On the low-lying zeros of elliptic curve L-functions

Transcription

1 On the low-lying zeros of elliptic curve L-functions Joint Work with Stephan Baier Liangyi Zhao Nanyang Technological University Singapore

2 The zeros of the Riemann zeta function The number of zeros ρ of ζ(s) with 0 Rρ 1 and 0 < Iρ T satisfies (1) N(T) = T ( T 2π log T 2π 2π ) + O(log T). RH gives all non-trivial zeros of ζ(s) have real part 1/2. Let... γ 2 γ 1 < 0 < γ 1 γ 2... be the imaginary parts of the zeros of ζ(s). By (1), we have ζ n = 1 2π γ nlog γ n n, as n. 1

3 Two Possibilities 1. Fixing an L-function and considering the distribution of the spacings of the imaginary parts of their zeros. This is in the direction of Montgomery s pair correlation conjecture. 2. Considering the distribution of zeros near the critical point (s = 1/2) on average over a family of L-functions. This is what we will do. We consider L-functions L(s, f) associated to a family F of arithmetic or analytic objects f F (like real Dirichlet characters, elliptic curves, cusp forms, etc.).

4 Statistic of low-lying zeros We associate the quantity D(f; φ) = γ f φ ( ) γf 2π log X to L(s, f), where φ is an even Schwarz class test function whose Fourier transform ˆφ has compact support, γ f runs through the imaginary parts of the nontrivial zeros L(s, f), and X is a parameter at our disposal. D(f; φ) represents the density of zeros of L(s, f) near the central point.

5 Let D(F; φ, w) = f F D(f; φ)w(f) be the average density, where w(f) is a suitable weight function. Let W X (F) = w(f) f F be the total weight. Katz and Sarnak made predictions for the average density, for natural families.

6 Notations Let E be an elliptic curve over Q, in the Weierstrass form y 2 = x 3 + ax + b, where a, b Z. We define λ E (p) = p + 1 E p and transform E to an ellipitic curve in global minimal Weierstrass form E : y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6. Let be the discriminant of E. The Hasse-Weil L-function associated with E a,b is given by L(s, E) = p ( 1 λe (p)p s + p 1 2s) 1 p ( 1 λe (p)p s) 1. λ E (p) = λ E (p) for p > 3. The infinite product converges absolutely and uniformly for Rs > 3/2 by the virtue of Hasse s theorem.

7 Birch-Swinnerton-Dyer Conjecture If r is the rank of an elliptic curve E over É, then the Hasse- Weil L-function L(E, s) has a zero of order r at the critical point s = 1. The residue lim s 1 (s 1) r L(E, s) has a concrete expression involving some invariants of E. Coates and Wiles proved that if E has complex multiplication and L(E,1) 0, then r = 0 (E has only a finite number of rational points). Gross and Zagier proved that if E is a modular elliptic curve such that L(E, s) has a first-order zero at s = 1, then r 1 (E has a rational point of infinite order).

8 Kolyvagin proved that a modular elliptic curve E for which L(E,1) 0 has rank 0, and a modular elliptic curve E for which L(E,1) has a first-order zero at s = 1 has rank 1. Wiles, Breuil, Conrad, Diamond and Taylor proved that all elliptic curves E over Q are modular (Taniyama-Shimura theorem), which extends the second and third result above to all elliptic curves over Q.

9 Low-lying zeros of Hasse-Weil L-functions Theorem 1 (Baier, Z.). Let F be the family of elliptic curves given by the Weierstrass equations E a,b : y 2 = x 3 + ax + b with a, b N. Let w C 0 (R+ R + ) and set w X (E a,b ) = w ( a A, b B), where A = X 1/3, B = X 1/2 (X a positive real number). Then (2) D(F; φ, w X ) [ˆφ(0) + 12 ] φ(0) W X (F) as X for φ with supp ˆφ ( 7/10, 7/10). Brumer: ±5/9 in place of ±7/10 under GRH. Heath-Brown: ±2/3 under GRH. Young: ±7/9 under GRH. Random matrix theory predicts (2) holds for arbitrary support of ˆφ.

10 Corollary 1. Assuming GRH for Hasse-Weil L-functions, the family of elliptic curves ordered as in Theorem 1 has average analytic rank r 1/2 + 10/7 = 27/14. Brumer: r 23/10. Heath-Brown: r 2. Young: r 25/14 Random Matrix Theory: r 1/2. Note that the majorant in Corollary 1 is strictly less than 2. Hence a positive proportion of elliptic curves have analytic rank either 0 or 1. By Kolyvagin s theorem, we have Corollary 2. Under the assumption of GRH for Hasse-Weil L-functions, a positive proportion of elliptic curves ordered as in Theorem 1 have algebraic ranks equal to analytic ranks.

11 We have the explicit formula General Approach D(E; φ) = ˆφ(0) log N log X φ(0) P 1(E; φ) P 2 (E; φ) + O where N is the conductor of the elliptic curve E, P 1 (E; φ) = ( ) log p 2log p λ E (p)ˆφ p>3 log X plog X, P 2 (E; φ) = ( ) λ E (p 2 2log p 2log p )ˆφ p>3 log X p 2 log X, X is some scaling parameter and λ E (p) = x mod p ( x 3 + ax + b ( 1 log X To prove Theorem 1, we need to show that the relevant average of P i (E; φ) over the family F satisfies the bound P i (F; φ, w X ) p AB log X. ). ),

12 Estimation of P 2 (F; φ, w X ) The Riemann hypothesis for symmetric square L-functions enabled Young to readily dispose of the contribution P 2 and infer D(E; φ) = ˆφ(0) log N log X + 1 ( ) loglog 2 φ(0) P 1(E; φ) + O. log X What is actually needed is P 2 averaged over the family of elliptic curves under consideration. It can be shown using careful evaluations of quadratic Gauss sums that the relevant average is indeed negligible.

13 Transformation of P 1 (F; φ, w X ) Using the Poisson summation formula, we have that P 1 (F; φ, w X ) = AB log X p>3 It suffices to estimate S(H, K, P) = ψ 4 (p) 2log p ( ) log p ˆφ p3/2 log X h k K k 2K 1 d k 2 ϕ(k 2 /d) where τ(χ) is the usual Gauss sum, ( ) k p χ mod k 2 /d e ( h3 k 2 p ) ŵ ( ha p, kb p τ(χ) χ(d 3 0 /d)q(d, k, χ), ). Q(d, k, χ) = P p<2p H/d 0 h 0 <2H/d 0 ψ 4 (p)χ(p) ( ) k p χ 3 (h 0 )e ( h3 0 d3 0 pk 2 ) U(h 0, d 0, k, p), U(h 0, d 0, k, p) is some smooth weight function of order of magnitute p 3/2, and d 0 is the least positive integer such that d d 3 0.

14 It suffices to show that S(H, K, P) X ε, for H (P/A) 1+ε, K (P/B) 1+ε, and P X 7/10 ε.

15 We need to bound 1 (3) k K d k 2 φ(k 2 /d) Case 1: χ 3 is principal χ mod k 2 /d χ 3 =χ 0 τ(χ)χ(d 3 0 /d)q(d, k, χ). For small d, the oscillations of the exponential factor is large, explore cancellation using a result due to M. Young to bound the sum. We further use rarity of characters χ with χ 3 = χ 0. If d is large, then the oscillations from the exponential sum is weak, so remove it using partial summation. We arrive at a sum involving mean-values of Legendre symbols. To dispose of this, we use Heath-Brown s large sieve inequality for real characters. We find that (3) is X ε if P X 7/10 ε. The appearance of this exponent marks the limit of our method.

16 Case 2: χ 3 is non-trivial If χ 3 is non-trivial with conductor l, we show, using Mellon tranform and residue theorems, that Q(d, k, χ) Q 1 (d, k, χ) + Q 2 (d, k, χ) + Q 3 (d, k, χ), where Q 1 is the contribution from a contour near the line σ = 1/2, Q 2 is contribution from non-trival zeros, and Q 3 (d, k, χ) = 0 if χψ 4 (k/ ) is also non-trivial and if that character is trivial then it is the contribution of the pole of ζ(s) at s = 1.

17 Hence, the contribution under considration is where T i := k K T 1 + T 2 + T 3 + E, d k 2 1 ϕ(k 2 /d) χ mod k 2 /d τ(χ) Q i (d, k, χ) for i = 1,2,3, and E is a negligible error term coming from proper prime powers. The term T 1 can be estimated by using a second moment estimate for Dirichlet L-functions. We find that T 1 X ε if P X 7/9 ε. The exponent 7/9 ε was obtained by M. Young under GRH for Dirichlet L-functions. The term T 2 is estimated by a general zero density theorem.

18 T 3, which is present only when both χ 3 and χψ 4 (k/ ) are trivial, is disposed by essentially trivial considerations. It should be noted that in the case when both of those characters are trivial, in certain range, the exponential sum estimate due to M. Young (which was needed above) is needed to establish the desired bound.

19 Possible Improvements An improvement of the exponent 7/10 is conceivable if we estimate a part of the sum with χ 3 = 1 in certain ranges using large sieve for sextic character, recently established by Baier and Young. We may be able to improve further if we use an alternative method for bounding another part of the sum with χ 3 1 sieve with square moduli, developed both jointly and independently by the authors.

20 Moreover, it would be great to have a completely unconditional majorant for the average analytic rank of all elliptic curves. Unfortunately, this seems to be out of reach due to the presence of the root number of L(E, s) in the approximate functional equation for L(E, s). Writing this root number explicitly, we obtain an expression that contains the term µ(4a b 2 ), µ(n) being the Möbius µ function, which is extremely difficult to handle.