) = ∑ n = ∏(1 − p ) for Re(s) > 1; Elliptic curve basics Modular forms, modular n≥1 p curves and Hecke symmetry Complex multiplication −s=2 Frobenius symmetry such that the function x(s) = p · G(s=2) · z(s) satisfies Monodromy Fine structure in characteristic p x(1 − s) = x(s): GEOMETRY AND NUMBERS Ching-Li Chai Sample arithmetic statements Diophantine equations Counting solutions of a diophantine equation Counting congruence solutions L-functions and distribution of prime numbers Zeta and L-values Sample of geometric structures and symmetries Elliptic curve basics Modular forms, modular curves and Hecke symmetry Complex multiplication Frobenius symmetry Monodromy Fine structure in characteristic p Figure: Riemann GEOMETRY AND Dirichlet L-functions NUMBERS Ching-Li Chai Sample arithmetic statements Diophantine equations Counting solutions of a diophantine equation Counting congruence solutions L-functions and distribution of prime numbers (ii) Similar properties hold for the Dirichlet L-function Zeta and L-values Sample of geometric −s structures and L(c;s) = ∑ c(n) · n Re(s) > 1 symmetries n2N;(n;N)=1 Elliptic curve basics Modular forms, modular curves and Hecke