ATOUROF FERMAT’S WORLD Ching-Li Chai
Samples of numbers
More samples in arithemetic ATOUROF FERMAT’S WORLD Congruent numbers Fermat’s infinite descent
Counting solutions
Ching-Li Chai Zeta functions and their special values Department of Mathematics Modular forms and University of Pennsylvania L-functions Elliptic curves, complex multiplication and Philadelphia, March, 2016 L-functions
Weil conjecture and equidistribution
ATOUROF Outline FERMAT’S WORLD Ching-Li Chai
1 Samples of numbers Samples of numbers More samples in 2 More samples in arithemetic arithemetic Congruent numbers
Fermat’s infinite 3 Congruent numbers descent
Counting solutions 4 Fermat’s infinite descent Zeta functions and their special values
5 Counting solutions Modular forms and L-functions
Elliptic curves, 6 Zeta functions and their special values complex multiplication and 7 Modular forms and L-functions L-functions Weil conjecture and equidistribution 8 Elliptic curves, complex multiplication and L-functions
9 Weil conjecture and equidistribution ATOUROF Some familiar whole numbers FERMAT’S WORLD Ching-Li Chai
Samples of numbers
More samples in §1. Examples of numbers arithemetic Congruent numbers
Fermat’s infinite 2, the only even prime number. descent 30, the largest positive integer m such that every positive Counting solutions Zeta functions and integer between 2 and m and relatively prime to m is a their special values prime number. Modular forms and L-functions 3 3 3 3 1729 = 12 + 1 = 10 + 9 , Elliptic curves, complex the taxi cab number. As Ramanujan remarked to Hardy, multiplication and it is the smallest positive integer which can be expressed L-functions Weil conjecture and as a sum of two positive integers in two different ways. equidistribution
ATOUROF Some familiar algebraic irrationals FERMAT’S WORLD Ching-Li Chai
Samples of numbers
More samples in arithemetic
Congruent numbers √ Fermat’s infinite 2, the Pythagora’s number, often the first irrational descent numbers one learns in school. Counting solutions √ Zeta functions and −1, the first imaginary number one encountered. their special values √ 1+ 5 Modular forms and 2 , the golden number, a root of the quadratic L-functions 2 polynomial x − x − 1. Elliptic curves, complex multiplication and L-functions
Weil conjecture and equidistribution ATOUROF Some familiar transcendental numbers FERMAT’S WORLD Ching-Li Chai
Samples of numbers −2 −6 −24 −n! More samples in 1 + 10 + 10 + 10 + 10 + ··· + 10 + ···, a arithemetic
Liouville number. Congruent numbers ∞ 1 Fermat’s infinite e = exp(1) = , the base of the natural logarithm. descent ∑ n! n=0 Counting solutions
π, area of a circle of radius 1. Zu Chungzhi (429–500) Zeta functions and gave two approximating fractions, their special values Modular forms and 22 355 L-functions and Elliptic curves, 7 113 complex multiplication and L-functions (both bigger than π), and obtained Weil conjecture and equidistribution 3.1415926 < π < 3.1415927.
ATOUROF Triangular numbers and Mersenne numbers FERMAT’S WORLD Ching-Li Chai §2. Some families of numbers Samples of numbers 1,3,6,10,15,21,28,36,45,55,66,78,91,..., the triangular More samples in arithemetic numbers, ∆ = n(n+1) . n 2 Congruent numbers
2,3,5,7,11,13,17,19,23,29,31,37,41,..., the prime Fermat’s infinite numbers. descent Counting solutions p 2 − 1 (p is a prime number) are the Mersenne numbers. Zeta functions and p their special values If Mp := 2 − 1 is a prime number (a Mersenne prime), Modular forms and then L-functions 1 p−1 p ∆M = M(M + 1) = 2 (2 − 1) Elliptic curves, 2 complex multiplication and is an even perfect number. For instance Mp is a Mersenne L-functions Weil conjecture and prime for p = 2,3,5,7,13,17,19,31,61,89,107,127,521 equidistribution and 74207281. Open question: are there infinitely many Mersenne primes? ATOUROF Fermat numbers FERMAT’S WORLD Ching-Li Chai
Samples of numbers 3,5,17,257,65537,4294967297 are the first few Fermat More samples in arithemetic numbers, r Congruent numbers F = 22 + 1. r Fermat’s infinite descent
Counting solutions
Not all Fermat numbers are primes; Euler found in 1732 that Zeta functions and their special values 32 2 + 1 = 4294967297 = 641 × 6700417. Modular forms and L-functions
Elliptic curves, If a prime number p is a Fermat number, then the regular complex multiplication and p-gon’s can be constructed with ruler and compass. For L-functions instance Fr is a prime number for r = 0,1,2,3,4,5,65537. Weil conjecture and equidistribution Open question: are there infinitely many Fermat primes?
ATOUROF FERMAT’S WORLD Ching-Li Chai
Samples of numbers
More samples in arithemetic
Congruent numbers
Fermat’s infinite descent
Counting solutions
Zeta functions and their special values
Modular forms and L-functions
Elliptic curves, complex multiplication and L-functions
Weil conjecture and equidistribution
Figure: Fermat ATOUROF Partition numbers FERMAT’S WORLD Ching-Li Chai
Samples of numbers The partition numbers p(n) is defined as the number of More samples in arithemetic unordered partitions of a whole integer n. Its generating series Congruent numbers is ∞ ∞ Fermat’s infinite p(n)xn = (1 − xm)−1 descent ∑ ∏ Counting solutions n=0 n=1 Zeta functions and e.g. (1,1,1,1),(2,1,1,1),(2,2),(3,1),(4) are the five ways to their special values Modular forms and partition 4, so p(4) = 5. L-functions
Elliptic curves, Ramanujan showed complex multiplication and L-functions 1 √ p(n) ∼ √ eπ 2n/3 Weil conjecture and 4n 3 equidistribution
ATOUROF FERMAT’S WORLD Ching-Li Chai
Samples of numbers
More samples in arithemetic
Congruent numbers
Fermat’s infinite descent
Counting solutions
Zeta functions and their special values
Modular forms and L-functions
Elliptic curves, complex multiplication and L-functions
Weil conjecture and equidistribution
Figure: Ramanujan ATOUROF FERMAT’S WORLD Ching-Li Chai Hans Rademacher proved the following exact formula for p(n): Samples of numbers √ π 2n−1/12 More samples in ∞ sinh √ arithemetic 1 1/2 d 3k p(n) = √ ∑ k Ak(n) √ Congruent numbers π 2 k=1 dn n − 1/24 Fermat’s infinite descent where √ Counting solutions π −1[s(m,k)−2nm/k] Ak(n) = ∑ e Zeta functions and 0≤m ATOUROF FERMAT’S WORLD Ching-Li Chai Samples of numbers More samples in arithemetic Congruent numbers Fermat’s infinite descent Counting solutions Zeta functions and their special values Modular forms and L-functions Elliptic curves, complex multiplication and L-functions Weil conjecture and equidistribution Figure: Rademacher ATOUROF Ramanujan numbers FERMAT’S WORLD Ching-Li Chai Samples of numbers 1,−24,252,−1472,4830,−6048,..., are the first few More samples in Ramanujan numbers, defined by arithemetic Congruent numbers 24 ∞ " ∞ # Fermat’s infinite descent τ(n)xn = x (1 − xn) ∑ ∏ Counting solutions n=1 n=1 Zeta functions and their special values Ramanujan conjectured that there is a constant C > 0 such that Modular forms and L-functions 11/2 τ(p) ≤ C p Elliptic curves, complex multiplication and for every prime number p. L-functions Weil conjecture and equidistribution Ramanujan’s conjecture was proved by Deligne in 1974 (as a consequence of his proof of the Weil conjecture) with C = 2. ATOUROF Bernouli numbers FERMAT’S WORLD Ching-Li Chai Samples of numbers More samples in arithemetic The Bernouli numbers are defined by Congruent numbers Fermat’s infinite ∞ descent x Bn n x = ∑ x Counting solutions e − 1 n=0 n! Zeta functions and their special values 1 1 1 1 1 B0 = 1, B1 = − 2 , B2 = 6 , B4 = − 30 , B6 = 42 , B8 = − 30 , Modular forms and 691 7 L-functions B12 = − , B14 = . 2730 6 Elliptic curves, complex multiplication and Remark. The Bernouli numbers are essentially the values of L-functions the Riemann zeta function at negative odd integers. Weil conjecture and equidistribution ATOUROF Heegner numbers FERMAT’S WORLD Ching-Li Chai Samples of numbers More samples in arithemetic −1,−2,−3,−7,−11,−19,−43,−67,−163 are the nine Congruent numbers Heegner numbers; they are the only negative integers −d such Fermat’s infinite that the class number of the imaginary quadratic fields descent √ Counting solutions Q( −d) is equal to one. Zeta functions and √ their special values π d For the larger Heegner numbers, e is close to an integer; e.g. Modular forms and L-functions √ π 67 Elliptic curves, e = 147197952743.99999866 complex multiplication and √ L-functions π 163 e = 161537412640768743.99999999999925007 Weil conjecture and equidistribution ATOUROF Two simple diophantine equations FERMAT’S WORLD Ching-Li Chai Samples of numbers §2. Some Diophantine equations More samples in arithemetic The equation Congruent numbers 2 2 2 x + y = z Fermat’s infinite descent has lots of integer solutions. The primitive ones with x odd Counting solutions and y even are given by the formula Zeta functions and their special values 2 2 2 2 Modular forms and s −t s +t L-functions x = st, y = 2 , z = 2 , s > t odd, gd(s,t) = 1 Elliptic curves, complex (Fermat) The equation multiplication and L-functions 4 4 2 Weil conjecture and x − y = z equidistribution has no non-trivial integer solution. 2 2 ATOUROF Primes of the form Ax + By FERMAT’S WORLD Ching-Li Chai Samples of numbers (Fermat) More samples in arithemetic 2 2 p = x + y ⇐⇒ p ≡ 1 (mod 4) Congruent numbers 2 2 Fermat’s infinite p = x + 2y ⇐⇒ p ≡ 1 or 3 (mod 8) descent 2 2 p = x + 3y ⇐⇒ p = 3 or p ≡ 1 (mod 3) Counting solutions Zeta functions and (Euler) their special values Modular forms and L-functions 2 2 p = x + 5y ⇐⇒ p ≡ 1,9 (mod 20) Elliptic curves, 2 2 complex 2p = x + 5y ⇐⇒ p ≡ 3,7 (mod 20) multiplication and L-functions Weil conjecture and equidistribution p = x2 + 14y2 or p = 2x2 + 7y2 ⇐⇒ p ≡ 1,9,15,23,25,39 (mod 56) ATOUROF FERMAT’S WORLD Ching-Li Chai Samples of numbers More samples in arithemetic Congruent numbers Fermat’s infinite descent Counting solutions Zeta functions and their special values Modular forms and L-functions Elliptic curves, complex multiplication and L-functions Weil conjecture and equidistribution Figure: Euler 2 2 ATOUROF Primes of the form Ax + By , continued FERMAT’S WORLD Ching-Li Chai Samples of numbers 2 2 More samples in p = x + 27y ⇐⇒ p ≡ 1 (mod 3) and 2 is a arithemetic cubic residue (mod p) Congruent numbers Fermat’s infinite descent Counting solutions Zeta functions and their special values 2 2 p = x + 64y ⇐⇒ p ≡ 1 (mod 4) and 2 is a Modular forms and L-functions biquadratic residue (mod p) Elliptic curves, complex multiplication and L-functions (Kronecker) Weil conjecture and equidistribution p = x2 + 31y2 ⇐⇒ (x3 − 10x)2 + 31(x2 − 1)2 ≡ 0 (mod p) for some integer x ATOUROF Some formulas discovered by Euler FERMAT’S WORLD Ching-Li Chai Samples of numbers π 1 1 1 = 1 − + − + ... More samples in 4 3 5 7 arithemetic Congruent numbers 1 1 1 π2 1 + 2 + 2 + 2 + ... = 6 Fermat’s infinite 2 3 4 descent Counting solutions 1 1 1 π4 1 + 24 + 34 + 44 + ... = 90 Zeta functions and their special values (1−2k+1) Modular forms and k k k L-functions • 1 − 2 + 3 − 4 + ... = − k+1 Bk+1 for k ≥ 1; in particular it vanishes if k is even. Elliptic curves, complex multiplication and L-functions 1 ( + 1 + 1 + 1 + ...) ∈ k ≥ π2k 1 22k 32k 42k Q for every integer 1. Weil conjecture and equidistribution ∞ n n n(3n+1)/2 ∏n=1 (1 − x ) = ∑n∈Z (−1) x ATOUROF Prelude to congruent numbers FERMAT’S WORLD Ching-Li Chai Samples of numbers More samples in arithemetic Congruent numbers §3. Congruent numbers. Fermat’s infinite The equation descent Counting solutions 2 3 2 Zeta functions and y z = x − n xz, n square free, their special values Modular forms and may or may not have a non-trivial (i.e. xyz 6= 0) integer L-functions n congruent number Elliptic curves, solution—depending on whether is a , to be complex discussed next. multiplication and L-functions Weil conjecture and equidistribution ATOUROF Congruent numbers: definition and examples FERMAT’S WORLD Ching-Li Chai Samples of numbers More samples in arithemetic A square free whole number n > 0 is a congruent number if Congruent numbers there is a right triangle with rational sides whose area is n. Fermat’s infinite descent For instance 5 is a congruent number, because Counting solutions 2 2 2 Zeta functions and (20/3) + (3/2) = (41/6) . Similarly 6 is a congruent number their special values 2 2 2 because 3 + 4 = 5 . Modular forms and L-functions Elliptic curves, 5,6,7,13,14,15,20,21,22,23,24,28,29,30,31,34,37,38,39, complex multiplication and 41,45,46,47 are the beginning of (the sequence of) congruent L-functions numbers. Weil conjecture and equidistribution ATOUROF 157 as a congruent numbers FERMAT’S WORLD Ching-Li Chai The number 157 is a congruent number, but the “simplest” Samples of numbers 2 2 2 non-trivial rational solution to a + b = c with a · b = 314 is More samples in arithemetic 157841 · 4947203 · 526771095761 Congruent numbers a = Fermat’s infinite 2 · 32 · 5 · 13 · 17 · 37 · 101 · 17401 · 46997 · 356441 descent Counting solutions 2 2 Zeta functions and 2 · 3 · 5 · 13 · 17 · 37 · 101 · 157 · 17401 · 46997 · 356441 their special values b = 157841 · 4947203 · 526771095761 Modular forms and L-functions Elliptic curves, complex 20085078913·1185369214457·9425458255024420419074801 c= multiplication and 2·32·5·13·17·37·101·17401·46997·356441·157841·4947203·526771095761 L-functions Weil conjecture and The point: although 157 is not a large number, the solution equidistribution involved may have very large numerators and denominators. In particular it may not be easy to either determine or search for congruent numbers. ATOUROF Congruent number problem FERMAT’S WORLD Ching-Li Chai Samples of numbers More samples in Congruence number problem: find an (easily checkable) arithemetic criterion for a square free positive integer to be a congruence Congruent numbers number. Fermat’s infinite descent Reformulation in terms of rational points on (a special kind Counting solutions Zeta functions and of) elliptic curves: a positive square free integer n is a their special values congruent number if and only if the equation Modular forms and L-functions 2 3 2 Elliptic curves, y = x − n x complex multiplication and L-functions has a solution (x,y) in rational numbers with y 6= 0. (Then there Weil conjecture and are infinitely many such rational solutions.) equidistribution ATOUROF Tunnell’s theorem on congruent numbers FERMAT’S WORLD Ching-Li Chai Samples of numbers More samples in Theorem (Tunnell 1983) If n is a square free positive arithemetic congruence number, then Congruent numbers Fermat’s infinite descent #{x,y,z∈ |n=2x2+y2+32z2}=(1/2)·#{(x,y,z∈ |n=2x2+y2+8z2} Z Z Counting solutions Zeta functions and if n is odd, and their special values Modular forms and L-functions #{x,y,z∈Z|n/2=4x2+y2+32z2}=(1/2)·#{(x,y,z∈Z|n/2=2x2+y2+8z2} Elliptic curves, complex multiplication and if n is even. Conversely if the Birch-Swinnerton-Dyer L-functions 2 3 2 conjecture holds (for the elliptic curve y = x − n x), then Weil conjecture and these equalities imply that n is a congruent number. equidistribution ATOUROF Fermat’s Last Theorem FERMAT’S WORLD Ching-Li Chai Samples of numbers More samples in (Fermat’s Last Theorem, now a theorem of arithemetic Wiles+Taylor-Wiles) Congruent numbers Fermat’s infinite descent p p p x + y + z = 0 Counting solutions Zeta functions and has no non-trivial integer solution if p is an odd prime number. their special values Modular forms and Proved by Wiles and Taylor/Wiles in 1994, more than 300 L-functions Elliptic curves, years after Fermat wrote the assertion at the margin of his complex multiplication and personal copy of the 1670 edition of Diophantus. L-functions Weil conjecture and Question/Discussion: Why should anyone care? equidistribution ATOUROF Fermat’s infinite descent illustrated FERMAT’S WORLD Ching-Li Chai Samples of numbers More samples in arithemetic §4. Fermat’s infinite descent Congruent numbers We will explain how to use Fermat’s method of infinite descent, Fermat’s infinite which he is jusifiably proud of, to show that the Diophantine descent equation Counting solutions 4 4 2 Zeta functions and x − y = z their special values Modular forms and has no non-trivial integer solution. L-functions Elliptic curves, complex May assume gcd(x,y,z) = 1. The either x,y are both odd, or x multiplication and is odd and y is even. We will consider only the first case that L-functions Weil conjecture and x,y are both odd. equidistribution ATOUROF Step 1. FERMAT’S WORLD Ching-Li Chai (x2 + y2) · (x2 − y2) = z2 =⇒ ∃u,v such that gcd(u,v) = 1, Samples of numbers 2 2 2 2 2 2 More samples in x + y = 2u , x − y = 2v and z = 2uv. arithemetic Congruent numbers 2 2 2v = (x + y) · (x − y) =⇒ ∃r,s such that x + y = r , Fermat’s infinite x − y = 2s2, v = rs (adjust the signs). descent Counting solutions 4 4 2 The original equation becomes r + 4s = 4u . Write r = 2t, Zeta functions and the equation becomes their special values Modular forms and L-functions 4 4 2 s + 4t = u Elliptic curves, complex multiplication and and we have L-functions Weil conjecture and equidistribution x = 2t2 + s2, y = 2t2 − s2, z = 4tsu, gcd(s,t,u) = 1. ATOUROF Step 2. FERMAT’S WORLD Ching-Li Chai Samples of numbers 4 4 2 More samples in From s + 4t = u , gcd(s,t,u) = 1, it is easy to see that u and arithemetic s are both odd. May assume u > 0. Congruent numbers 2 2 2 2 2 Fermat’s infinite 4t = (u − s )(u + s ) =⇒ ∃a,b such that u − s = 2b , descent 2 2 2 u + s = 2a , t = ab, gcd(a,b) = 1. Counting solutions Zeta functions and 2 2 2 their special values t = ab =⇒ ∃x1,y1 such that a = x1, b = y1 and t = x1y1.. It 4 4 Modular forms and follows that u = x1 + y1 and L-functions Elliptic curves, 4 4 2 complex x1 − y1 = s . multiplication and L-functions Weil conjecture and Let z1 = s. Then (x1,y1,z1) is an integer solution of the original equidistribution 4 4 2 equation x − y = z , with |x1| strictly smaller. ATOUROF Fermat’s infinite descent, continuted FERMAT’S WORLD Ching-Li Chai Samples of numbers More samples in arithemetic Conclusion. Starting with a non-trivial solution, we obtain an Congruent numbers Fermat’s infinite infinite sequence of non-trivial solutions descent (x,y,z),(x1,y1,z1),(x2,y2,z2),(x3,y3,z3),... such that the Counting solutions |x| > |x1| > |x2| > |x3| > ···. That’s impossible. Q.E.D. Zeta functions and their special values Modular forms and (We leave it to the reader to check that if we start with a L-functions 4 4 2 non-trivial solution of x − y = z such that x is odd and y is Elliptic curves, complex even, the same argument will also lead us to another non-trivial multiplication and solution such that the absolute value of x decreases. ) L-functions Weil conjecture and equidistribution ATOUROF An intrinsic description of infinite descent FERMAT’S WORLD Ching-Li Chai 4 4 2 Samples of numbers Remark. Consider algebraic varieties X1 : x − y = z and 4 4 2 More samples in X2 : s + 4t = u ; and maps f : X1 → X2 arithemetic Congruent numbers 4 4 f : (x,y,z) 7→ (s,t,u) = (z,xy,x + y ) Fermat’s infinite descent and g : X2 → X1 Counting solutions Zeta functions and their special values g : (s,t,u) 7→ (s2 + 2t2,s2 − 2t2,4stu) Modular forms and L-functions The varieties X1 and X2 correspond to elliptic curves E1,E2 Elliptic curves, complex over Q with√ complex multiplication; they become isomorphic multiplication and 4 L-functions over Q( −4). Weil conjecture and √ equidistribution The maps√ f ,g correspond to “multiplication by (1 + −1) and (1 − −1)” respectively. Their composition is “multiplication by 2”, defined over Q. ATOUROF Sum of two squares FERMAT’S WORLD Ching-Li Chai §5. Counting solutions. Samples of numbers Notation. For each integer k ≥ 1, let rk(n) be the number of More samples in k arithemetic k-tuples (x1,...,xn) ∈ Z such that Congruent numbers x2 + ... + x2 = n. Fermat’s infinite 1 k descent Counting solutions Write n = 2f · n · n , where every prime divisor of n is ≡ 1 1 2 1 Zeta functions and (mod 4) and every prime divisor of n2 is ≡ 3 (mod 4)). their special values Modular forms and L-functions Fermat showed that r (n) > 0 (i.e. n is a sum of two 2 Elliptic curves, squares) if and only if every prime divisor p of n occurs complex 2 multiplication and in n2 to an even power. L-functions Weil conjecture and Assume this is the case, Jacobi obtained equidistribution r2(n) = 4d(n1) where d(n1) is the number of divisors of n1. ATOUROF Sum of four squares FERMAT’S WORLD Ching-Li Chai Samples of numbers More samples in arithemetic Lagrange showed that r4(n) > 0 for every n ∈ Z. Congruent numbers Fermat’s infinite Jacobi obtained descent 0 r4(n) = 8σ (n) Counting solutions 0 Zeta functions and where σ (n) is the sum of divisors of n which are not their special values divisible by 4. More explicitly, Modular forms and L-functions Elliptic curves, 8 · ∑d|n d n odd complex r4(n) = multiplication and 24 · ∑d|n,d odd d n even L-functions Weil conjecture and equidistribution ATOUROF Sum of three squares FERMAT’S WORLD Ching-Li Chai Legendre showed that n is a sum of three squares if and Samples of numbers only if n is not of the form 4a(8m + 7), and a More samples in r3(4 n) = r3(n). arithemetic Congruent numbers Let Rk(n) be the number of primitive solutions of 2 2 Fermat’s infinite x1 + ··· + xk = n, i.e. gcd(x1,...,xk) = 1. Then descent Counting solutions bn/4c s 24∑ n ≡ 1,2 (mod 4) Zeta functions and s=1 n their special values R (n) = 3 Modular forms and bn/2c s L-functions 8∑s=1 n n ≡ 3 (mod 8) Elliptic curves, complex More conceptually, if n is squre free, then multiplication and L-functions √ · h( ( −n)) n ≡ ( ) Weil conjecture and 24 Q √ for 3 mod 8 equidistribution r3(n) = 12 · h(Q( −n)) for n ≡ 1,2,5,6 (mod 8) 0 for n ≡ 7 (mod 8) √ √ where h(Q( −n) is the class number of Q( −n). ATOUROF Mod p points for a CM elliptic curve FERMAT’S WORLD Ching-Li Chai For the elliptic curve E be the elliptic curve E = {y2 = x3 + x}, √ √ Samples of numbers which hascomplex multiplication by [ −1] with −1 acting √ Z More samples in by (x,y) 7→ (−x, −1y), we have arithemetic √ √ Congruent numbers #E( p) = 1 + p − ap, −2 p ≤ ap ≤ 2 p Fermat’s infinite F descent for all prime numbers p. Counting solutions Zeta functions and (This is a general perperty of elliptic curves, due to Hasse. The their special values CM property allows us to give an explicit formula for ap’s.) Modular forms and L-functions For odd p we have Elliptic curves, complex multiplication and u3 + u L-functions ap = ∑ Weil conjecture and p equidistribution u∈Fp 0 if p ≡ 3 (mod 4) = −2a if p = a2 + 4b2 with a ≡ 1 (mod 4) ATOUROF The Riemann zeta function FERMAT’S WORLD Ching-Li Chai Samples of numbers §6. Zeta functions and their special values More samples in arithemetic The Riemann zeta function ζ(s) is a meromorphic function on Congruent numbers with only a simple pole at s = 0 (and holomorphic C Fermat’s infinite elsewhere), descent Counting solutions −s −s −1 ζ(s) = n = (1 − p ) for Re(s) > 1, Zeta functions and ∑ ∏ their special values n≥1 p Modular forms and L-functions −s/2 such that the function ξ(s) = π · Γ(s/2) · ζ(s) satisfies Elliptic curves, complex multiplication and ξ(1 − s) = ξ(s). L-functions Weil conjecture and Z ∞ equidistribution −t s−1 Here Γ(s) = e t dt for Re(s) > 0, extended to C by 0 Γ(s + 1) = sΓ(s). ATOUROF FERMAT’S WORLD Ching-Li Chai Samples of numbers More samples in arithemetic Congruent numbers Fermat’s infinite descent Counting solutions Zeta functions and their special values Modular forms and L-functions Elliptic curves, complex multiplication and L-functions Weil conjecture and equidistribution Figure: Riemann ATOUROF Dirichlet L-functions FERMAT’S WORLD Ching-Li Chai Samples of numbers More samples in arithemetic Similar properties hold for the Dirichlet L-function Congruent numbers −s Fermat’s infinite L(χ,s) = ∑ χ(n) · n Re(s) > 1 descent n∈N,(n,N)=1 Counting solutions Zeta functions and × × their special values for a primitive Dirichlet character χ : (Z/NZ) → C1 . Modular forms and L-functions × (Here C1 denotes the set of all complex numbers with absolute Elliptic curves, × complex value 1, (Z/NZ) is the set of all integers modulo N which are multiplication and prime to N, and χ is a function compatible with the rules of L-functions Weil conjecture and multiplication for both its source and target.) equidistribution ATOUROF FERMAT’S WORLD Ching-Li Chai Samples of numbers More samples in arithemetic Congruent numbers Fermat’s infinite descent Counting solutions Zeta functions and their special values Modular forms and L-functions Elliptic curves, complex multiplication and L-functions Weil conjecture and equidistribution Figure: Dirichlet ATOUROF L-functions and distribution of prime numbers FERMAT’S WORLD Ching-Li Chai Samples of numbers Theme. Zeta and L-values often contain deep More samples in arithemetic arithmetic/geometric information. Congruent numbers Fermat’s infinite Dirichlet’s theorem for primes in arithmetic progression descent ↔ L(χ,1) 6= 0 ∀ Dirichlet character χ. Counting solutions The prime number theorem Zeta functions and their special values ↔ zero free region of ζ(s) near {Re(s) = 1}. Modular forms and Riemann’s hypothesis ↔ (estimate of) the second term L-functions Elliptic curves, in the asymptotic expansion of complex multiplication and π(x) := #{p prime|p ≤ x} L-functions Note: the first/main term in expansion of π(x) is Weil conjecture and R x dt x x 2x 6x equidistribution Li(x) := 2 logt ∼ logx + (logx)2 + (logx)3 + (logx)4 + ··· ATOUROF More magical properties of zeta values FERMAT’S WORLD Ching-Li Chai Samples of numbers More samples in arithemetic Certain values of zeta or L-functions tend to be rational or Congruent numbers algebraic numbers, or becomes rational/algebraic after Fermat’s infinite suitable transcendental factors are removed. descent These special zeta values contains deep information such Counting solutions Zeta functions and as class numbers, Mordell-Weil group, Selmer group, their special values Tate-Shafarevich group, etc. Modular forms and √ L-functions Examples. (a) Leibniz’s formula: [ −1] is a PID (because Elliptic curves, Z √ complex the formula implies that the class number h(Q( −1)) is 1). multiplication and L-functions (b) Bk/k appears in the formula for the number of Weil conjecture and equidistribution (isomorphism classes of) exotic (4k − 1)-spheres. ATOUROF Bernouli numbers as zeta values FERMAT’S WORLD Ching-Li Chai Samples of numbers More samples in Recall that the Bernoulli numbers Bn are defined by arithemetic Congruent numbers x Bn n Fermat’s infinite x = ∑ · x descent e − 1 n∈ n! N Counting solutions Zeta functions and B0 = 1, B1 = −1/2, B2 = 1/6, B4 = −1/30, B6 = 1/42, their special values B8 = −1/30, B10 = 5/66, B12 = −691/2730. Modular forms and L-functions Elliptic curves, (i) (Euler) ζ(1 − k) = −Bk/k ∀ even integer k > 0. complex multiplication and L-functions (ii) (Leibniz’s formula, 1678; Madhava, ∼ 1400) Weil conjecture and 1 1 1 π equidistribution 1 − 3 + 5 − 7 + ··· = 4 ATOUROF Euler’s evaluation of zeta values. FERMAT’S WORLD Ching-Li Chai Insert a factor tk into the fomal infinite series for ζ(−n) and ! Samples of numbers ∞ ∞ k k n More samples in evaluate at t = 1: ζ(−k) = ∑ n = ∑ n t arithemetic n=1 n=1 t=1 Congruent numbers d k n k n From t t = n t , we get Fermat’s infinite dt descent ! Counting solutions d k ∞ d k t ζ(−k) = t tn = t Zeta functions and ∑ their special values dt n=1 dt 1 − t t=1 t=1 Modular forms and L-functions x d d Elliptic curves, Let t = e , so t dt = dx and complex multiplication and L-functions d k ex B ζ(−k) = = − k+1 Weil conjecture and dx 1 − ex k + 1 equidistribution x=0 for k > 0. Esp. ζ(−k) ∈ Q , ζ(−2k) = 0 ∀ k > 0. ATOUROF Congruence of zeta values FERMAT’S WORLD Ching-Li Chai Samples of numbers More samples in arithemetic Theme. Special zeta or L-values satisfy strong congruence Congruent numbers Fermat’s infinite properties—causing them to have p-adic avatars (called p-adic descent L-functions) which interpolate complex L-functions Counting solutions Zeta functions and Example. (Kummer congruence) their special values Modular forms and L-functions (i) ζ(m) ∈ Z(p) for m ≤ 0 with m 6≡ 1 (mod p − 1) 0 0 Elliptic curves, (ii) ζ(m) ≡ ζ(m )(mod p) for all m,m ≤ 0 with complex 0 multiplication and m ≡ m 6≡ 1 (mod p − 1). L-functions Weil conjecture and equidistribution ATOUROF Examples of p-adic properties of zeta values FERMAT’S WORLD Ching-Li Chai EXAMPLESOF KUMMERCONGRUENCE. Samples of numbers 1 More samples in ζ(−1) = − ; −1 ≡ 1 (mod p − 1) only for p = 2,3. arithemetic 22 · 3 Congruent numbers 691 ζ(−11) = ; −11 ≡ 1 (mod p − 1) holds Fermat’s infinite 23 · 32 · 5 · 7 · 13 descent only for p = 2,3,5,7,13. Counting solutions 1 Zeta functions and ζ(−5) = − ≡ ζ(−1)(mod 5), and we have their special values 22 · 32 · 7 Modular forms and −1 ≡ −5 (mod 5). L-functions (This congruence holds because 3 · 7 ≡ 1 (mod 5).) Elliptic curves, complex multiplication and EXAMPLE. (Kummer’s criterion) The prime factor 691 of the L-functions (− ) Weil conjecture and numerator√ of ζ 11 implies that 691 divides the class number equidistribution − / of Q(e2π 1 691). (One such congruence for a ζ(m), e.g. m = −11, sufficies.) ATOUROF FERMAT’S WORLD Ching-Li Chai Samples of numbers More samples in arithemetic Congruent numbers Fermat’s infinite descent Counting solutions Zeta functions and their special values Modular forms and L-functions Elliptic curves, complex multiplication and L-functions Weil conjecture and equidistribution Figure: Kummer ATOUROF Modular forms and Hecke symmetry FERMAT’S WORLD Ching-Li Chai Samples of numbers More samples in §7. Modular forms and L-functions arithemetic Congruent numbers Let Γ ⊂ SL2(Z) be a congruence subgroup of SL2(Z), i.e. Γ Fermat’s infinite descent contains all elements which are ≡ I2 (mod N) for some N. Counting solutions (a) A holomorphic function f (τ) on the upper half plane H is Zeta functions and their special values said to be a modular form of weight k and level Γ if Modular forms and L-functions −1 k a b Elliptic curves, f ((aτ + b)(cτ + d) ) = (cτ + d) · f (τ) ∀γ = ∈ Γ complex c d multiplication and L-functions and has moderate growth at all cusps. Weil conjecture and equidistribution ATOUROF Modular forms and L-functions FERMAT’S WORLD Ching-Li Chai Samples of numbers (c) The L-function More samples in arithemetic −s Lf (s) = ∑ an n Congruent numbers n≥1 Fermat’s infinite descent attached to a cusp form Counting solutions √ Zeta functions and 2π −1τ their special values f = an e ∀τ ∈ ∑ H Modular forms and n≥1 L-functions Elliptic curves, of weight k for Γ = SL2( ), which is a common eigenvector of complex Z multiplication and all Hecke correspondences, admits an Euler product L-functions Weil conjecture and −s k−1−2s −1 equidistribution Lf (s) = ∏(1 − ap p + p ) p ATOUROF The Ramanujan τ function FERMAT’S WORLD Ching-Li Chai Example. Weight 12 cusp forms for SL2(Z) are constant Samples of numbers More samples in multiples of arithemetic Congruent numbers ∆ = q · (1 − qm)24 = τ(n)qn ∏ ∑ Fermat’s infinite m≥1 n descent Counting solutions and Zeta functions and Tp(∆) = τ(p) · ∆ ∀p, their special values Modular forms and p 0 L-functions where Tp is the Hecke operator represented by . Elliptic curves, 0 1 complex multiplication and L-functions L( ,s) = a · n−s Let ∆ ∑n≥1 n . We have Weil conjecture and equidistribution L(∆,s) = ∏(1 − τ(p)p−s + p11−2s)−1. p ATOUROF How to count number of sum of squares FERMAT’S WORLD Ching-Li Chai Samples of numbers More samples in arithemetic Method. Explicitly identify k-th power the theta series Congruent numbers √ k m2 k 2π −1τ n Fermat’s infinite θ (τ) = ∑ q where q = e = ∑ rk(n)q descent m∈N n∈N Counting solutions Zeta functions and with a modular form obtained in a different way, such as their special values Eisenstein series. Modular forms and L-functions (Because θ(τ) is a modular form of weight 1/2, its k-th power Elliptic curves, complex is a modular form of weight k/2. Modular forms of a given multiplication and weight for a given congruence subgroup form a finite L-functions Weil conjecture and dimensional vector space.) equidistribution ATOUROF Counting congruence solutions FERMAT’S WORLD Ching-Li Chai Samples of numbers More samples in (a) Count the number of congruence solutions of a given arithemetic diophantine equation modulo a (fixed) prime number p Congruent numbers Fermat’s infinite descent (b) Identify the L-function for a given diophantine equation Counting solutions (basically the generating function for the number of Zeta functions and their special values congruence solutions modulo p as p varies) Modular forms and with L-functions an L-function coming from harmonic analysis. (The latter is Elliptic curves, complex associated to a modular form). multiplication and L-functions Weil conjecture and Remark. (b) is an essential aspect of the Langlands program. equidistribution ATOUROF Elliptic curves basics FERMAT’S WORLD Ching-Li Chai Samples of numbers §9. Elliptic curves, complex multiplication and L-functions More samples in arithemetic Congruent numbers Equivalent definitions of an elliptic curve E : Fermat’s infinite descent a projective curve with an algebraic group law; Counting solutions a projective curve of genus one together with a rational Zeta functions and their special values point (= the origin); Modular forms and L-functions over C: a complex torus of the form Eτ = C/Zτ + Z, Elliptic curves, where τ ∈ H := upper-half plane; complex × multiplication and over a field F with 6 ∈ F : given by an affine equation L-functions Weil conjecture and 2 3 equidistribution y = 4x − g2 x − g3, g2,g3 ∈ F . ATOUROF Weistrass theory FERMAT’S WORLD Ching-Li Chai For Eτ = C/Zτ + Z, let Samples of numbers xτ (z) = ℘(τ,z) More samples in arithemetic 1 1 1 Congruent numbers = 2 + ∑ 2 − 2 z (z − mτ − n) (mτ + n) Fermat’s infinite (m,n)6=(0,0) descent Counting solutions y (z) = d ℘(τ,z) τ dz Zeta functions and their special values Then E satisfies the Weistrass equation τ Modular forms and L-functions 2 3 yτ = 4xτ − g2(τ)xτ − g3(τ) Elliptic curves, complex multiplication and with L-functions 1 Weil conjecture and g (τ) = 60 equidistribution 2 ∑ (mτ + n)4 (0,0)6=(m,n)∈Z2 1 g (τ) = 140 3 ∑ (mτ + n)6 (0,0)6=(m,n)∈Z2 ATOUROF The j-invariant FERMAT’S WORLD Ching-Li Chai Elliptic curves are classified by their j-invariant Samples of numbers 3 g2 More samples in j = 1728 arithemetic g3 − 27g2 2 3 Congruent numbers Fermat’s infinite Over C, j(Eτ ) depends only on the lattice Zτ + Z of Eτ . So j(τ) descent is a modular function for SL2(Z): Counting solutions Zeta functions and aτ + b their special values j = j(τ) Modular forms and cτ + d L-functions Elliptic curves, for all a,b,c,d ∈ with ad − bc = 1. complex Z multiplication and L-functions We have a Fourier expansion Weil conjecture and equidistribution 1 j(τ) = + 744 + 196884q + 21493760q2 + ··· , q √ 2π −1τ where q = qτ = e . ATOUROF Elliptic curves with complex multiplication FERMAT’S WORLD Ching-Li Chai Samples of numbers An elliptic E over C is said to have complex multiplication if its 0 More samples in endomorphism algebra End (E) is an imaginary quadratic field. arithemetic Congruent numbers Example. Consequences of Fermat’s infinite descent • j( /OK) is an algebraic integer C Counting solutions • K · j( / ) = the Hilbert class field of K. C OK Zeta functions and their special values √ π 67 Modular forms and e = 147197952743.9999986624542245068292613··· L-functions √ j −1+ −67 = − = − 15 · 3 · 3 · 3 Elliptic curves, 2 147197952000 2 3 5 11 complex multiplication and √ L-functions π 163 e = 262537412640768743.99999999999925007259719... Weil conjecture and √ equidistribution −1+ −163 j 2 = −262537412640768000 = −218 · 33 · 53 · 233 · 293 ATOUROF A CM curve and its associated modular form FERMAT’S WORLD Ching-Li Chai Samples of numbers More samples in We have seen that the number of congruent points on the arithemetic 2 3 elliptic curve E = y = x + x is given by Congruent numbers Fermat’s infinite descent #E(Fp) = 1 + p − ap Counting solutions Zeta functions and and for odd p we have their special values 3 Modular forms and u + u L-functions a = p ∑ p Elliptic curves, u∈Fp complex multiplication and 0 if p ≡ 3 (mod 4) L-functions = 2 2 Weil conjecture and −2a if p = a + 4b with a ≡ 1 (mod 4) equidistribution ATOUROF A CM curve and its associated modular form, FERMAT’S WORLD continued Ching-Li Chai Samples of numbers L(E,s) E The L-function attached to with More samples in arithemetic −s 1−2s −1 −s ∏ (1 − app + p ) = ∑an · n Congruent numbers p odd n Fermat’s infinite descent is equal to a Hecke L-function L(ψ,s), where the Hecke Counting solutions Zeta functions and character ψ is the given by their special values Modular forms and 0 if 2|N(a) L-functions ψ(a) = √ λ if a = (λ), λ ∈ 1 + 4 + 2 −1 Elliptic curves, Z Z complex multiplication and n L-functions The function fE(τ) = ∑n an · q is a modular form of weight 2 Weil conjecture and and level 4, and equidistribution N(a) a2+b2 fE(τ) = ∑ψ(a) · q = ∑ a · q a a≡1 (mod 4) b≡0 (mod 2) ATOUROF Estimates of Fourier coefficients by Weil FERMAT’S WORLD conjecture Ching-Li Chai Samples of numbers More samples in §10. Weil conjecture and equidistribution arithemetic √ Congruent numbers Let ∆(τ) = ∑ τ(n)e2π −1 be the normalized cusp form of n≥1 Fermat’s infinite weight 12 whose Fourier coefficients are the Ramanujan descent numbers τ(n); they are eigenvalues of Hecke operators Tn. Counting solutions Zeta functions and their special values (Eichler & Shimura) (p) = + for each prime τ αp αp Modular forms and number p, where αp is the eigenvalues of a “Frobenius L-functions Elliptic curves, operator for p”. complex multiplication and (Deligne) Deligne showed that the Ramanujan conjecture L-functions 11/2 |τ(p)| ≤ C · p is a consequence of Weil’s conjecture Weil conjecture and 11/2 equidistribution (which asserts that |αp| = p in this case). Then he proved Weil’s conjecture in 1974. ATOUROF Equidistribution FERMAT’S WORLD Ching-Li Chai Remark. Similar estimates of Fourier coefficients of modular Samples of numbers More samples in forms also follows from Weil’s conjecture. This gives the best arithemetic possible estimates of Fourier coefficients by algebraic methods. Congruent numbers (Estimates obtained by analytic methods so far are very far off.) Fermat’s infinite descent Counting solutions Question In what sense is the above estimate “best possible”? Zeta functions and √ their special values ANSWER. The family of real numbers {τ(p)/ p} is Modular forms and equidistributed√ in [−2,2] with respect to the measure L-functions 1 4 − t2dt, i.e. Elliptic curves, 2π complex multiplication and Z 2 L-functions 1 √ 1 p 2 lim f (ap/ p) = f (t) 4 − t dt Weil conjecture and x→∞ ∑ equidistribution #{p : p ≤ x} p≤x 2π −2 for every continuous function f (t) on [−2,2].