Newman's conjecture in function fields

作者：Chang Alan; Mehrle David; Miller Steven J^*; Reiter Tomer; Stahl Joseph; Yott Dylan

来源：Journal of Number Theory, 2015, 157: 154-169.

DOI：10.1016/j.jnt.2015.04.028

• Newman's conjecture
• Zeros of the L-functions
• Function fields

Text. De Bruijn and Newman introduced a deformation of the completed Riemann zeta function zeta, and proved there is a real constant A which encodes the movement of the nontrivial zeros of zeta under the deformation. The Riemann hypothesis is equivalent to the assertion that Lambda <= 0. Newman, however, conjectured that Lambda >= 0, remarking, "the new conjecture is a quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so". Andrade, Chang and Miller extended the machinery developed by Newman and Polya to L-functions for function fields. In this setting we must consider a modified Newman's conjecture: sup(f is an element of F) Lambda(f) >= 0, for a family of L-functions.
We extend their results by proving this modified Newman's conjecture for several families of L-functions. In contrast with previous work, we are able to exhibit specific L-functions for which Lambda(D) = 0, and thereby prove a stronger statement: max(L is an element of F) Lambda(L) = 0. Using geometric techniques, we show a certain deformed L-function must have a double root, which implies Lambda = 0. For a different family, we construct particular elliptic curves with p + 1 points over F-p. By the Weil conjectures, this has either the maximum or minimum possible number of points over F-p2n. The fact that #E(F-p2n) attains the bound tells us that the associated L-function satisfies Lambda = 0.
Video. For a video summary of this paper, please visit http://youtu.be/hM6-pjq7Gi0.

• 出版日期2015-12