Finite conductor models for zeros near the central point of elliptic curve L-functions
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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 and N. C. Snaith Maine - Québec Number Theory Conference University of Maine, October -,
2 Introduction
3 Riemann Zeta Function ζ(s) = n= n s = p prime ( p s ), Re(s) >. Functional Equation: ( s ξ(s) = Γ π ) s ζ(s) = ξ( s). Riemann Hypothesis (RH): All non-trivial zeros have Re(s) = ; can write zeros as +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) = ; can write zeros as +iγ.
5 5 Intro Theory/Models Data Mordell-Weil Group Elliptic curve y = x 3 + ax + b with rational solutions P = (x, y ) and Q = (x, y ) and connecting line y = mx + b. R Q P R P E P Q E P 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 = x 3 + ax + b, associate L-function L(s, E) = n= a E (n) n s = p prime L E (p s ), where a E (p) = p #{(x, y) (Z/pZ) : y 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 = /.
7 7 Intro Theory/Models Data One parameter family E : y = 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 = 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 g i be even Schwartz functions whose Fourier Transform is compactly supported, L(s, f) an L-function with zeros + iγ f and conductor Q f : D n,f (g) = g j,...,jn j i ±j k ( γ f,j log Q f π ) g n ( γ f,jn log Q f π 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) = ( ) ( ) log Qf log lim N F N π γ Qf j ;f g n π γ j n;f f F N g j,...,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 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: 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 7 million spacings b/w adjacent zeros of ζ(s), starting at the th zero (from Odlyzko) versus RMT prediction.
13 Orthogonal Random Matrix Models RMT: SO(N): N eigenvalues in pairs e ±iθ j, probability measure on [,π] N : dǫ (θ) j<k (cosθ k cosθ j ) j dθ j. Independent Model: {( Ir r A N,r = g) } : g SO(N r). Interaction Model: Sub-ensemble of SO(N) with the last r of the N eigenvalues equal +: j, k N r: dε r (θ) j<k (cosθ k cosθ j ) j ( cosθ j ) r j dθ j,
14 Random Matrix Models and One-Level Densities Fourier transform of -level density: ˆρ (u) = δ(u)+ η(u). Fourier transform of -level density (Rank, Indep): ˆρ,Independent (u) = [δ(u)+ ] η(u)+. Fourier transform of -level density (Rank, Interaction): ˆρ,Interaction (u) = [δ(u)+ ] η(u)+ + ( u )η(u).
15 Comparing the RMT Models Theorem: M 4 For small support, one-param family of rank r over Q(T): ( ) log CEt lim ϕ N F N π γ E t,j E t F N j = ϕ(x)ρ G (x)dx + rϕ() 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 : SO(even)
18 RMT: Theoretical Results (N ) st normalized evalue above : SO(odd)
19 Rank Curves: st Norm Zero: 4 One-Param of Rank Figure 4a: 9 rank curves from 4 rank families, log(cond) [3.6, 9.98], median =.35, mean =.36
20 Rank Curves: st Norm Zero: 4 One-Param of Rank Figure 4b: 996 rank curves from 4 rank families, log(cond) [5., 6.], median =.8, mean =.86.
21 Spacings b/w Norm Zeros: Rank 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 curves from the 4 one-param families of rank over Q(T); 7 rank curves from the one-param families of rank over Q(T). 863 Rank Curves 7 Rank Curves t-statistic Median z z.8.3 Mean z z StDev z z.49.5 Median z 3 z..9 Mean z 3 z.4..8 StDev z 3 z.5.47 Median z 3 z Mean z 3 z StDev z 3 z.5.5
22 Spacings b/w Norm Zeros: Rank 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 curves from the one-param families of rank over Q(T); 3 rank 4 curves from the one-param families of rank over Q(T). 64 Rank Curves 3 Rank 4 Curves t-statistic Median z z.6.7 Mean z z StDev z z.5.4 Median z 3 z..8 Mean z 3 z StDev z 3 z Median z 3 z Mean z 3 z StDev z 3 z.44.4
23 Rank Curves from Rank & Rank 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; 7 rank curves from the one-param families of rank over Q(T); 64 rank curves from the one-param families of rank over Q(T). 7 Rank Curves 64 Rank Curves t-statistic Median z z.3.6 Mean z z StDev z z.5.5 Median z 3 z.9. Mean z 3 z StDev z 3 z Median z 3 z Mean z 3 z StDev z 3 z.5.44
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. Discretize Jacobi ensembles. L(/, E) discretized. Study matrices in SO(N 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.
25 Modeling lowest zero of L E (s,χ d ) with < d < 4, Lowest zero for L E (s,χ d ) (bar chart), lowest eigenvalue of SO(N) with N eff (solid), standard N (dashed).
26 Modeling lowest zero of L E (s,χ d ) with < d < 4, Lowest zero for L E (s,χ d ) (bar chart); lowest eigenvalue of SO(N): N eff = (solid) with discretisation, and N eff =.3 (dashed) without discretisation.
27 Example of Decreasing repulsion: r r = x
28 Example of Decreasing repulsion: r r = x
29 Example of Decreasing repulsion: r r = x
30 Example of Decreasing repulsion: r r = x
31 Example of Decreasing repulsion: r r = x
32 Example of Decreasing repulsion: r r = x
33 Example of Decreasing repulsion: r r = x
34 Example of Decreasing repulsion: r r = x
35 Example of Decreasing repulsion: r r = x
36 Example of Decreasing repulsion: r r = x
37 Example of Decreasing repulsion: r r = x
38 Example of Decreasing repulsion: r r = x
39 Example of Decreasing repulsion: r r = x
40 Example of Decreasing repulsion: r r = x
41 Example of Decreasing repulsion: r r = x
42 Example of Decreasing repulsion: r r = x
43 Example of Decreasing repulsion: r r = x
44 Example of Decreasing repulsion: r r = x
45 Example of Decreasing repulsion: r r = x
46 Example of Decreasing repulsion: r r = x
47 Example of Decreasing repulsion: r r = x
48 Example of Decreasing repulsion: r r = x
49 Example of Decreasing repulsion: r r = x
50 Example of Decreasing repulsion: r r =.83333e x
51 Example of Decreasing repulsion: r r = x
52 Numerics (J. Stopple):,3,83 negative fundamental discriminants d [, ] Histogram of normalized zeros (γ, about 4 million). Red: main term. Blue: includes O(/ log X) terms. Green: all lower order terms.
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