Finite conductor models for zeros near the central point of elliptic curve L-functions

Transcription

1 Finite conductor models for zeros near the central point of elliptic curve L-functions Steven J Miller Williams College Steven.J.Miller@williams.edu Joint with E. Dueñez, D. Huynh, J. P. Keating, N. C. Snaith, appendix by Siman Marshall Maine - Québec Number Theory Conference University of Maine, October 1 st, 2011

2 Introduction

3 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.

4 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 (GRH): All non-trivial zeros have Re(s) = 1 2 ; can write zeros as 1 2 +iγ.

5 5 Intro Old Theory/Models Data/New Model Ratios Conjecture / Excised Orthogonal Ensemble Explanation 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

6 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.

7 7 Intro Old Theory/Models Data/New Model Ratios Conjecture / Excised Orthogonal Ensemble Explanation 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 + 1.

8 Measures of Spacings: n-level Density and Families Let g i be even Schwartz functions whose Fourier Transform is compactly supported, L(s, f) an L-function with zeros 1 + iγ 2 f and conductor Q f : D n,f (g) = g 1 j 1,...,jn j i ±j k ( γ f,j1 log Q f 2π ) g 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) )

9 n-level Density n-level density: F = F N a family of L-functions ordered by conductors, g k an even Schwartz function: D n,f (g) = 1 ( ) ( ) log Qf log lim N F N 2π γ Qf j 1 ;f g n 2π γ j n;f f F N g 1 j 1,...,jn j i ±j k As N, n-level density converges to g( x )ρ n,g(f) ( x )d x = ĝ( 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 1 of a classical compact group.

10 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: 1 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.

11 Theory and Models

12 Zeros of ζ(s) vs GUE 70 million spacings b/w adjacent zeros of ζ(s), starting at the 10 20th zero (from Odlyzko) versus RMT prediction.

13 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 +1: 1 j, k N r: dε 2r (θ) j<k (cosθ k cosθ j ) 2 j (1 cosθ j ) 2r j dθ j,

14 Random Matrix Models and One-Level Densities Fourier transform of 1-level density: ˆρ 0 (u) = δ(u)+ 1 2 η(u). Fourier transform of 1-level density (Rank 2, Indep): ˆρ 2,Independent (u) = [δ(u)+ 12 ] η(u)+2. Fourier transform of 1-level density (Rank 2, Interaction): ˆρ 2,Interaction (u) = [δ(u)+ 12 ] η(u)+2 + 2( u 1)η(u).

15 Comparing the RMT Models Theorem: M 04 For small support, one-param family of rank r over Q(T): 1 ( ) 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.

16 Data

17 RMT: Theoretical Results (N ) st normalized evalue above 1: SO(even)

18 RMT: Theoretical Results (N ) st normalized evalue above 1: SO(odd)

19 Rank 0 Curves: 1st Norm Zero: 14 One-Param of Rank Figure 4a: 209 rank 0 curves from 14 rank 0 families, log(cond) [3.26, 9.98], median = 1.35, mean = 1.36

20 Rank 0 Curves: 1st Norm Zero: 14 One-Param of Rank Figure 4b: 996 rank 0 curves from 14 rank 0 families, log(cond) [15.00, 16.00], median =.81, mean =.86.

21 Spacings b/w Norm Zeros: Rank 0 One-Param Families over Q(T) All curves have log(cond) [15, 16]; z j = imaginary part of j th normalized zero above the central point; 863 rank 0 curves from the 14 one-param families of rank 0 over Q(T); 701 rank 2 curves from the 21 one-param families of rank 0 over Q(T). 863 Rank 0 Curves 701 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

22 Spacings b/w Norm Zeros: Rank 2 one-param families over Q(T) All curves have log(cond) [15, 16]; z j = imaginary part of the j th norm zero above the central point; 64 rank 2 curves from the 21 one-param families of rank 2 over Q(T); 23 rank 4 curves from the 21 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

23 Rank 2 Curves from Rank 0 & Rank 2 Families over Q(T) All curves have log(cond) [15, 16]; z j = imaginary part of the j th norm zero above the central point; 701 rank 2 curves from the 21 one-param families of rank 0 over Q(T); 64 rank 2 curves from the 21 one-param families of rank 2 over Q(T). 701 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

24 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(1/2, E) discretized. Study matrices in SO(2N eff ) with Λ A (1) ce N. Painlevé VI differential equation solver. Use explicit formulas for densities of Jacobi ensembles. Key input: Selberg-Aomoto integral for initial conditions.

25 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).

26 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.

27 Ratio s Conjecture and Excised Orthogonal Ensembles

28 History Farmer (1993): Considered T 0 ζ(s+α)ζ(1 s +β) ζ(s +γ)ζ(1 s +δ) dt, conjectured (for appropriate values) T (α+δ)(β +γ) T 1 α β(δ β)(γ α) (α+β)(γ +δ) (α+β)(γ +δ).

29 History Farmer (1993): Considered T 0 ζ(s+α)ζ(1 s +β) ζ(s +γ)ζ(1 s +δ) dt, conjectured (for appropriate values) T (α+δ)(β +γ) T 1 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 ( 1 +α, f) 2 ω f L ( 1 +γ, f). f F 2

30 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.

31 Inputs for 1-level density Approximate Functional Equation: L(s, f) = m x a m m s +ǫx L(s) n y a n n 1 s; ǫ sign of the functional equation, X L (s) ratio of Γ-factors from functional equation.

32 Inputs for 1-level density Approximate Functional Equation: L(s, f) = m x a m m s +ǫx L(s) n y a n n 1 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 = 1 R F 2π 2πi ( )g( ) f F γ R F (r) = R α F(α,γ). α=γ=r (c) (1 c)

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

34 Procedure (Recipe) Use approximate functional equation to expand numerator. Expand denominator by generalized Mobius function: cusp form 1 L(s, f) = µ f (h), h s h where µ f (h) is the multiplicative function equaling 1 for h = 1, λ f (p) if n = p, χ 0 (p) if h = p 2 and 0 otherwise.

35 Procedure (Recipe) Use approximate functional equation to expand numerator. Expand denominator by generalized Mobius function: cusp form 1 L(s, f) = µ f (h), h s h where µ f (h) is the multiplicative function equaling 1 for h = 1, 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.

36 Procedure (Recipe) Use approximate functional equation to expand numerator. Expand denominator by generalized Mobius function: cusp form 1 L(s, f) = µ f (h), h s h where µ f (h) is the multiplicative function equaling 1 for h = 1, 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).

37 Procedure (Recipe) Use approximate functional equation to expand numerator. Expand denominator by generalized Mobius function: cusp form 1 L(s, f) = µ f (h), h s h where µ f (h) is the multiplicative function equaling 1 for h = 1, 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.

38 Procedure ( Illegal Steps ) Use approximate functional equation to expand numerator. Expand denominator by generalized Mobius function: cusp form 1 L(s, f) = µ f (h), h s h where µ f (h) is the multiplicative function equaling 1 for h = 1, 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.

39 1-Level Prediction from Ratio s Conjecture A E (α,γ) ( = Y 1 E (α,γ) ( λ(p m )ωe m λ(p) pm(1/2+α) p M m=0 ( ( 1+ p λ(p 2m ) λ(p) p + 1 pm(1+2α) p 1+α+γ where p M + 1 p 1+2γ m=0 Y E (α,γ) = m=1 )) λ(p 2m ) p m(1+2α) λ(p m )ω m+1 ) ) E p 1/2+γ p m(1/2+α) m=0 λ(p 2m+1 ) p m(1+2α) ζ(1+2γ)l E (sym 2, 1+2α) ζ(1+α+γ)l E (sym 2, 1+α+γ). Huynh, Morrison and Miller confirmed Ratios prediction, which is

40 1-Level Prediction from Ratio s Conjecture 1 ( γd L) X g d F(X) γ π d [ ( ) 1 M d = 2LX g(τ) 2 log + Γ (1 + iπτ ) + Γ (1 iπτ ) ] dτ 2π Γ L Γ L d F(X) + 1 g(τ) ζ ( 1 + 2πiτ ) + L ( E sym 2, 1 + 2πiτ ) (M l 1) log M ( L ζ L L E L 2+ l=1 M 2iπτ ) dτ l L 1 log M 1 g(τ) dτ + g(τ) log p λ(p 2k+2 ) λ(p 2k ) dτ L k=0 (k+1)(1+ πiτ M L ) L (p + 1) (k+1)(1+ 2πiτ ) p M k=0 p L 1 [ ( ) M d 2iπτ/L Γ(1 iπτ LX g(τ) L ) ζ(1 + 2iπτ )L L E (sym 2, 1 2iπτ ) L 2π Γ(1 + d F(X) iπτ L ) L E (sym 2, 1) A E ( iπτ L, iπτ ) ] dτ + O(X 1/2+ε ); L

41 Numerics (J. Stopple): 1,003,083 negative fundamental discriminants d [10 12, ] Histogram of normalized zeros (γ 1, about 4 million). Red: main term. Blue: includes O(1/ log X) terms. Green: all lower order terms.

42 Excised Orthogonal Ensemble: Preliminaries Characteristic polynomial of A SO(2N) is Λ A (e iθ, N) := det(i Ae iθ ) = N (1 e i(θk θ) )(1 e i( θk θ) ), k=1 with e ±iθ 1,...,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 (1, N) exp(x).

43 One-Level Densities One-level density R G(N) 1 for a (circular) ensemble G(N): R G(N) 1 (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 1 (θ 1 ) = CX 2πi c+i c i where C X is a normalization constant and R J N 1 (θ 1 ; r 1/2, 1/2) = N 2 Nr exp( rx) R J N r 1 (θ 1 ; r 1/2, 1/2)dr π 0 π 0 N w (r 1/2, 1/2) (cosθ j ) j=1 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 w (α,β) (cosθ) = (1 cosθ) α+1/2 (1+cosθ) β+1/2, α = r 1/2 and β = 1/2.

44 Results With C X normalization constant and P(N, r,θ) defined in terms of Jacobi polynomials, R T X 1 (θ) = CX 2πi N 1 j=0 c+i c i exp( rx) 2 N2 +2Nr N r Γ(2+j)Γ(1/2+j)Γ(r + 1/2+j) Γ(r + N + j) (1 cosθ) r Γ(N + 1)Γ(N + r) P(N, r,θ) dr. 2N + r 1 Γ(N + r 1/2)Γ(N 1/2) 2 1 r Residue calculus implies R T X 1 (θ) = 0 for d(θ,x) < 0 and R T X 1 (θ) = R SO(2N) 1 (θ)+c X b k exp((k + 1/2)X) for d(θ,x) 0, k=0 where d(θ,x) := (2N 1) log 2+log(1 cosθ) X and b k are coefficients arising from the residues. As X, θ fixed, R T X 1 (θ) R SO(2N) 1 (θ).

45 Another Explanation (This section is by Simon Marshall)

46 Waldspurger s special value formula Can explain observed repulsion using Waldspurger s formula and some complex analysis. Waldspurger s formula L(1/2, E χ d ) = κ E c E ( d ) 2 d 1/2, where c E ( d ) Z.

47 Waldspurger s special value formula Can explain observed repulsion using Waldspurger s formula and some complex analysis. Waldspurger s formula L(1/2, E χ d ) = κ E c E ( d ) 2 d 1/2, where c E ( d ) Z. If L(1/2, E χ d ) 0, then L(1/2, E χ d ) E d 1/2.

48 Waldspurger s special value formula Can explain observed repulsion using Waldspurger s formula and some complex analysis. Waldspurger s formula L(1/2, E χ d ) = κ E c E ( d ) 2 d 1/2, where c E ( d ) Z. If L(1/2, E χ d ) 0, then L(1/2, E χ d ) E d 1/2. By combining this with Jensen s formula obtain Theorem: Marshall, 11 Define γ d to be the height of the lowest nonreal zero of L(s, E χ d ). If L(1/2, E χ d ) 0 then we have ln γ d ln d 1/4+O E,ǫ (ln ln d 1+ǫ ).

49 Connection with large-scale zero repulsion Repulsion γ d d 1/4+o(1) on a much smaller scale than the mean zero spacing (ln d ) 1. Why should it imply anything about the rescaled limiting density of zeros?

50 Connection with large-scale zero repulsion Repulsion γ d d 1/4+o(1) on a much smaller scale than the mean zero spacing (ln d ) 1. Why should it imply anything about the rescaled limiting density of zeros? The answer is because it holds for every member of a (conjecturally) large family of L-functions. The family of nonvanishing quadratic twists Define D 0 (E) and L D by D 0 (E) = {d d fundamental, L(1/2, E χ d ) 0}, L D = { γ d ln d : d D 0 (E), D/2 d D}. By minimal rank conjecture, expect D 0 (E) to have D elements of size at most D, so L D D.

51 The limiting distribution vanishes at the origin Suppose L D has limiting distribution of the form ρ(x)dx, ρ a smooth function on [0, ). If ρ vanishes to order r at the origin, then for sufficiently large D we have Prob( x L D : x D 1/(r+1) ) δ > 0.

52 The limiting distribution vanishes at the origin Suppose L D has limiting distribution of the form ρ(x)dx, ρ a smooth function on [0, ). If ρ vanishes to order r at the origin, then for sufficiently large D we have Prob( x L D : x D 1/(r+1) ) δ > 0. Contradicts γ d d 1/4+o(1) if r 2, thus r 3.

53 The limiting distribution vanishes at the origin Suppose L D has limiting distribution of the form ρ(x)dx, ρ a smooth function on [0, ). If ρ vanishes to order r at the origin, then for sufficiently large D we have Prob( x L D : x D 1/(r+1) ) δ > 0. Contradicts γ d d 1/4+o(1) if r 2, thus r 3. Disagrees with the Katz-Sarnak heuristics, as they predict a limiting distribution which does not vanish at the origin. Similar results hold in the case of first order vanishing, using the Gross-Zagier formula in place of Waldspurger. One again finds a disagreement with Katz-Sarnak.