Moduli spaces of stable pairs and non-abelian zeta functions of curves via wall-crossing

Abstract

Abstract. In this paper we study and relate the non-abelian zeta functions introduced by Weng and invariants of the moduli spaces of arbitrary rank stable pairs over curves. We prove a wall-crossing formula for the latter invariants and obtain an explicit formula for these invariants in terms of the motive of a curve. Previously, formulas for these invariants were known only for rank 2 due to Thaddeus and for rank 3 due to Muñoz. Using these results we obtain an explicit formula for the non-abelian zeta functions, we check the uniformity conjecture by Weng for the ranks 2 and 3, and we prove the counting miracle conjecture.

Motives and the Hodge conjecture for moduli spaces of pairs

“…This misses the geometrical description, but it might be applied for arbitrary rank. Indeed, very recently, after the submission of this paper, we have been communicated that this has been undertaken in [19].…”

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

“…When F is a function field, a totally different approach has been used, thanks to an unexpected work of Mozgovoy-Reineke [14]. The uniformity of zetas for functional fields has been finally verified in the paper [22] of Zagier and myself, as a direct consequence of Theorem 7.2 of [14] and Theorem 2 of [22].…”

“…Indeed, in [14], based on the theories of Hall algebra and wall-crossing, Mozgovoy-Reineke are able to obtain a close formula for ζ X,F q ;n (s) in terms of partitions of n and abelian zeta function ζ X/F q (s) of X/F q . On the other hand, by examining the Lie structures involved in great details in [22], Zagier and myself are able to obtain the explicit formula for ζ SL n X,F q (s) as stated in the theorem above.…”

“…Finally, we demonstrate that the related bounds in lower ranks in turn play a central role in establishing the Riemann hypothesis for higher rank zetas. 1 Special uniformity of zetas Special uniformity for zeta functions of curves over finite fields is conjectured in [19] and established in [22], with the help of the result in [14]. In this section, we recall some basic constructions involved.…”

Higher-rank zeta functions and SL n -zeta functions for curves

“…2) In ref. 5, as recalled in Section 6, using the theory of Hall algebras and wall-crossing techniques, a formula for ζX ,n (s) of the same general shape is proved. 3) A short calculation, given in Section 7, shows that the two formulas agree.…”