The moduli space of odd spin curves of genus 9 is unirational, realized birationally as a locally trivial projective bundle over a finite quotient of the moduli space of n-pointed odd stable spin curves of lower genus.
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Using the Chenevier-Lannes classification of automorphic representations and a conjectural correspondence to ℓ-adic Galois representations, the Euler characteristics of overline M_{3,n} and M_{3,n} (n≤14) and of local systems V_λ on A_3 (|λ|≤16) are computed in the Grothendieck group of such Galois/
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The unirationality of $S_9^-$ and moduli spaces of pointed spin curves
The moduli space of odd spin curves of genus 9 is unirational, realized birationally as a locally trivial projective bundle over a finite quotient of the moduli space of n-pointed odd stable spin curves of lower genus.
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Cohomology of moduli spaces via a result of Chenevier and Lannes
Using the Chenevier-Lannes classification of automorphic representations and a conjectural correspondence to ℓ-adic Galois representations, the Euler characteristics of overline M_{3,n} and M_{3,n} (n≤14) and of local systems V_λ on A_3 (|λ|≤16) are computed in the Grothendieck group of such Galois/