The p-rank strata of the moduli space of hyperelliptic curves

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• 作者：Jeffrey D. Achter^a ; ^{j.achter@colostate.edu"" rel=""nofollow} ; Rachel Pries ; ^a ; ^{pries@math.colostate.edu"" rel=""nofollow}
• 关键词：p-Rank ; Moduli ; Hyperelliptic ; Jacobian ; Monodromy
• 刊名：Advances in Mathematics
• 出版年：2011
• 出版时间：1 August 2011
• 年：2011
• 卷：227
• 期：5
• 页码：1846-1872
• 全文大小：282 K

文摘

We prove results about the intersection of the p-rank strata and the boundary of the moduli space of hyperelliptic curves in characteristic p3. This yields a strong technique that allows us to analyze the stratum of hyperelliptic curves of genus g and p-rank f. Using this, we prove that the endomorphism ring of the Jacobian of a generic hyperelliptic curve of genus g and p-rank f is isomorphic to if g4. Furthermore, we prove that the -monodromy of every irreducible component of is the symplectic group if g3, and ℓ≠p is an odd prime (with mild hypotheses on ℓ when f=0). These results yield numerous applications about the generic behavior of hyperelliptic curves of given genus and p-rank over finite fields, including applications about Newton polygons, absolutely simple Jacobians, class groups and zeta functions.