Biases in Moments of Satake Parameters and Models for L-function Zeros

Transcription

1 Biases in Moments of Satake Parameters and Models for L-function Zeros Steven J. Miller (Williams) and Kevin Yang (Harvard) Maine-Québec Number Theory Conference Orono, Maine, October 3, 2015

2 Bias Conjecture for Elliptic Curves With Blake Mackall (Williams), Christina Rapti (Bard) and Karl Winsor (Michigan) s:

3 Last Summer: Families and Moments A one-parameter family of elliptic curves is given by E : y 2 = x 3 + A(T )x + B(T ) where A(T ), B(T ) are polynomials in Z[T ]. Each specialization of T to an integer t gives an elliptic curve E(t) over Q. The r th moment of the Fourier coefficients is A r,e (p) = a E(t) (p) r. t mod p

4 Negative Bias in the First Moment A 1,E (p) and Family Rank (Rosen-Silverman) If Tate s Conjecture holds for E then lim X 1 X p X A 1,E (p) log p p = rank(e/q). By the Prime Number Theorem, A 1,E (p) = rp + O(1) implies rank(e/q) = r.

5 Bias Conjecture Second Moment Asymptotic (Michel) For families E with j(t ) non-constant, the second moment is A 2,E (p) = p 2 + O(p 3/2 ). The lower order terms are of sizes p 3/2, p, p 1/2, and 1.

6 Bias Conjecture Second Moment Asymptotic (Michel) For families E with j(t ) non-constant, the second moment is A 2,E (p) = p 2 + O(p 3/2 ). The lower order terms are of sizes p 3/2, p, p 1/2, and 1. In every family we have studied, we have observed: Bias Conjecture The largest lower term in the second moment expansion which does not average to 0 is on average negative.

7 Preliminary Evidence and Patterns Let n 3,2,p equal the number of cube roots of 2 modulo p, [ ( 3 ) and set ] ) ] 2 c 0 (p) = p, c 1 (p) =, c 3/2 (p) = p x(p) p ( 4x 3 +1 p ) + ( 3 p ). [ x mod p ( x 3 x p Family A 1,E (p) A 2,E (p) y 2 = x 3 + Sx + T 0 p 3 p 2 y 2 = x ( 3) 3 (9T + 1) 2 0 y 2 = x 3 ± 4(4T + 2)x 0 { 2p 2 2p p 2 mod 3 0 p 1 mod 3 { 2p 2 2p p 1 mod 4 0 p 3 mod 4 y 2 = x 3 + (T + 1)x 2 + Tx 0 p 2 2p 1 y 2 = x 3 + x 2 + 2T p 2 2p ( ) 3 p y 2 = x 3 + Tx p p 2 n 3,2,p p 1 + c 3/2 (p) y 2 = x 3 T 2 x + T 2 2p p 2 p c 1 (p) c 0 (p) y 2 = x 3 T 2 x + T 4 2p p 2 p c 1 (p) c 0 (p) y 2 = x 3 + Tx 2 (T + 3)x + 1 2c p,1;4 p p 2 4c p,1;6 p 1 where c p,a;m = 1 if p a mod m and otherwise is 0.

8 Lower order terms and average rank 1 N 2N t=n 2 N γ t 2N t=n ( ) log R φ γ t 2π p log p log R 1 p 2 φ = φ(0) + φ(0) 2 N 2N t=n ( ) 2 log p a t (p) 2 + O log R φ(x) 0 gives upper bound average rank. p ( ) log p 1 log R p φ log p a t (p) log R ( log log R log R ). Expect big-oh term Ω(1/ log R).

9 Implications for Excess Rank Katz-Sarnak s one-level density statistic is used to measure the average rank of curves over a family. More curves with rank than expected have been observed, though this excess average rank vanishes in the limit. Lower-order biases in the moments of families explain a small fraction of this excess rank phenomenon.

10 Methods for Obtaining Explicit Formulas For a family E : y 2 = x 3 + A(T )x + B(T ), we can write a E(t) (p) = ( ) x 3 + A(t)x + B(t) p ( where p ( ) x p x mod p ) is the Legendre symbol modp given by = 1 if x is a non-zero square modulo p 0 if x 0 mod p 1 otherwise.

11 Lemmas on Legendre Symbols Linear and Quadratic Legendre Sums x mod p x mod p ( ) ax + b p ( ) ax 2 + bx + c p = 0 if p a = ( ) a p ( (p 1) a p ) if p b 2 4ac if p b 2 4ac Average Values of Legendre Symbols ( ) The value of for x Z, when averaged over all x p primes p, is 1 if x is a non-zero square, and 0 otherwise.

12 Rank 0 Families Theorem (MMRW 14): Rank 0 Families Obeying the Bias Conjecture For families of the form E : y 2 = x 3 + ax 2 + bx + ct + d, ( ( ) ( )) 3 a A 2,E (p) = p 2 2 3b p p p The average bias in the size p term is 2 or 1, according to whether a 2 3b Z is a non-zero square.

13 Families with Rank Theorem (MMRW 14): Families with Rank For families of the form E : y 2 = x 3 + at 2 x + bt 2, ( ( A 2,E (p) = p 2 p 1 + ) ( 3 + p )) ( 3a ( p x(p) )) 2 x 3 +ax p. These include families of rank 0, 1, and 2. The average bias in the size p terms is 3 or 2, according to whether 3a Z is a non-zero square.

14 Families with Rank Theorem (MMRW 14): Families with Complex Multiplication For families of the form E : y 2 = x 3 + (at + b)x, ( ( )) 1 A 2,E (p) = (p 2 p) 1 +. p The average bias in the size p term is 1. The size p 2 term is not constant, but is on average p 2, and an analogous Bias Conjecture holds.

15 Families with Unusual Distributions of Signs Theorem (MMRW 14): Families with Unusual Signs For the family E : y 2 = x 3 + Tx 2 (T + 3)x + 1, ( ( )) 3 A 2,E (p) = p 2 p p The average bias in the size p term is 2. The family has an usual distribution of signs in the functional equations of the corresponding L-functions.

16 The Size p 3/2 Term Theorem (MMRW 14): Families with a Large Error For families of the form E : y 2 = x 3 + (T + a)x 2 + (bt + b 2 ab + c)x bc, A 2,E (p) = p 2 3p 1 + p ( ) cx(x + b)(bx c) p x mod p The size p 3/2 term is given by an elliptic curve coefficient and is thus on average 0. The average bias in the size p term is 3.

17 General Structure of the Lower Order Terms The lower order terms appear to always have no size p 3/2 term or a size p 3/2 term that is on average 0; exhibit their negative bias in the size p term; be determined by polynomials in p, elliptic curve coefficients, and congruence classes of p (i.e., values of Legendre symbols).

18 Bias Conjecture for Elliptic Curves With Megumi Asada and Eva Fourakis (Williams) s:

19 Summary of Results Dirichlet characters of prime level: bias +1. Holomorphic cusp forms: bias 1/2. r th Symmetric Power F r,x,δ,q : bias +1/48.

20 Dirichlet Family F q Definition Prime q Z and F q = {χ χ 0 (q)} is the family of nontrivial Dirichlet characters of conductor q. The second moment at p is M 2 (F q ; p) := χ F q χ 2 (p). Goal: Compute asymptotics for the sum M 2,X (F q ) = M 2 (F q ; p) = χ 2 (p). p<x p<x χ F q

21 Results for F q Theorem Family F q has positive bias in the second moment of +1.

22 Results for F q Theorem Family F q has positive bias in the second moment of +1. Have M 2 (F q ; p) := χ F q χ 2 (p). From orthogonality relations: { q 2 if p ±1(q); M 2 (F q ; p) = 1 if p ±1(q), Thus p<x M 2 (F q ; p) = Main term size π(x). p<x (q 2) p<x p ±1(q) p ±1(q) 1.

23 Cuspidal Newforms Fix level q = 1. For weight k, consider an orthonormal basis B k,q (χ 0 ) of H k,q (χ 0 ), the space of holomorphic cusp forms on the surface Γ 0 \h of level k and trivial nebentypus. Family F X := k<x k 0(2) B k,q=1 (χ 0 ).

24 An Important Tool: Petersson Trace Formula For any n, m 1, we have Γ(k 1) (4πp) k 1 f B k,q (χ 0 ) λ f (p) 2 = δ(p, p)+2πi k c 0(q) S c (p, p) c ( ) 4πp J k 1 c where λ f (n) is the n-th Hecke eigenvalue of f, δ(m, n) is Kronecker s delta, S c (m, n) is the classical Kloosterman sum, and J k 1 (t) is the k-bessel function.

25 Cusp Newform: F <X We gain asymptotic control over J k 1 (t) by averaging over even weights k.

26 Cusp Newform: F <X We gain asymptotic control over J k 1 (t) by averaging over even weights k. M 2 (F X ; p) = M 2 (H k,1 (χ 0 ); p) = k <X λ f (p) 2 k <X f B k,1 (χ 0 ) where k <X denotes summing over even k.

27 Cusp Newform: F <X We gain asymptotic control over J k 1 (t) by averaging over even weights k. M 2 (F X ; p) = M 2 (H k,1 (χ 0 ); p) = k <X λ f (p) 2 k <X f B k,1 (χ 0 ) where k <X denotes summing over even k. Theorem Let ϕ C0 (R >0) be real-valued, and let X > 1. Then 4 ( ) ( ) k 1 t ϕ J k 1 (t) = ϕ + t ( ) t X X 6X 3 ϕ(2) X k 0(2)

28 Cusp Newform: F <X To handle S c (m, n), we instead compute M 2 (F X ; δ) = p<x δ M 2 (F X ; p) log p. After several substitutions and iterations of integration by parts, M 2 (F X ; δ) = 1 2 X 1+δ X ( ) 1+δ X 1+δ + O 2 log 2 X δ log 3 X δ yields a bias of 1/2.

29 Varying the Level: F X ; δ; ɛ Can also vary the level: M 2 (F X ; δ; ε) = M 2 (F q,x ; δ) q<x ε = q<x ε p<x δ k <X f B k,q (χ 0 ) λ f (p) 2 log p = 1 2 X 1+δ+ε X ( ) 1+δ+ε X 1+δ+ε + O. 2 log 2 X δ log 3 X δ

30 Symmetric Lift Family Fix a square-free level q and study for δ > 0 F r,x,δ,q = [ ] Sym r H k,q (χ 0 ). k<x δ Second moment: for ε > 0: M 2,ε (F r,x,δ,q ) = 1 ϕ(q) p<x ε k<x δ f H k,q (χ 0) λ 2 Sym f(p), r find bias of +1/48 in M 2,ε (F r,x,δ ) = lim q q sq free M 2,ε (F r,x,δ,q ).

31 Finite Conductor Models at Central Point With Owen Barrett and Blaine Talbut (Chicago) and Gwyn Moreland (Michigan) s: Excised Orthogonal Ensemble joint with Eduardo Dueñez, Duc Khiem Huynh, Jon Keating and Nina Snaith. Numerical experiments ongoing with Nathan Ryan.

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

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

34 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

35 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.

36 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

37 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

38 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

39 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.

40 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).

41 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.

42 Effective Matrix Size: Families with Unitary Symplectic Monodromy L-function attached to quadratic Dirichlet character. L(χ, s) = p< (1 χ(p)p s ) 1. L-function attached to symmetric power. L(Sym r f, s) = p< L p(sym r f, s). Compute 1-level Density: Study distribution of zeros D 1,ϕ (F) = #F 1 f F ρ f =1/2+iγ f ϕ(γ f log Q 2π )

43 Integral Representation of One-Level Density We bound conductors of families by a parameter X For quadratic Dirichlet characters, we have: Theorem The One-Level Density is represented by the integral kernel K (τ) = 1 sin(2πτ) + 1 cos(2πτ) ( ) 1 2πτ Λ 1 + O log X log 2 X for Λ < 0. Similarly for the family of quadratic twists of Sym r f.

44 Deducing Effective Matrix Size Matching with integral kernel of matrix groups. π K N 1,USp(2N)(t) = 1 sin(2πt) + 1 cos(2πt) πt 2N π K N 1,SO(2N+1)(t), same leading term. Note π N (K 1,SO(2N+1) K 1,USp(2( N)) ) 1 cos(2πt) 2N Unitary Symplectic Families behave like SO(2N + 1) for bounded X. Similarly for quadratic twists of Sym 2 f.

45 Excised Orthogonal Ensemble As before, let F be those quadratic twists of L(E, s). Idea: interpret L(E, 1 + it) as an integral kernel. 2 Taylor Series expansion: L(E, s) = L(E, 1 2 ) + L (E, 1 2 )(s 1 2 ) +... Goal: match power series coefficients with that of ch H (e iθ ). Amalgamate integral kernels together: attach to F a product distribution E F L(E, 1 + it)dt. 0 2

46 Excised Orthogonal Ensemble (continued) We deduce Theorem Let F X be those quadratic( twists of an elliptic curve ) E/Q of L conductor N < X. If sup (n) n (E, 1 2 ) ch(n) (1) < δ, then L D 1,FX D < ε. 1,MN(X) 2

47 References

48 References Biases: 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, Lower order terms in the 1-level density for families of holomorphic cuspidal newforms, Acta Arithmetica 137 (2009), Moments of the rank of elliptic curves (with Siman Wong), Canad. J. of Math. 64 (2012), no. 1, mwmomentsranksec812final.pdf

49 References Central: Investigations of zeros near the central point of elliptic curve L-functions, Experimental Mathematics 15 (2006), no. 3, 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). 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).