The derived Riemann hypothesis for curves over finite fields

Let XX be an integral regular projective curve of genus gg over Fq\Bbb F_q, and let nm=(n0,n1,,nm){\bf n}_m=(n_0,n_1,\ldots,n_m) be an (m+1)(m+1)-tuple of positive integers. Let ζ^X,Fq(nm)(s)\widehat\zeta^{\,({\bf n}_m)}_{X,\Bbb F_q}(s) denote the nm{\bf n}_m-derived zeta function of XX.

nm{\bf n}_m-derived Riemann hypothesis. The function ζ^X,Fq(nm)(s)\widehat\zeta^{\,({\bf n}_m)}_{X,\Bbb F_q}(s) satisfies the Riemann hypothesis: all its zeros lie on

(s)=12.\Re(s)=\frac{1}{2}.

The conjecture asserts the analogue of the classical curve Riemann hypothesis for every derived zeta function in this family. The supplied text does not state whether it has been resolved.

Sources & referencesView supporting material

Primary source

Lin Weng, “Derived Zeta Functions for Curves over Finite Fields”, arXiv:2203.11488 (2022).

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