Counting the number of trigonal curves of genus 5 over finite fields
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Counting the number of trigonal curves of genus 5 over finite fields Thomas Wennink1 Received: 8 July 2019 / Accepted: 20 December 2019 © The Author(s) 2020
Abstract The trigonal curves of genus 5 can be represented by projective plane quintics that have one singularity of delta invariant one. Combining this with a partial sieve method for plane curves we count the number of such curves over any finite field. The main application is that this gives the motivic Euler characteristic of the moduli space of trigonal curves of genus 5. Keywords Moduli space of curves · Trigonal curves · Plane curves · Finite fields · Trace of Frobenius Mathematics Subject Classification 11G20 · 14H10 · 14H50
1 Introduction Inside the moduli space M5 of smooth curves of genus 5 there is a subvariety T5 parameterizing trigonal curves. The main result of this article is to count the number of points |T5 (Fq )| for all finite fields Fq . Theorem 1 The number of Fq -isomorphism classes of smooth trigonal curves of genus 5 over a finite field Fq , weighted by the size of their Fq -automorphism group, is given by |T5 (Fq )| = q 11 + q 10 − q 8 + 1. This is a step forward in the open problem of computing the cohomology of M5 , the moduli space of stable genus 5 curves. Consider the stratification of M5 by topological type. The moduli spaces Mg,n that appear in the stratification satisfy g ≤ 5, n ≤ 10 − 2g. When g ≤ 3 their Sn -equivariant counts over finite fields Fq are known and are polynomials in q (see [10] for genus 0, [8] or [2] for genus 1, [4] for genus 2 and [3] for genus 3). The missing parts are M4,1 , M4,2 and M5 . If their Sn -equivariant counts are polynomials as well then arguing as in Sect. 3 of [5] we get the cohomology and Hodge structure of M5 . We expect the point counts of these spaces to be polynomials in q because of Theorem F in [6]. There is a conjectural correspondence due to Langlands (see Sect. 1.3 in [7]) that
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Thomas Wennink [email protected] University of Liverpool, Liverpool, England
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Geometriae Dedicata
translates this classification to one on motives of proper smooth stacks over Z. Assuming this correspondence, by Theorem F, the cohomology of M4,1 must be completely of Tate Hodge type while the cohomologies of M4,2 and M5 could in principle also have Tate twists of the cusp form motive S[12]. By Poincaré duality it is sufficient to check if this motive appears in H 11 . It would then appear in H 0,11 , which is not possible since M4,2 and M5 do not carry holomorphic 11-forms as they are unirational. Therefore, assuming Langlands’ correspondence, we conclude that the cohomologies of M4,1 , M4,2 and M5 are of Tate Hodge type, which implies that their Sn -equivariant point counts over finite fields Fq are polynomials in q. (And the same is true of M4,1 , M4,2 and M5 by applying the stratification by topological type and the known results for g ≤ 3 that we cited earlier). We can decompose the moduli space M5 into the moduli of hyperelliptic curves (whose point count over Fq is known to be q
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