Riemann Hypothesis for Non-Abelian Zeta Functions of Curves over Finite Fields

Abstract

In this paper, we develop some basic techniques towards the Riemann hypothesis for higher rank non-abelian zeta functions of an integral regular projective curve of genus g over a finite field F q . As an application of the Riemann hypothesis for these genuine zeta functions, we obtain some explicit bounds on the fundamental non-abelian αand β-invariants of X/F q in terms of X and n, q and g: α X,Fq ;n (mn) = V q h 0 (X,V) − 1 #Aut(V) and β X,Fq

Derived Zeta Functions for Curves over Finite Fields

“…Remark. In the classical setting, namely m = 0, this special counting miracle is conjectured by the author in [5], and proved first for elliptic curves by Zagier and myself in [8], and later established by Sugahara (see the appendix in [7]). An independent proof for the classical can be found in [3] and [7] as well.…”

“…The importance of this special uniformity of zetas should never be underestimated. For examples, it has been used in [7] to establish the rank three Riemann hypothesis, and as to be seen below plays an important role in defining what we call n m -derived zeta functions for curves over finite fields.…”

“…We start with m = 0, then this theorem is proved by Yoshida. For details, please refer to Theorem 2.2 of [7] and the related discussions.…”

“…The interested reader may find a proof of this lemma, together with a generalization from Lemma 4.5 of [7].…”