From the Manhattan Project to Elliptic Curves: Introduction to Random Matrix Theory

Transcription

1 From the Manhattan Project to Elliptic Curves: Introduction to Random Matrix Theory Steven J Miller Dept of Math/Stats, Williams College sjm1@williams.edu, Steven.Miller.MC.96@aya.yale.edu Williams College, July 27, 2016

2 Introduction

3 Goals Determine correct scale and statistics to study eigenvalues and zeros of L-functions. See similar behavior in different systems. Discuss the tools and techniques needed to prove the results.

4 Fundamental Problem: Spacing Between Events General Formulation: Studying system, observe values at t 1, t 2, t 3,... Question: What rules govern the spacings between the t i?

5 5 Intro Classical RMT Toeplitz Block Circulant Checkerboard Matrices L-Functions Qs and Refs Fundamental Problem: Spacing Between Events General Formulation: Studying system, observe values at t 1, t 2, t 3,... Question: What rules govern the spacings between the t i? Examples: Spacings between Energy Levels of Nuclei. Eigenvalues of Matrices. Zeros of L-functions. Summands in Zeckendorf Decompositions. Primes. n k α mod 1.

6 Fundamental Problem: Spacing Between Events General Formulation: Studying system, observe values at t 1, t 2, t 3,... Question: What rules govern the spacings between the t i? Examples: Spacings between Energy Levels of Nuclei. Eigenvalues of Matrices. Zeros of L-functions. Summands in Zeckendorf Decompositions. Primes. n k α mod 1.

7 7 Intro Classical RMT Toeplitz Block Circulant Checkerboard Matrices L-Functions Qs and Refs Sketch of proofs In studying many statistics, often three key steps: 1 Determine correct scale for events. 2 Develop an explicit formula relating what we want to study to something we understand. 3 Use an averaging formula to analyze the quantities above. It is not always trivial to figure out what is the correct statistic to study!

8 Eigenvalue Review: I z C : z = x + iy with i = 1; z = z H = x iy. Eigenvalue/Eigenvector: λ C, v 0 : A v = λ v. Can find by det(a λi) = 0 but computational nightmare! Real Symmetric: A = A T ; Hermitian: A = A H (complex conjugate transpose). Length of v is v H v ; v 2 = v H v ; v w = v H w.

9 Eigenvalue Review: II A real implies eigenvalues real: If A v = λ v then v H A H v = v H A v ( ) H ( A v v = v H A v ) ( λ v ) H v = v H ( λ v ) λ v H v = λ v H v λ v 2 = λ v 2, and thus as length is non-zero have λ = λ and is real, and then get coefficients of v real. A complex Hermitian: similar proof shows eigenvalues real (coefficients can be complex).

10 Eigenvalue Review: III Orthogonal: Q T Q = QQ T = I; Unitary: U H U = UU H = I. Spectral Theorem: If A is real symmetric or complex Hermitian than can diagonalize (real symmetric: A = Q T ΛQ, complex Hermitian A = U H ΛU). Proof: Trivial if distinct eigenvalues as each has an eigenvector, mutually orthogonal, choose unit length and let these be columns of Q: v1 T A T v2 = v 1 T A v 2 ( A v1 ) T v2 = v 1 T ( A v 2 ) λ 1 v1 T v2 = λ 2 v1 T v2.

11 Classical Random Matrix Theory With Olivia Beckwith, Leo Goldmakher, Chris Hammond, Steven Jackson, Cap Khoury, Murat Koloğlu, Gene Kopp, Victor Luo, Adam Massey, Eve Ninsuwan, Vincent Pham, Karen Shen, Jon Sinsheimer, Fred Strauch, Nicholas Triantafillou, Wentao Xiong

12 Origins of Random Matrix Theory Classical Mechanics: 3 Body Problem intractable.

13 Origins of Random Matrix Theory Classical Mechanics: 3 Body Problem intractable. Heavy nuclei (Uranium: 200+ protons / neutrons) worse! Get some info by shooting high-energy neutrons into nucleus, see what comes out. Fundamental Equation: Hψ n = E n ψ n H : matrix, entries depend on system E n : energy levels ψ n : energy eigenfunctions

14 Origins of Random Matrix Theory Statistical Mechanics: for each configuration, calculate quantity (say pressure). Average over all configurations most configurations close to system average. Nuclear physics: choose matrix at random, calculate eigenvalues, average over matrices (real Symmetric A = A T, complex Hermitian A T = A).

15 Random Matrix Ensembles A = a 11 a 12 a 13 a 1N a 12 a 22 a 23 a 2N a 1N a 2N a 3N a NN = AT, a ij = a ji Fix p, define Prob(A) = p(a ij ). 1 i j N This means Prob(A : a ij [α ij,β ij ]) = 1 i j N βij Want to understand eigenvalues of A. x ij =α ij p(x ij )dx ij.

16 Eigenvalue Distribution δ(x x 0 ) is a unit point mass at x 0 : f(x)δ(x x0 )dx = f(x 0 ).

17 Eigenvalue Distribution δ(x x 0 ) is a unit point mass at x 0 : f(x)δ(x x0 )dx = f(x 0 ). To each A, attach a probability measure: µ A,N (x) = 1 N N δ i=1 ( x λ ) i(a) 2 N

18 Eigenvalue Distribution δ(x x 0 ) is a unit point mass at x 0 : f(x)δ(x x0 )dx = f(x 0 ). To each A, attach a probability measure: b a µ A,N (x) = 1 N µ A,N (x)dx = # N δ i=1 ( x λ ) i(a) 2 N { λ i : λ i(a) 2 N [a, b] } N

19 Eigenvalue Distribution δ(x x 0 ) is a unit point mass at x 0 : f(x)δ(x x0 )dx = f(x 0 ). To each A, attach a probability measure: b a µ A,N (x) = 1 N µ A,N (x)dx = k th moment = # N δ i=1 ( x λ ) i(a) 2 N { λ i : λ i(a) 2 N [a, b] } N N i=1 λ i(a) k = Trace(Ak ) 2 k N k k N. k 2 +1

20 Wigner s Semi-Circle Law Wigner s Semi-Circle Law N N real symmetric matrices, entries i.i.d.r.v. from a fixed p(x) with mean 0, variance 1, and other moments finite. Then for almost all A, as N µ A,N (x) { 2 π 1 x 2 if x 1 0 otherwise.

21 SKETCH OF PROOF: Eigenvalue Trace Lemma Want to understand the eigenvalues of A, but choose the matrix elements randomly and independently. Eigenvalue Trace Lemma Let A be an N N matrix with eigenvalues λ i (A). Then Trace(A k ) = N λ i (A) k, n=1 where Trace(A k ) = N N a i1 i 2 a i2 i 3 a in i 1. i 1 =1 i k =1

22 SKETCH OF PROOF: Correct Scale Trace(A 2 ) = By the Central Limit Theorem: N λ i (A) 2. i=1 N N Trace(A 2 ) = a ij a ji = N λ i (A) 2 N 2 i=1 i=1 j=1 N N a 2 ij N 2 i=1 j=1 Gives NAve(λ i (A) 2 ) N 2 or Ave(λ i (A)) N.

23 SKETCH OF PROOF: Averaging Formula Recall k-th moment of µ A,N (x) is Trace(A k )/2 k N k/2+1. Average k-th moment is Trace(A k ) p(a 2 k N k/2+1 ij )da ij. i j Proof by method of moments: Two steps Show average of k-th moments converge to moments of semi-circle as N ; Control variance (show it tends to zero as N ).

24 SKETCH OF PROOF: Averaging Formula for Second Moment Substituting into expansion gives N 2 N i=1 N a 2 ji p(a 11 )da 11 p(a NN )da NN j=1 Integration factors as a ij = a 2 ijp(a ij )da ij (k,l) (i,j) k<l a kl = p(a kl )da kl = 1. Higher moments involve more advanced combinatorics (Catalan numbers).

25 SKETCH OF PROOF: Averaging Formula for Higher Moments Higher moments involve more advanced combinatorics (Catalan numbers). 1 2 k N k/2+1 N i 1 =1 N a i1 i 2 a ik i 1 p(a ij )da ij. i k =1 Main contribution when the a il i l+1 s matched in pairs, not all matchings contribute equally (if did would get a Gaussian and not a semi-circle; this is seen in Real Symmetric Palindromic Toeplitz matrices). Distribution of eigenvalues of real symmetric palindromic Toeplitz matrices and circulant matrices (with Adam Massey and John Sinsheimer), Journal of Theoretical Probability 20 (2007), no. 3, i j

26 Numerical examples Distribution of eigenvalues Gaussian, N=400, 500 matrices Matrices: Gaussian p(x) = 1 2π e x2 /2

27 Numerical examples 2500 The eigenvalues of the Cauchy distribution are NOT semicirular Cauchy Distribution: p(x) = 1 π(1+x 2 ) I. Zakharevich, A generalization of Wigner s law, Comm. Math. Phys. 268 (2006), no. 2,

28 GOE Conjecture GOE Conjecture: As N, the probability density of the spacing b/w consecutive normalized eigenvalues approaches a limit independent of p. Until recently only known if p is a Gaussian. GOE(x) π 2 xe πx2 /4.

29 Numerical Experiment: Uniform Distribution Let p(x) = 1 2 for x x 104 The local spacings of the central 3/5 of the eigenvalues of x300 uniform matrices, normalized in batches of : uniform on [ 1, 1]

30 Cauchy Distribution Let p(x) = 1 π(1+x 2 ) The local spacings of the central 3/5 of the eigenvalues of x100 Cauchy matrices, normalized in batches of : Cauchy

31 Cauchy Distribution Let p(x) = 1 π(1+x 2 ). 3.5 x 104 The local spacings of the central 3/5 of the eigenvalues of x300 Cauchy matrices, normalized in batches of : Cauchy

32 Random Graphs Degree of a vertex = number of edges leaving the vertex. Adjacency matrix: a ij = number edges b/w Vertex i and Vertex j. A = These are Real Symmetric Matrices.

33 McKay s Law (Kesten Measure) with d = 3 Density of Eigenvalues for d-regular graphs f(x) = { d 2π(d 2 x 2 ) 4(d 1) x 2 x 2 d 1 0 otherwise.

34 McKay s Law (Kesten Measure) with d = 6 Fat Thin: fat enough to average, thin enough to get something different than semi-circle (though as d recover semi-circle).

35 3-Regular Graph with 2000 Vertices: Comparison with the GOE Spacings between eigenvalues of 3-regular graphs and the GOE:

36 Real Symmetric Toeplitz Matrices Chris Hammond and Steven J. Miller

37 Toeplitz Ensembles Toeplitz matrix is of the form b 0 b 1 b 2 b N 1 b 1 b 0 b 1 b N 2 b 2 b 1 b 0 b N b 1 N b 2 N b 3 N b 0 Will consider Real Symmetric Toeplitz matrices. Main diagonal zero, N 1 independent parameters. Normalize Eigenvalues by N.

38 Eigenvalue Density Measure µ A,N (x)dx = 1 N N i=1 ( δ x λ ) i(a) dx. N The k th moment of µ A,N (x) is Let M k (A, N) = 1 N k 2 +1 M k N i=1 have M 2 = 1 and M 2k+1 = 0. λ k i (A) = Trace(Ak ) N k = lim N E A [M k (A, N)];

39 Even Moments M 2k (N) = 1 N k+1 1 i 1,,i 2k N Main Term: b j s matched in pairs, say E(b i1 i 2 b i2 i 3 b i2k i 1 ). b im i m+1 = b in i n+1, x m = i m i m+1 = i n i n+1. Two possibilities: i m i m+1 = i n i n+1 or i m i m+1 = (i n i n+1 ). (2k 1)!! ways to pair, 2 k choices of sign.

40 Main Term: All Signs Negative (else lower order contribution) M 2k (N) = 1 N k+1 1 i 1,,i 2k N E(b i1 i 2 b i2 i 3 b i2k i 1 ). Let x 1,...,x k be the values of the i j i j+1 s, ǫ 1,...,ǫ k the choices of sign. Define x 1 = i 1 i 2, x 2 = i 2 i 3,.... i 2 = i 1 x 1 i 3 = i 1 x 1 x 2. i 1 = i 1 x 1 x 2k k x x 2k = (1+ǫ j )η j x j = 0, η j = ±1. j=1

41 Even Moments: Summary Main Term: paired, all signs negative. M 2k (N) (2k 1)!!+O k ( 1 N ). Bounded by Gaussian.

42 The Fourth Moment ij ij ij li jk jk jk li li kl kl kl M 4 (N) = 1 E(b N 3 i1 i 2 b i2 i 3 b i3 i 4 b i4 i 1 ) Let x j = i j i j+1. 1 i 1,i 2,i 3,i 4 N

43 The Fourth Moment Case One: x 1 = x 2, x 3 = x 4 : i 1 i 2 = (i 2 i 3 ) and i 3 i 4 = (i 4 i 1 ). Implies i 1 = i 3, i 2 and i 4 arbitrary. Left with E[b 2 x 1 b 2 x 3 ]: N 3 N times get 1, N times get p 4 = E[b 4 x 1 ]. Contributes 1 in the limit.

44 The Fourth Moment M 4 (N) = 1 E(b N 3 i1 i 2 b i2 i 3 b i3 i 4 b i4 i 1 ) 1 i 1,i 2,i 3,i 4 N Case Two: Diophantine Obstruction: x 1 = x 3 and x 2 = x 4. i 1 i 2 = (i 3 i 4 ) and i 2 i 3 = (i 4 i 1 ). This yields i 1 If i 2, i 4 2N 3 choices. = i 2 + i 4 i 3, i 1, i 2, i 3, i 4 {1,...,N}. and i 3 < N 3, i 1 > N: at most ( )N3 valid

45 The Fourth Moment Theorem: Fourth Moment: Let p 4 be the fourth moment of p. Then M 4 (N) = 2 2 ( ) O p 4. N 500 Toeplitz Matrices,

46 Main Result Theorem: HM 05 For real symmetric Toeplitz matrices, the limiting spectral measure converges in probability to a unique measure of unbounded support which is not the Gaussian. If p is even have strong convergence). Massey, Miller and Sinsheimer 07 proved that if first row is a palindrome converges to a Gaussian.

47 Not rescaled. Looking at middle 11 spacings, 1000 Toeplitz matrices ( ), entries iidrv from the standard normal. Poissonian Behavior?

48 Block Circulant Ensemble With Murat Koloğlu, Gene Kopp, Fred Strauch and Wentao Xiong.

49 The Ensemble of m-block Circulant Matrices Symmetric matrices periodic with period m on wrapped diagonals, i.e., symmetric block circulant matrices. 8-by-8 real symmetric 2-block circulant matrix: c 0 c 1 c 2 c 3 c 4 d 3 c 2 d 1 c 1 d 0 d 1 d 2 d 3 d 4 c 3 d 2 c 2 d 1 c 0 c 1 c 2 c 3 c 4 d 3 c 3 d 2 c 1 d 0 d 1 d 2 d 3 d 4. c 4 d 3 c 2 d 1 c 0 c 1 c 2 c 3 d 3 d 4 c 3 d 2 c 1 d 0 d 1 d 2 c 2 c 3 c 4 d 3 c 2 d 1 c 0 c 1 d 1 d 2 d 3 d 4 c 3 d 2 c 1 d 0 Choose distinct entries i.i.d.r.v.

50 Oriented Matchings and Dualization Compute moments of eigenvalue distribution (as m stays fixed and N ) using the combinatorics of pairings. Rewrite: M n (N) = = 1 N n i 1,...,i n N E(a i1 i 2 a i2 i 3 a ini 1 ) 1 η( )m N n 2 +1 d1 ( ) m dl ( ). where the sum is over oriented matchings on the edges {(1, 2),(2, 3),...,(n, 1)} of a regular n-gon.

51 Oriented Matchings and Dualization c 0 c 1 c 2 c 3 c 4 d 3 c 2 d 1 c 1 d 0 d 1 d 2 d 3 d 4 c 3 d 2 c 2 d 1 c 0 c 1 c 2 c 3 c 4 d 3 c 3 d 2 c 1 d 0 d 1 d 2 d 3 d 4 c 4 d 3 c 2 d 1 c 0 c 1 c 2 c 3 d 3 d 4 c 3 d 2 c 1 d 0 d 1 d 2 c 2 c 3 c 4 d 3 c 2 d 1 c 0 c 1 d 1 d 2 d 3 d 4 c 3 d 2 c 1 d 0 Figure: An oriented matching in the expansion for M n (N) = M 6 (8).

52 Contributing Terms As N, the only terms that contribute to this sum are those in which the entries are matched in pairs and with opposite orientation.

53 Only Topology Matters Think of pairings as topological identifications; the contributing ones give rise to orientable surfaces. Contribution from such a pairing is m 2g, where g is the genus (number of holes) of the surface. Proof: combinatorial argument involving Euler characteristic.

54 Computing the Even Moments Theorem: Even Moment Formula M 2k = k/2 g=0 ε g (k)m 2g + O k ( 1 N ), with ε g (k) the number of pairings of the edges of a (2k)-gon giving rise to a genus g surface. J. Harer and D. Zagier (1986) gave generating functions for the ε g (k).

55 55 Intro Classical RMT Toeplitz Block Circulant Checkerboard Matrices L-Functions Qs and Refs Harer and Zagier where k/2 g=0 1+2 ε g (k)r k+1 2g = (2k 1)!! c(k, r) c(k, r)x k+1 = k=0 ( ) r 1+x. 1 x Thus, we write M 2k = m (k+1) (2k 1)!! c(k, m).

56 A multiplicative convolution and Cauchy s residue formula yield the characteristic function of the distribution. φ(t) = = (it) 2k M 2k (2k)! k=0 1 2πim = 1 m e t2 2m z =2 2z 1 m l=1 ( m l = 1 ( t 2 /2m) k c(k, m) m k! k=0 ( (1+z ) 1 m 1 1) 1 z 1 ) ( ) 1 t 2 l 1. (l 1)! m e t2 z/2m dz z

57 57 Intro Classical RMT Toeplitz Block Circulant Checkerboard Matrices L-Functions Qs and Refs Results Fourier transform and algebra yields Theorem: Koloğlu, Kopp and Miller The limiting spectral density function f m (x) of the real symmetric m-block circulant ensemble is given by f m (x) = mx 2 e 2 m 2πm r=0 1 m r (2r)! (2r + 2s)! (r + s)!s! s=0 ( m r + s + 1 ) ( 1 2) s (mx 2 ) r. Fixed m equals m m GOE, as m converges to the semicircle distribution.

58 Results (continued) Figure: Plot for f 1 and histogram of eigenvalues of 100 circulant matrices of size

59 Results (continued) Figure: Plot for f 2 and histogram of eigenvalues of block circulant matrices of size

60 Results (continued) Figure: Plot for f 3 and histogram of eigenvalues of block circulant matrices of size

61 Results (continued) Figure: Plot for f 4 and histogram of eigenvalues of block circulant matrices of size

62 Results (continued) Figure: Plot for f 8 and histogram of eigenvalues of block circulant matrices of size

63 Results (continued) Figure: Plot for f 20 and histogram of eigenvalues of block circulant matrices of size

64 Results (continued) Figure: Plot of convergence to the semi-circle. The Limiting Spectral Measure for Ensembles of Symmetric Block Circulant Matrices (with Murat Koloǧlu, Gene S. Kopp, Frederick W. Strauch and Wentao Xiong), Journal of Theoretical Probability 26 (2013), no. 4,

65 Checkerboard Martrices Joint with Paula Burkhardt, Peter Cohen, Jonathan Dewitt, Max Hlavacek, Eyvindur A. Palsson, Aaditya Sharma, Carsten Sprunger, Yen Nhi Truong Vu, Roger Van Peski and Kevin Yang.

66 Checkerboard Matrices An N N matrix A is a random real symmetric checkerboard matrix if w b 0,1 w b 0,3 w b 0,N 1 b 0,1 w b 1,2 w b 1,4 w A = w b 1,2 w b 2,3 w b 2,N b 0,N 1 w b 2,N 1 w b 4,N 1 w, where the b i,j are i.i.d.r.v, and w is a random variable. It is k-checkerboard if w occurs every k entries in row starting with the entry at the index (i mod k) of the ith row.

67 Evolving Results Last summer proved all but k eigenvalues in bulk and converge to semi-circle, remaining k in blip of size N/k. Evolving Results: Variance in blip like 1/N. There are two peaks; for k = 2 from the (1, 1,...,1) and the (1, 1, 1, 1,...,1, 1) eigenvectors. The limiting spectral distribution in blip is a double delta spike for any k.

68 Evolving Results Last summer proved all but k eigenvalues in bulk and converge to semi-circle, remaining k in blip of size N/k. Evolving Results: Variance in blip like 1/N. False. Is order k. There are two peaks; for k = 2 from the (1, 1,...,1) and the (1, 1, 1, 1,...,1, 1) eigenvectors. False. The limiting spectral distribution in blip is a double delta spike for any k. False. Goes to nice k k GOE!

69 Key ideas Weights: To concentrate in blip, let f(n) = (n 1)x n nx n 1, and set µ blip A;N (x) = 1 ( ) ( ( λ f δ x λ N )). k N/k k Combinatorics: Reduce to k k GOE. λ

70 Results Figure: 2-Checkerboard: N = 100, 100 trials.

71 Results Figure: 2-Checkerboard: N = 150, 100 trials.

72 Results Figure: 2-Checkerboard: N = 200, 100 trials.

73 Results Figure: 2-Checkerboard: N = 250, 100 trials.

74 Results Figure: 2-Checkerboard: N = 300, 100 trials.

75 Results Figure: 2-Checkerboard: N = 350, 100 trials.

76 Introduction to L-Functions

77 77 Intro Classical RMT Toeplitz Block Circulant Checkerboard Matrices L-Functions Qs and Refs Riemann Zeta Function ζ(s) = n=1 1 n s = p prime ( 1 1 p s ) 1, Re(s) > 1.

78 Riemann Zeta Function ζ(s) = n=1 1 n s = p prime ( 1 1 p s ) 1, Re(s) > 1. Unique Factorization: n = p r 1 1 p rm m.

79 Riemann Zeta Function ζ(s) = n=1 1 n s = p prime ( 1 1 p s ) 1, Re(s) > 1. Unique Factorization: n = p r 1 1 p rm m. p ( 1 1 p s ) 1 = [ = n s + ( 1 2 s ) n s. ][ s + ( 1 3 s ) 2 + ]

80 Riemann Zeta Function (cont) ζ(s) = 1 = ( 1 1 ) 1, Re(s) > 1 n s p s n p π(x) = #{p : p is prime, p x} Properties of ζ(s) and Primes:

81 Riemann Zeta Function (cont) ζ(s) = 1 = ( 1 1 ) 1, Re(s) > 1 n s p s n p π(x) = #{p : p is prime, p x} Properties of ζ(s) and Primes: lim s 1 +ζ(s) =, π(x).

82 Riemann Zeta Function (cont) ζ(s) = 1 = ( 1 1 ) 1, Re(s) > 1 n s p s n p π(x) = #{p : p is prime, p x} Properties of ζ(s) and Primes: lim s 1 +ζ(s) =, π(x). ζ(2) = π2, π(x). 6

83 Riemann Zeta Function ζ(s) = n=1 1 n s = p prime ( 1 1 p s ) 1, Re(s) > 1. Functional Equation: ( s ξ(s) = Γ π 2) s 2 ζ(s) = ξ(1 s). Riemann Hypothesis (RH): All non-trivial zeros have Re(s) = 1 2 ; can write zeros as 1 2 +iγ. Observation: Spacings b/w zeros appear same as b/w eigenvalues of Complex Hermitian matrices A T = A.

84 General L-functions L(s, f) = n=1 a f (n) n s = p prime L p (s, f) 1, Re(s) > 1. Functional Equation: Λ(s, f) = Λ (s, f)l(s, f) = Λ(1 s, f). Generalized Riemann Hypothesis (RH): All non-trivial zeros have Re(s) = 1 2 ; can write zeros as 1 2 +iγ. Observation: Spacings b/w zeros appear same as b/w eigenvalues of Complex Hermitian matrices A T = A.

85 Elliptic Curves: Mordell-Weil Group Elliptic curve y 2 = x 3 + ax + b with rational solutions P = (x 1, y 1 ) and Q = (x 2, y 2 ) and connecting line y = mx + b. R Q P R P E P Q E 2P Addition of distinct points P and Q Adding a point P to itself E(Q) E(Q) tors Z r

86 Elliptic curve L-function E : y 2 = x 3 + ax + b, associate L-function L(s, E) = n=1 a E (n) n s = p prime L E (p s ), where a E (p) = p #{(x, y) (Z/pZ) 2 : y 2 x 3 + ax + b mod p}.

87 Elliptic curve L-function E : y 2 = x 3 + ax + b, associate L-function L(s, E) = n=1 a E (n) n s = p prime L E (p s ), where a E (p) = p #{(x, y) (Z/pZ) 2 : y 2 x 3 + ax + b mod p}. Birch and Swinnerton-Dyer Conjecture Rank of group of rational solutions equals order of vanishing of L(s, E) at s = 1/2.

88 Properties of zeros of L-functions infinitude of primes, primes in arithmetic progression. Chebyshev s bias: π 3,4 (x) π 1,4 (x) most of the time. Birch and Swinnerton-Dyer conjecture. Goldfeld, Gross-Zagier: bound for h(d) from L-functions with many central point zeros. Even better estimates for h(d) if a positive percentage of zeros of ζ(s) are at most 1/2 ǫ of the average spacing to the next zero.

89 Distribution of zeros ζ(s) 0 for Re(s) = 1: π(x), π a,q (x). GRH: error terms. GSH: Chebyshev s bias. Analytic rank, adjacent spacings: h(d).

90 Explicit Formula (Contour Integration) ζ (s) ζ(s) = d ds logζ(s) = d ds log p ( 1 p s ) 1

91 Explicit Formula (Contour Integration) ζ (s) ζ(s) = d ds logζ(s) = d ds log p = d log ( 1 p s) ds p ( 1 p s ) 1 = p log p p s 1 p s = p log p p s + Good(s).

92 Explicit Formula (Contour Integration) ζ (s) ζ(s) = d ds logζ(s) = d ds log p = d log ( 1 p s) ds = p p log p p s 1 p s = p log p p s ( 1 p s ) 1 + Good(s). Contour Integration: ζ (s) x s ζ(s) s ds vs log p p ( ) s x ds p s.

93 Explicit Formula (Contour Integration) ζ (s) ζ(s) = d ds logζ(s) = d ds log p = d log ( 1 p s) ds = p p log p p s 1 p s = p log p p s ( 1 p s ) 1 + Good(s). Contour Integration: ζ (s) ζ(s) φ(s)ds vs p log p φ(s)p s ds.

94 Explicit Formula (Contour Integration) ζ (s) ζ(s) = d ds logζ(s) = d ds log p = d log ( 1 p s) ds = p p log p p s 1 p s = p log p p s ( 1 p s ) 1 + Good(s). Contour Integration (see Fourier Transform arising): ζ (s) ζ(s) φ(s)ds vs log p φ(s)e σ log p e it log p ds. Knowledge of zeros gives info on coefficients. p

95 Explicit Formula: Example Dirichlet L-functions: Let φ be an even Schwartz function and L(s,χ) = n χ(n)/ns a Dirichlet L-function from a non-trivial character χ with conductor m and zeros ρ = 1 + iγ 2 χ. Then ( ) log(m/π) φ γ χ = φ(y)dy 2π ρ ( ) log p log p χ(p) log(m/π) φ log(m/π) p 1/2 2 p 2 p log p log(m/π) φ ( 2 log p log(m/π) ) χ 2 (p) p ( 1 ) + O. log m

96 Takeaways Very similar to Central Limit Theorem. Universal behavior: main term controlled by first two moments of Satake parameters, agrees with RMT. First moment zero save for families of elliptic curves. Higher moments control convergence and can depend on arithmetic of family.

97 Modeling lowest zero of L E11 (s,χ d ) with 0 < d < 400, Lowest zero for L E11 (s,χ d ) (bar chart), lowest eigenvalue of SO(2N) with N eff (solid), standard N 0 (dashed).

98 Modeling lowest zero of L E11 (s,χ d ) with 0 < d < 400, Lowest zero for L E11 (s,χ d ) (bar chart); lowest eigenvalue of SO(2N): N eff = 2 (solid) with discretisation, and N eff = 2.32 (dashed) without discretisation. The lowest eigenvalue of Jacobi Random Matrix Ensembles and Painlevé VI, (with E. Dueñez, D. K. Huynh, J. Keating and N. Snaith), Journal of Physics A: Mathematical and Theoretical 43 (2010) (27pp). Models for zeros at the central point in families of elliptic curves (with E. Dueñez, D. K. Huynh, J. Keating and N. Snaith), J. Phys. A: Math. Theor. 45 (2012) (32pp).

99 Open Questions and References

100 Open Questions: Low-lying zeros of L-functions Generalize excised ensembles for higher weight GL 2 families where expect different discretizations. Obtain better estimates on vanishing at the central point by finding optimal test functions for the second and higher moment expansions. Further explore L-function Ratios Conjecture to predict lower order terms in families, compute these terms on number theory side. See Dueñez-Huynh-Keating-Miller-Snaith, Miller, and the Ratios papers.

101 Publications: Random Matrix Theory 1 Distribution of eigenvalues for the ensemble of real symmetric Toeplitz matrices (with Christopher Hammond), Journal of Theoretical Probability 18 (2005), no. 3, Distribution of eigenvalues of real symmetric palindromic Toeplitz matrices and circulant matrices (with Adam Massey and John Sinsheimer), Journal of Theoretical Probability 20 (2007), no. 3, The distribution of the second largest eigenvalue in families of random regular graphs (with Tim Novikoff and Anthony Sabelli), Experimental Mathematics 17 (2008), no. 2, Nuclei, Primes and the Random Matrix Connection (with Frank W. K. Firk), Symmetry 1 (2009), ; doi: /sym Distribution of eigenvalues for highly palindromic real symmetric Toeplitz matrices (with Steven Jackson and Thuy Pham), Journal of Theoretical Probability 25 (2012), The Limiting Spectral Measure for Ensembles of Symmetric Block Circulant Matrices (with Murat Koloǧlu, Gene S. Kopp, Frederick W. Strauch and Wentao Xiong), Journal of Theoretical Probability 26 (2013), no. 4, Distribution of eigenvalues of weighted, structured matrix ensembles (with Olivia Beckwith, Karen Shen), submitted December 2011 to the Journal of Theoretical Probability, revised September The expected eigenvalue distribution of large, weighted d-regular graphs (with Leo Goldmahker, Cap Khoury and Kesinee Ninsuwan), preprint.

102 Publications: L-Functions 1 The low lying zeros of a GL(4) and a GL(6) family of L-functions (with Eduardo Dueñez), Compositio Mathematica 142 (2006), no. 6, Low lying zeros of L functions with orthogonal symmetry (with Christopher Hughes), Duke Mathematical Journal 136 (2007), no. 1, Lower order terms in the 1-level density for families of holomorphic cuspidal newforms, Acta Arithmetica 137 (2009), The effect of convolving families of L-functions on the underlying group symmetries (with Eduardo Dueñez), Proceedings of the London Mathematical Society, 2009; doi: /plms/pdp Low-lying zeros of number field L-functions (with Ryan Peckner), Journal of Number Theory 132 (2012), The low-lying zeros of level 1 Maass forms (with Levent Alpoge), preprint The n-level density of zeros of quadratic Dirichlet L-functions (with Jake Levinson), submitted September 2012 to Acta Arithmetica. 8 Moment Formulas for Ensembles of Classical Compact Groups (with Geoffrey Iyer and Nicholas Triantafillou), preprint 2013.

103 Publications: Elliptic Curves 1 1- and 2-level densities for families of elliptic curves: evidence for the underlying group symmetries, Compositio Mathematica 140 (2004), Variation in the number of points on elliptic curves and applications to excess rank, C. R. Math. Rep. Acad. Sci. Canada 27 (2005), no. 4, Investigations of zeros near the central point of elliptic curve L-functions, Experimental Mathematics 15 (2006), no. 3, Constructing one-parameter families of elliptic curves over Q(T) with moderate rank (with Scott Arms and Álvaro Lozano-Robledo), Journal of Number Theory 123 (2007), no. 2, Towards an average version of the Birch and Swinnerton-Dyer Conjecture (with John Goes), Journal of Number Theory 130 (2010), no. 10, The lowest eigenvalue of Jacobi Random Matrix Ensembles and Painlevé VI, (with Eduardo Dueñez, Duc Khiem Huynh, Jon Keating and Nina Snaith), Journal of Physics A: Mathematical and Theoretical 43 (2010) (27pp). 7 Models for zeros at the central point in families of elliptic curves (with Eduardo Dueñez, Duc Khiem Huynh, Jon Keating and Nina Snaith), J. Phys. A: Math. Theor. 45 (2012) (32pp). 8 Effective equidistribution and the Sato-Tate law for families of elliptic curves (with M. Ram Murty), Journal of Number Theory 131 (2011), no. 1, Moments of the rank of elliptic curves (with Siman Wong), Canad. J. of Math. 64 (2012), no. 1,

104 Publications: L-Function Ratio Conjecture 1 A symplectic test of the L-Functions Ratios Conjecture, Int Math Res Notices (2008) Vol. 2008, article ID rnm146, 36 pages, doi: /imrn/rnm An orthogonal test of the L-Functions Ratios Conjecture, Proceedings of the London Mathematical Society 2009, doi: /plms/pdp A unitary test of the L-functions Ratios Conjecture (with John Goes, Steven Jackson, David Montague, Kesinee Ninsuwan, Ryan Peckner and Thuy Pham), Journal of Number Theory 130 (2010), no. 10, An Orthogonal Test of the L-functions Ratios Conjecture, II (with David Montague), Acta Arith. 146 (2011), An elliptic curve family test of the Ratios Conjecture (with Duc Khiem Huynh and Ralph Morrison), Journal of Number Theory 131 (2011), Surpassing the Ratios Conjecture in the 1-level density of Dirichlet L-functions (with Daniel Fiorilli). submitted September 2012 to Proceedings of the London Mathematical Society.