Newman s Conjecture for function field L-functions

Transcription

1 Newman s Conjecture for function field L-functions David Mehrle Presenters: t < Λ t Λ Λ Joseph Stahl September 28, 2014 Joint work with: Tomer Reiter, Dylan Yott Advisors: Steve Miller, Alan Chang

2 The Riemann Hypothesis and the ζ Function ζ(s) := n 1 1 n s = p prime (1 p s ) 1 ξ(s) = ξ(1 s), ξ(s) := s(s 1) ( s π s/2 Γ ζ(s) 2 2) Conjecture: Riemann Hypothesis ξ(s) = 0 = R(s) = 1 2

3 Pólya s idea If x R, then Ξ(x) R. ( ) 1 Ξ(x) = ξ 2 + ix Riemann Hypothesis true all zeros of Ξ(x) are real. Ξ Φ(u) = 1 2π 0 Ξ(x) cos ux dx Ξ t (x) = 0 e tu2 Φ(u) cos ux du

4 Newman s Conjecture Theorem (De Bruijn, Newman) There exists Λ R such that if t < Λ, Ξ t has a nonreal zero, and if t Λ, Ξ t has only real zeros. Riemann Hypothesis Λ 0 Conjecture (Newman) Λ 0. The new conjecture is a quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so. - Newman Fact It is known that Λ

5 Function Field Analogy Slogan: Function fields behave a lot like number fields! Number Fields and Function Fields Field K (Q) F q (T ) Ring of Integers O K (Z) F q [T ] Primes p O K ((p) Z) π F q [T ] irreducible Zeta Function ζ K (ζ Q = ζ) Weil Zeta Function Riemann Hypothesis Weil Conjectures

6 Newman setup in Function Fields Idea: Mimic Pólya s setup q = p n, D F q [x], L(s, χ D ) := L(s, χ D ) = f monic ( c n q s ) n, cn = n=0 f monic deg f=n χ D (f) f s χ D (f) L(s, χ D ) is a polynomial in q s of degree deg D 1. L(s, χ D ) ξ(s, χ D ) Ξ(x, χ D ) Ξ t (x, χ D ) g ( Ξ t (x, χ D ) := Φ 0 + Φ n e tn2 e inx + e inx) n=1

7 Newman s Conjecture in Function Fields Lemma (Andrade, Chang, Miller 2013) If Ξ t (x, χ D ) has only real zeros for some t R, then for all t > t, Ξ t (x, χ D ) has only real zeros. Lemma (Andrade, Chang, Miller 2013) There exists Λ D [, 0] such that 1. if t Λ, then Ξ t (x, χ D ) has only real zeros, 2. if t < Λ, then Ξ t (x, χ D ) has a non-real zero.

8 Examples Example D = x 5 + x 4 + x 3 + 2x + 2 F 5 [x] : Ξ t (x, D) = 10e 4t cos 2x 2 5e t cos x 1 Λ D Example D(T ) = T 3 + T F 3 [T ] = Ξ t (x, χ D ) = 3e t cos x Λ D =

9 Newman s Conjecture in Function Fields Conjecture Fix q a power of an odd prime. Then sup Λ D 0. D F q[t ] good Conjecture Fix g N. Then sup D good, deg D=2g+1 q=p k, p 3 Λ D 0.

10 Newman s Conjecture in Function Fields Conjecture Fix D Z[T ] square-free. Let p be prime, and let D p F p [T ] be the polynomial obtained by reducing D modulo p. Then sup D p good p 3 Λ Dp 0.

11 Previous Work Theorem (Andrade, Chang, Miller 2013) Let D Z[x] be square-free with deg D = 3. For each odd prime p, we can reduce D to D p F p [x]. Then sup p Λ Dp = 0. proof sketch. Step 1: Show that Λ Dp = log a p(d) 2 p where a p (D) is the trace of Frobenius. Step 2: Use the Sato Tate conjecture.

12 Motivation For Our Strategy If q = p n is a square, then 2 q Z, so a q (D) can actually equal 2 q. In this case, Λ D = log 1 = 0. Weil conjectures = E/F p with the average number of points will acheive the maximum and minimum number of points possible over particular extensions of F p. Judicious choices of D and p (such that y 2 = D(x) has p + 1 points over F p ) will give us Newman s conjecture in certain cases!

13 The Weil Conjectures for Curves Theorem (Weil Conjectures) Let X be a curve over F q. The Hasse-Weil zeta function of X is defined as Z(X, s) = exp ( m 1 Nm m (q s ) m), where N m is the number of points of X over F q m. 1. Z(X, s) = 2. Z(X, n s) = ±q f(x) Z(X, s). P (T ) (1 T )(1 qt ), P Z[T ] 3. Let α be a root of P. Then α = q 1/2. Example: Z(P 1, s) = 1 (1 T )(1 qt )

14 A Key Lemma and a Key Observation Lemma (Andrade, Chang, Miller 2013) Λ D = 0 L(s, χ D ) has a double root. Observation L(s, χ D ) is the numerator of the zeta function Z(X, s), X : y 2 = D(x). More precisely, Z(X, s) = Z(P 1, s)l(s, χ D ). Proof (Idea) Use the Euler products for Z(X, s), L(s, χ D ). Z(X, s) = L(s, χ D ) = π π monic, irred. (1 N(π)) s ( 1 χ(π)n(π) s ) 1

15 Results Theorem The L-function corresponding to D(x) = x q x has a double root. This implies that Λ D = 0 (considering D over F q ). Proof Sketch The curve X : y 2 = x q x carries an action of F q that commutes with Frobenius. These actions reduce to actions at the level of cohomology H l (X). For X : y2 = x q x, Z(X, s) = Z(P 1, s)l(s, χ D ). Next, recall that the L-function is defined as a Gauss sum. A result of Nick Katz = L(s, χ D ) = (T 2 q ± 1) g.

16 Results Corollary If F is a family of good polynomials over various finite fields, and contains at least one polynomial over F q of the form x q x for some q, then sup Λ D = 0. D F In particular, F = {D F q [T ] D good} and F = {D deg D = 2g + 1, 2g + 1 = p k for some p} are such families. This sup is really a max!

17 Continuing Previous Results Theorem Let D Z[T ] be a square-free monic cubic polynomial. Then there exists a number field K/Q such that sup Λ Dp = max Λ Dp = 0, p O K p O K where D p denotes reduction modulo the prime ideal p.

18 Future Directions Degree greater than 3 case of the third Newman s conjecture. Fix a number field K/Q and a square-free monic cubic D O K [T ]. Does there exist a prime p O K such that Λ Dp = 0 or a sequence of primes {p n } n N such that Λ Dpi 0? Does Newman s conjecture hold for the family F = {D deg D = 2g + 1, g N} when 2g + 1 is not a power of a prime?

19 Acknowledgements Presented by: David Mehrle Joseph Stahl Joint work with: Tomer Reiter Dylan Yott Advised by: Steven J. Miller Alan Chang Special thanks to: The PROMYS Program The SMALL REU Boston University Williams College Funded by: NSF Grants DMS , DMS , the PROMYS Program, and Williams College