Lower Order Biases in Fourier Coefficients of Elliptic Curve and Cuspidal Newform families

Transcription

1 Lower Order Biases in Fourier Coefficients of Ellitic Curve and Cusidal Newform families Jared Lichtman, Steven Miller, Eric Winsor & Jianing Yang with Ryan Chen & Yujin H. Kim Advisor: Steven J. Miller Maine-Québec Number Theory Conference Univ. o. Maine, October XIV, MMXVII

2 Ellitic Curves over Q Interested in ellitic curves over Q E/Q : y 2 = x 3 + ax + b { } where a, b Q and 4a b 2 0, and their reduction mod. Definition (Good reduction) An ellitic curve E/Q : y 2 = x 3 + ax + b has good reduction at a rime if 4a b 2 0 (mod ). The reduction E(F ) is defined as y 2 = x 3 + [a]x + [b], where [a], [b] are the reductions of a and b (mod ).

3 Hasse s Theorem Recall E(F ) := {(x, y) : y 2 = x 3 + ax + b} #E(F ) = ( )) x (1 3 + ax + b + 1 x F = + 1 ( ) x 3 + ax + b. x F Define the Frobenius trace as a E () := + 1 #E(F ), have Hasse bound a E () 2.

4 Families and Moments A one-arameter family of ellitic curves is given by E : y 2 = x 3 + A(T )x + B(T ) where A(T ), B(T ) are olynomials in Z[T ]. Each secialization of T to an integer t gives an ellitic curve E(t) over Q. The r th moment (note not normalizing by 1/) is A r,e () = a E(t) () r, t mod where a E(t) () = + 1 #E t (F ) is the Frobenius trace of E(t).

5 Negative Bias in the First Moment First moment related to the rank of the ellitic curve family. A 1,E () and Family Rank (Rosen-Silverman) Given technical assumtions (Tate s conjecture) related to L-functions associated with E, lim X 1 X X A 1,E () log = rank(e/q).

6 Bias Conjecture The j(t )-invariant is j(t ) = A(T ) 3. 4A(T ) 3 +27B(T ) 2 Second Moment Asymtotic (Michel) For families with j(t ) non-constant, the second moment is A 2,E () = 2 + O( 3/2 ), with lower order terms of sizes 3/2,, 1/2, and 1.

7 Bias Conjecture The j(t )-invariant is j(t ) = A(T ) 3. 4A(T ) 3 +27B(T ) 2 Second Moment Asymtotic (Michel) For families with j(t ) non-constant, the second moment is A 2,E () = 2 + O( 3/2 ), with lower order terms of sizes 3/2,, 1/2, and 1. In every family studied, observe: Bias Conjecture The largest lower term in the second moment exansion which does not average to 0 is on average negative.

8 Relation with Excess Rank Lower order negative bias increases the bound for average rank in families through statistics of zero densities near the central oint. Unfortunately only a small amount, not enough to exlain observed excess rank.

9 1-Parameter Families

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

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

12 Lemma (SMALL 14) Consider a one-arameter family of ellitic curves of the form E : y 2 = P(x)T + Q(x), where P(x), Q(x) Z[x] have degrees at most 3. Then the second moment can be exanded as A 2,E () = ( ) Q(x) 2 ( ) P(x) 2 P(x) 0 x() ( ) P(x)P(y) + (x,y) 0 where (x, y) = (P(x)Q(y) P(y)Q(x)) 2.

13 Second Moments of Linear-coefficient Families We comuted exlicit formulas for the second moments of some one-arameter families with linear coefficients in T : Family y 2 = (ax + b)(cx 2 + dx + e + T ) y 2 = (ax 2 + bx + c)(dx + e + T ) y 2 = x(ax 2 + bx + c + dtx) ( ( A 2,E )) () ( ( )) ( 2 ) { ( ( )) 2 b ac 1 1 ( 1 ) ac if ad 2bc if ad 2bc if b 2 4ac if b 2 4ac y 2 = x(ax + b)(cx + d + Tx) 1

14 Numerics for Higher Even Moments Want to comute all higher moments; however, going beyond the second leads to intractable Legendre sums. Have some numerical results for higher moments. For examle, the 4 th moment of y 2 x 3 + (t + 1)x 2 + tx:

15 Families with Constant j(t )

16 Constant j(t ) invariant families Question: What haens in families with constant j(t )? E(T ) : y 2 = x 3 + A(T )x has j(t ) = 1728, T Z. E(T ) : y 2 = x 3 + B(T ) has j(t ) = 0. For these families we can comute any moment. Comutation is fast when j(t ) is constant.

17 j = 0 Curves Consider E : y 2 = x 3 + B over F. If 2 (mod 3), then a E () = 0. Gauss Six-Order Theorem If 1 (mod 3), can write = a 2 + 3b 2, a 2 (mod 3), b > 0, and 2a B is a sextic residue in F 2a B cubic, non-sextic residue a E () = a ± 3b B quadratic, non-sextic a ± 3b B non-quadratic, non-cubic.

18 Moments of One-Parameter j = 0 Families For r 0, comute k th moment of E T : y 2 = x 3 AT r. Have A k () = 0 when 3(4), and moments determined only by r (mod 6): { 0 k is odd r 1, 5(6) : A k () = ( (2a) k + (a 3b) k + (a + 3b) k) k is even 1 3 r 2, 4(6) : { A k () = 1 ( 3 ( 2a) k + (a 3b) k + (a + 3b) k) A quadratic residue ( (2a) k + ( a 3b) k + ( a + 3b) k) A quadratic nonresidue 1 3 r 3 : A k () = { ( ( 2a) k + (2a) k) A cubic residue ( (a ± 3b) k + ( a 3b) k) A cubic nonresidue.

19 j = 1728 Curves Consider E : y 2 = x 3 Ax over F. If 3 (mod 4), then a E () = 0. Gauss Four-Order Theorem If 1 (mod 4), then write = a 2 + b 2, where b is even and a + b 1 (mod 4). We have: 2a A is a quartic residue a E () = 2a A quadratic, non-quartic residue ±2b A not a quadratic residue.

20 Moments of One-Parameter j = 1728 Families For r 0, consider E(T ) : y 2 = x 3 AT r x. When 3 (mod 4), all moments are 0. Have { 0 k is odd r 1, 3(4) : A k () = ( 1)2 k 1 (a k + b k ) k is even 0 k is odd r 2(4) : A k () = ( 1)(2a) k A quadratic residue, k is even ( 1)(2b) k A quadratic nonresidue, k is even For r 0(4), we get similar but more elaborate results.

21 Bias in L-functions of Cusidal Newforms

22 Cusidal Newforms Definition (Holomorhic Form of Weight k, level N) A holomorhic function f (z) : H C, of moderate growth, ( for) which ( ) az + b a b f = (cz + d) k f (z), Γ cz + d c d 0 (N) where {( ) } a b Γ 0 (N) = SL c d 2 (Z) : c 0 (mod N). Modular forms are eriodic and have a Fourier exansion, if constant term equals 0 called a cus form. A cusidal newform of level N is a cus form that cannot be reduced to a cus form of level M, where M N.

23 Averaging over Weights Let F X,δ,N be the family of cusidal newforms of weights smaller than some ositive X δ of a square-free level N. Averaging over rimes less than X σ, define the r th moment of the family F X,δ,N as: M r,σ (F X,δ,N ) = 1 π(x σ ) 1 <X σ k<x δ H k (N) k<x δ f H k (N) λ r f ().

24 Averaging over Weights Let F X,δ,N be the family of cusidal newforms of weights smaller than some ositive X δ of a square-free level N. Averaging over rimes less than X σ, define the r th moment of the family F X,δ,N as: M r,σ (F X,δ,N ) = 1 π(x σ ) 1 <X σ k<x δ H k (N) k<x δ f H k (N) λ r f (). Study the asymtotic behavior of the moments as N : M r,σ (F X,δ ) = lim M r,σ (F X,δ,N ). N

25 Averaging over Weights Theorem (SMALL 17) Let F X,δ,N be the family of cusidal newforms of weights k X δ of a square-free level N, and M r,σ (F X,δ ) the limiting r th moment of the family as the level N. Then log log X C r/2 + C σ r/2 1 π(x σ ) even r ( ) M r,σ (F X,δ ) = 1 +O + 1 X 2δ π(x σ ) 0 odd r, ( where C n = 1 2n ) n+1 n is the n th Catalan number. Bias for cusidal newforms is a ositive integer, instead of the negative bias in ellitic curve families.

26 An Imortant Tool: Petersson Trace Formula Petersson Trace Formula For any n, m 1, we have Γ(k 1) (4π) k 1 f H k,n (χ 0) λ f () 2 = δ(, )+2πi k c 0(N) S c (, ) c ( ) 4π 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.

27 An Imortant Tool: Petersson Trace Formula [ILS] gives the following bound for the Petersson Trace Formula: { k 1 ϕ(n) δn, 12 λ f (n) = n n 9 7 k N 6 ( ) 7 + O (n, N) n 6 k 3 N 3 0 else f Hk (N) where level N and n are square-free, (n, N 2 ) N, and ϕ(n) denotes the Euler totient function. We also find the following relation that allows us to comute higher moments of cusidal newform families. λ f () r = C(r l, l)λ f ( r 2l ) 0 l r/2 where C(n, k) = ( ) ( n+k k n+k k 1) are numbers in the Catalan s Triangle.

28 Questions for Further Study Does the Bias Conjecture hold for ellitic families with constant j-invariant? Are there cusidal newform families with negative biases in their moments? Does the average bias always occur in the terms of size or 1? How is the Bias Conjecture formulated for all higher even moments? Can they be modeled by olynomials? What other families obey the Bias Conjecture? Kloosterman sums? Higher genus curves?

29 References B. Mackall, S.J. Miller, C. Rati, K. Winsor, Lower-Order Biases in Ellitic Curve Fourier Coefficients in Families, Frobenius Distributions: Lang-Trotter and Sato-Tate Conjectures (David Kohel and Igor Sharlinski, editors), Contemorary Mathematics 663, AMS, Providence, RI 2016.htts: //web.williams.edu/mathematics/sjmiller/ublic_html/math/aers/biascirm30.df S.J. Miller, 1- and 2-level densities for families of ellitic curves: evidence for the underlying grou symmetries, Comositio Mathematica 140 (2004), htt://arxiv.org/df/math/ S.J. Miller, Variation in the number of oints on ellitic curves and alications to excess rank, C. R. Math. Re. Acad. Sci. Canada 27 (2005), no. 4, htt://arxiv.org/abs/math/ S.J. Miller, Investigations of zeros near the central oint of ellitic curve L-functions, Exerimental Mathematics 15 (2006), no. 3, htt://arxiv.org/df/math/ S.J. Miller, Lower order terms in the 1-level density for families of holomorhic cusidal newforms, Acta Arithmetica 137 (2009), htt://arxiv.org/df/ v4. S.J. Miller, S. Wong, Moments of the rank of ellitic curves, Canad. J. of Math. 64 (2012), no. 1, htt://web.williams.edu/mathematics/sjmiller/ublic_html/math/aers/ mwmomentsranksec812final.df

30 Thank you! Questions? Work suorted by NSF Grants DMS and DMS , Dartmouth College, Princeton University and Williams College.