From the Manhattan Project to Elliptic Curves

Transcription

1 From the Manhattan Project to Elliptic Curves Steven J Miller Dept of Math/Stats, Williams College sjm@williams.edu, Steven.Miller.MC.96@aya.yale.edu Joint with E. Dueñez, D. Huynh, J. P. Keating, N. C. Snaith CNTA XV, Université Laval, July 9, 208

2 Introduction

3 Riemann Zeta Function ζ(s) = n= n s = p prime ( p s ), Re(s) >. Functional Equation: ( s ξ(s) = Γ π 2) s 2 ζ(s) = ξ( s). Riemann Hypothesis (RH): All non-trivial zeros have Re(s) = 2 ; can write zeros as 2 +iγ.

4 General L-functions L(s, f) = n= a f (n) n s = p prime L p (s, f), Re(s) >. Functional Equation: Λ(s, f) = Λ (s, f)l(s, f) = Λ( s, f). Generalized Riemann Hypothesis (GRH): All non-trivial zeros have Re(s) = 2 ; can write zeros as 2 +iγ.

5 5 Mordell-Weil Group Elliptic curve y 2 = x 3 + ax + b with rational solutions P = (x, y ) 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

6 Elliptic curve L-function E : y 2 = x 3 + ax + b, associate L-function L(s, E) = n= a E (n)/ 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 = /2.

7 7 One parameter family E : y 2 = x 3 + A(T)x + B(T), A(T), B(T) Z[T]. Silverman s Specialization Theorem Assume (geometric) rank of E/Q(T) is r. Then for all t Z sufficiently large, each E t : y 2 = x 3 + A(t)x + B(t) has (geometric) rank at least r. Average rank conjecture For a generic one-parameter family of rank r over Q(T), expect in the limit half the specialized curves have rank r and half have rank r +.

8 Measures of Spacings: n-level Density and Families Let φ i be even Schwartz functions whose Fourier Transform is compactly supported, L(s, f) an L-function with zeros + iγ 2 f and conductor Q f : D n,f (φ) = φ j,...,jn j i ±j k ( γ f,j log Q f 2π ) φ n ( γ f,jn log Q f 2π )

9 Measures of Spacings: n-level Density and Families Let φ i be even Schwartz functions whose Fourier Transform is compactly supported, L(s, f) an L-function with zeros + iγ 2 f and conductor Q f : D n,f (φ) = φ j,...,jn j i ±j k ( γ f,j log Q f 2π ) φ n ( γ f,jn log Q f 2π ) Properties of n-level density: Individual zeros contribute in limit.

10 Measures of Spacings: n-level Density and Families Let φ i be even Schwartz functions whose Fourier Transform is compactly supported, L(s, f) an L-function with zeros + iγ 2 f and conductor Q f : D n,f (φ) = φ j,...,jn j i ±j k ( γ f,j log Q f 2π ) φ n ( γ f,jn log Q f 2π ) Properties of n-level density: Individual zeros contribute in limit. Most of contribution is from low zeros.

11 Measures of Spacings: n-level Density and Families Let φ i be even Schwartz functions whose Fourier Transform is compactly supported, L(s, f) an L-function with zeros + iγ 2 f and conductor Q f : D n,f (φ) = φ j,...,jn j i ±j k ( γ f,j log Q f 2π ) φ n ( γ f,jn log Q f 2π ) Properties of n-level density: Individual zeros contribute in limit. Most of contribution is from low zeros. Average over similar L-functions (family).

12 n-level Density n-level density: F = F N a family of L-functions ordered by conductors, φ k an even Schwartz function: D n,f (φ) = ( ) ( ) log Qf log lim N F N 2π γ Qf j ;f φ n 2π γ j n;f f F N φ j,...,jn j i ±j k As N, n-level density converges to φ( x )ρ n,g(f) ( x )d x = φ( u) ρ n,g(f) ( u)d u. Conjecture (Katz-Sarnak) (In the limit) Scaled distribution of zeros near central point agrees with scaled distribution of eigenvalues near of a classical compact group.

13 Correspondences Similarities between L-Functions and Nuclei: Zeros Schwartz test function Support of test function Energy Levels Neutron Neutron Energy.

14 Testing Random Matrix Theory Predictions Know the right model for large conductors, searching for the correct model for finite conductors. In the limit must recover the independent model, and want to explain data on: Excess Rank: Rank r one-parameter family over Q(T): observed percentages with rank r First (Normalized) Zero above Central Point: Influence of zeros at the central point on the distribution of zeros near the central point.

15 Theory and Models

16 Orthogonal Random Matrix Models RMT: SO(2N): 2N eigenvalues in pairs e ±iθ j, probability measure on [0,π] N : dǫ 0 (θ) j<k (cosθ k cosθ j ) 2 j dθ j. Independent Model: {( I2r 2r A 2N,2r = g) } : g SO(2N 2r). Interaction Model: Sub-ensemble of SO(2N) with the last 2r of the 2N eigenvalues equal +: j, k N r: dε 2r (θ) j<k (cosθ k cosθ j ) 2 j ( cosθ j ) 2r j dθ j,

17 Random Matrix Models and One-Level Densities Fourier transform of -level density: ˆρ 0 (u) = δ(u)+ 2 η(u). Fourier transform of -level density (Rank 2, Indep): ˆρ 2,Independent (u) = [δ(u)+ 2 ] η(u)+2. Fourier transform of -level density (Rank 2, Interaction): ˆρ 2,Interaction (u) = [δ(u)+ 2 ] η(u)+2 + 2( u )η(u).

18 Comparing the RMT Models Theorem: M 04 For small support, one-param family of rank r over Q(T): ( ) log CEt lim ϕ N F N 2π γ E t,j E t F N j = ϕ(x)ρ G (x)dx + rϕ(0) where G = { SO if half odd SO(even) if all even SO(odd) if all odd. Supports Katz-Sarnak, B-SD, and Independent model in limit.

19 Data

20 RMT: Theoretical Results (N ) st normalized evalue above : SO(even)

21 RMT: Theoretical Results (N ) st normalized evalue above : SO(odd)

22 Rank 0 Curves: st Norm Zero: 4 One-Param of Rank Figure 4a: 209 rank 0 curves from 4 rank 0 families, log(cond) [3.26, 9.98], median =.35, mean =.36

23 Rank 0 Curves: st Norm Zero: 4 One-Param of Rank Figure 4b: 996 rank 0 curves from 4 rank 0 families, log(cond) [5.00, 6.00], median =.8, mean =.86.

24 Rank 2 Curves from y 2 = x 3 T 2 x + T 2 (Rank 2 over Q(T)) st Normalized Zero above Central Point Figure 5a: 35 curves, log(cond) [7.8, 6.], µ =.85, µ =.92, σ µ =.4

25 Rank 2 Curves from y 2 = x 3 T 2 x + T 2 (Rank 2 over Q(T)) st Normalized Zero above Central Point Figure 5b: 34 curves, log(cond) [6.2, 23.3], µ =.37, µ =.47, σ µ =.34

26 Spacings b/w Norm Zeros: Rank 0 One-Param Families over Q(T) All curves have log(cond) [5, 6]; z j = imaginary part of j th normalized zero above the central point; 863 rank 0 curves from the 4 one-param families of rank 0 over Q(T); 70 rank 2 curves from the 2 one-param families of rank 0 over Q(T). 863 Rank 0 Curves 70 Rank 2 Curves t-statistic Median z 2 z Mean z 2 z StDev z 2 z Median z 3 z Mean z 3 z StDev z 3 z Median z 3 z Mean z 3 z StDev z 3 z

27 Spacings b/w Norm Zeros: Rank 2 one-param families over Q(T) All curves have log(cond) [5, 6]; z j = imaginary part of the j th norm zero above the central point; 64 rank 2 curves from the 2 one-param families of rank 2 over Q(T); 23 rank 4 curves from the 2 one-param families of rank 2 over Q(T). 64 Rank 2 Curves 23 Rank 4 Curves t-statistic Median z 2 z Mean z 2 z StDev z 2 z Median z 3 z Mean z 3 z StDev z 3 z Median z 3 z Mean z 3 z StDev z 3 z

28 Rank 2 Curves from Rank 0 & Rank 2 Families over Q(T) All curves have log(cond) [5, 6]; z j = imaginary part of the j th norm zero above the central point; 70 rank 2 curves from the 2 one-param families of rank 0 over Q(T); 64 rank 2 curves from the 2 one-param families of rank 2 over Q(T). 70 Rank 2 Curves 64 Rank 2 Curves t-statistic Median z 2 z Mean z 2 z StDev z 2 z Median z 3 z Mean z 3 z StDev z 3 z Median z 3 z Mean z 3 z StDev z 3 z

29 Summary of Data The repulsion of the low-lying zeros increased with increasing rank, and was present even for rank 0 curves. As the conductors increased, the repulsion decreased. Statistical tests failed to reject the hypothesis that, on average, the first three zeros were all repelled equally (i. e., shifted by the same amount).

30 Convergence to the RMT limit: What s the right matrix size? RMT + Katz-Sarnak: Limiting behavior for random matrices as N and L-functions as conductors tend to infinity agree. How well do the classical matrix groups model local statistics of L-functions outside the scaling limit? (Arithmetic enters!)

31 Convergence to the RMT limit L: 70 million ζ(s) nearest-neighbor spacings (Odlyzko). R: Difference b/w ζ(s) and asymptotic CUE curve (dots) compared to difference b/w CUE of size N 0 and asymptotic curve (dashed line) (from Bogomolny et. al.).

32 Convergence to the RMT limit: Incorporating Finite Matrix Size Difference b/w nearest-neighbor spacing of ζ(s) zeros and asymptotic CUE for a billion zeros in window near (dots) compared to theory that takes into account arithmetic of lower order terms (full line) (from Bogomolny et. al.). New model should incorporate finite matrix size...

33 New Model for Finite Conductors Replace conductor N with N effective. Arithmetic info, predict with L-function Ratios Conj. Do the number theory computation. Excised Orthogonal Ensembles. L(/2, E) discretized. Study matrices in SO(2N eff ) with Λ A () ce N. Painlevé VI differential equation solver. Use explicit formulas for densities of Jacobi ensembles. Key input: Selberg-Aomoto integral for initial conditions.

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

35 Modeling lowest zero of L E (s,χ d ) with 0 < d < 400, Lowest zero for L E (s,χ d ) (bar chart); lowest eigenvalue of SO(2N): N eff = 2 (solid) with discretisation, and N eff = 2.32 (dashed) without discretisation.

36 Ratio s Conjecture

37 History Farmer (993): Considered T 0 ζ(s+α)ζ( s +β) ζ(s +γ)ζ( s +δ) dt, conjectured (for appropriate values) T (α+δ)(β +γ) T α β(δ β)(γ α) (α+β)(γ +δ) (α+β)(γ +δ).

38 History Farmer (993): Considered T 0 ζ(s+α)ζ( s +β) ζ(s +γ)ζ( s +δ) dt, conjectured (for appropriate values) T (α+δ)(β +γ) T α β(δ β)(γ α) (α+β)(γ +δ) (α+β)(γ +δ). Conrey-Farmer-Zirnbauer (2007): conjecture formulas for averages of products of L-functions over families: R F = L ( +α, f) 2 ω f L ( +γ, f). f F 2

39 Uses of the Ratios Conjecture Applications: n-level correlations and densities; mollifiers; moments; vanishing at the central point; Advantages: RMT models often add arithmetic ad hoc; predicts lower order terms, often to square-root level.

40 Inputs for -level density Approximate Functional Equation: L(s, f) = m x a m m s +ǫx L(s) n y a n n s; ǫ sign of the functional equation, X L (s) ratio of Γ-factors from functional equation.

41 Inputs for -level density Approximate Functional Equation: L(s, f) = m x a m m s +ǫx L(s) n y a n n s; ǫ sign of the functional equation, X L (s) ratio of Γ-factors from functional equation. Explicit Formula: g Schwartz test function, ( ω f g γ log N ) f = R F 2π 2πi ( )g( ) f F γ R F (r) = R α F(α,γ). α=γ=r (c) ( c)

42 Procedure (Recipe) Use approximate functional equation to expand numerator.

43 Procedure (Recipe) Use approximate functional equation to expand numerator. Expand denominator by generalized Mobius function: cusp form L(s, f) = µ f (h), h s h where µ f (h) is the multiplicative function equaling for h =, λ f (p) if n = p, χ 0 (p) if h = p 2 and 0 otherwise.

44 Procedure (Recipe) Use approximate functional equation to expand numerator. Expand denominator by generalized Mobius function: cusp form L(s, f) = µ f (h), h s h where µ f (h) is the multiplicative function equaling for h =, λ f (p) if n = p, χ 0 (p) if h = p 2 and 0 otherwise. Execute the sum over F, keeping only main (diagonal) terms.

45 Procedure (Recipe) Use approximate functional equation to expand numerator. Expand denominator by generalized Mobius function: cusp form L(s, f) = µ f (h), h s h where µ f (h) is the multiplicative function equaling for h =, λ f (p) if n = p, χ 0 (p) if h = p 2 and 0 otherwise. Execute the sum over F, keeping only main (diagonal) terms. Extend the m and n sums to infinity (complete the products).

46 Procedure (Recipe) Use approximate functional equation to expand numerator. Expand denominator by generalized Mobius function: cusp form L(s, f) = µ f (h), h s h where µ f (h) is the multiplicative function equaling for h =, λ f (p) if n = p, χ 0 (p) if h = p 2 and 0 otherwise. Execute the sum over F, keeping only main (diagonal) terms. Extend the m and n sums to infinity (complete the products). Differentiate with respect to the parameters.

47 Procedure ( Illegal Steps ) Use approximate functional equation to expand numerator. Expand denominator by generalized Mobius function: cusp form L(s, f) = µ f (h), h s h where µ f (h) is the multiplicative function equaling for h =, λ f (p) if n = p, χ 0 (p) if h = p 2 and 0 otherwise. Execute the sum over F, keeping only main (diagonal) terms. Extend the m and n sums to infinity (complete the products). Differentiate with respect to the parameters.

48 -Level Prediction from Ratio s Conjecture A E (α,γ) ( = Y E (α,γ) ( λ(p m )ωe m λ(p) pm(/2+α) p M m=0 ( ( + p λ(p 2m ) λ(p) p + pm(+2α) p +α+γ where p M + p +2γ m=0 Y E (α,γ) = m= )) λ(p 2m ) p m(+2α) λ(p m )ω m+ ) ) E p /2+γ p m(/2+α) m=0 λ(p 2m+ ) p m(+2α) ζ(+2γ)l E (sym 2, +2α) ζ(+α+γ)l E (sym 2, +α+γ). Huynh, Morrison and Miller confirmed Ratios prediction, which is

49 -Level Prediction from Ratio s Conjecture ( γd L) X g d F(X) γ π d [ ( ) M d = 2LX g(τ) 2 log + Γ ( + iπτ ) + Γ ( iπτ ) ] dτ 2π Γ L Γ L d F(X) + g(τ) ζ ( + 2πiτ ) + L ( E sym 2, + 2πiτ ) (M l ) log M ( L ζ L L E L 2+ l= M 2iπτ ) dτ l L log M g(τ) dτ + g(τ) log p λ(p 2k+2 ) λ(p 2k ) dτ L k=0 (k+)(+ πiτ M L ) L (p + ) (k+)(+ 2πiτ ) p M k=0 p L [ ( ) M d 2iπτ/L Γ( iπτ LX g(τ) L ) ζ( + 2iπτ )L L E (sym 2, 2iπτ ) L 2π Γ( + d F(X) iπτ L ) L E (sym 2, ) A E ( iπτ L, iπτ ) ] dτ + O(X /2+ε ); L

50 Numerics (J. Stopple):,003,083 negative fundamental discriminants d [0 2, ] Histogram of normalized zeros (γ, about 4 million). Red: main term. Blue: includes O(/ log X) terms. Green: all lower order terms.

51 Excised Orthogonal Ensembles

52 Excised Orthogonal Ensemble: Preliminaries Characteristic polynomial of A SO(2N) is Λ A (e iθ, N) := det(i Ae iθ ) = N ( e i(θk θ) )( e i( θk θ) ), k= with e ±iθ,...,e ±iθ N the eigenvalues of A. Motivated by the arithmetical size constraint on the central values of the L-functions, consider Excised Orthogonal Ensemble T X : A SO(2N) with Λ A (, N) exp(x).

53 One-Level Densities One-level density R G(N) for a (circular) ensemble G(N): R G(N) (θ) = N... P(θ,θ 2,...,θ N )dθ 2...dθ N, where P(θ,θ 2,...,θ N ) is the joint probability density function of eigenphases.

54 One-Level Densities One-level density R G(N) for a (circular) ensemble G(N): R G(N) (θ) = N... P(θ,θ 2,...,θ N )dθ 2...dθ N, where P(θ,θ 2,...,θ N ) is the joint probability density function of eigenphases. The one-level density excised orthogonal ensemble: R T X (θ ) := C X N π 0 π 0 H(log Λ A (, N) X) j<k(cosθ j cosθ k ) 2 dθ 2 dθ N, Here H(x) denotes the Heaviside function { for x > 0 H(x) = 0 for x < 0, and C X is a normalization constant

55 One-Level Densities One-level density R G(N) for a (circular) ensemble G(N): R G(N) (θ) = N... P(θ,θ 2,...,θ N )dθ 2...dθ N, where P(θ,θ 2,...,θ N ) is the joint probability density function of eigenphases. The one-level density excised orthogonal ensemble: R T X (θ ) = CX 2πi c+i c i where C X is a normalization constant and R J N (θ ; r /2, /2) = N 2 Nr exp( rx) R J N r (θ ; r /2, /2)dr π 0 π 0 N w (r /2, /2) (cosθ j ) j= j<k(cosθ j cosθ k ) 2 dθ 2 dθ N is the one-level density for the Jacobi ensemble J N with weight function 55 w (α,β) (cosθ) = ( cosθ) α+/2 (+cosθ) β+/2, α = r /2 and β = /2.

56 Results With C X normalization constant and P(N, r,θ) defined in terms of Jacobi polynomials, R T X (θ) = CX 2πi N j=0 c+i c i exp( rx) 2 N2 +2Nr N r Γ(2+j)Γ(/2+j)Γ(r + /2+j) Γ(r + N + j) ( cosθ) r Γ(N + )Γ(N + r) P(N, r,θ) dr. 2N + r Γ(N + r /2)Γ(N /2) 2 r Residue calculus implies R T X (θ) = 0 for d(θ,x) < 0 and R T X (θ) = R SO(2N) (θ)+c X b k exp((k + /2)X) for d(θ,x) 0, k=0 where d(θ,x) := (2N ) log 2+log( cosθ) X and b k are coefficients arising from the residues. As X, θ fixed, R T X (θ) R SO(2N) (θ).

57 57 Numerical check.2 R_: formula R_: data Figure: One-level density of excized SO(2N), N = 2 with cut-off Λ A (, N) 0.. The red curve uses our formula. The blue crosses give the empirical one-level density of 200,000 numerically generated matrices.

58 Theory vs Experiment Figure: Cumulative probability density of the first eigenvalue from numerically generated matrices A SO(2N std ) with Λ A (, N std ) 2.88 exp( N std /2) and N std = 2 red dots compared with the first zero of even quadratic twists L E (s,χ d ) with prime fundamental discriminants 0 < d 400,000 blue crosses. The random matrix data is scaled so that the means of the two distributions agree.

59 Conclusion and References

60 Conclusion and Future Work In the limit: Birch and Swinnerton-Dyer, Katz-Sarnak appear true. Finite conductors: model with Excised Ensembles (cut-off on characteristic polynomials due to discretization at central point). Future Work: Joint with Owen Barrett and Nathan Ryan (and possibly some of his students): looking at other GL2 families (and hopefully higher) to study the relationship between repulsion at finite conductors and central values (effect of weight, level).

61 References - and 2-level densities for families of elliptic curves: evidence for the underlying group symmetries, Compositio Mathematica 40 (2004), Investigations of zeros near the central point of elliptic curve L-functions, Experimental Mathematics 5 (2006), no. 3, Lower order terms in the -level density for families of holomorphic cuspidal newforms, Acta Arithmetica 37 (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: 0.2/plms/pdp08. 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 (200) (27pp). 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 (202) 5207 (32pp).