ATOUR OF FERMAT’S WORLD Ching-Li Chai Samples of numbers More samples in arithemetic ATOUR OF 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 ATOUR OF 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 ATOUR OF 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 ATOUR OF Some familiar algebraic irrationals FERMAT’S WORLD Ching-Li Chai Samples of numbers More samples in arithemetic Congruent numbers p Fermat’s infinite 2, the Pythagora’s number, often the first irrational descent numbers one learns in school. Counting solutions p Zeta functions and −1, the first imaginary number one encountered. their special values p 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 ATOUR OF 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 p, 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 p), and obtained Weil conjecture and equidistribution 3:1415926 < p < 3:1415927: ATOUR OF 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, D = 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 DM = 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? ATOUR OF 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? ATOUR OF 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 ATOUR OF 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 p(n) ∼ p ep 2n=3 Weil conjecture and 4n 3 equidistribution ATOUR OF 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 ATOUR OF FERMAT’S WORLD Ching-Li Chai Hans Rademacher proved the following exact formula for p(n): Samples of numbers p p 2n−1=12 More samples in ¥ sinh p arithemetic 1 1=2 d 3k p(n) = p ∑ k Ak(n) p Congruent numbers p 2 k=1 dn n − 1=24 Fermat’s infinite descent where p Counting solutions p −1[s(m;k)−2nm=k] Ak(n) = ∑ e Zeta functions and 0≤m<k;(m;k)=1 their special values Modular forms and and the Dedekind sum s(m;k) is by definition L-functions Elliptic curves, complex s(m;k) = sawt(j=k)sawt(mj=k) multiplication and ∑ L-functions 1≤j≤k−1 Weil conjecture and equidistribution 1 sawt(x) = x − bxc − 2 if x 62 Z; sawt(x) = 0 if x 2 Z ATOUR OF 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 ATOUR OF 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 t(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 t(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. ATOUR OF 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 ATOUR OF 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 p Counting solutions Q( −d) is equal to one. Zeta functions and p their special values p d For the larger Heegner numbers, e is close to an integer; e.g. Modular forms and L-functions p p 67 Elliptic curves, e = 147197952743:99999866 complex multiplication and p L-functions p 163 e = 161537412640768743:99999999999925007 Weil conjecture and equidistribution ATOUR OF 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.