Monodromy of the p-Rank Strata of the Moduli Space of Curves

Achter, Jeffrey D.; Pries, Rachel

doi:10.1093/imrn/rnn053

Cited by 20 publications

(52 citation statements)

References 25 publications

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“…In fact, such theorems are already known in the case M = M g [2] and M = H g [3, ğ3.7]. Thus, in a family of curves Xwith random p-divisible group, parameterized by a moduli scheme M, the average number of surjections from Jac(X)(F q ) to (Z/pZ) d is 1.…”

Section: Relation With Geometry Of Moduli Spacesmentioning

confidence: 97%

Random Dieudonné modules, random -divisible groups, and random curves over finite fields

Cais¹,

Ellenberg²,

Zureick-Brown

2013

J. Inst. Math. Jussieu

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Link to this article: http://journals.cambridge.org/abstract_S1474748012000862How to cite this article: Bryden Cais, Jordan S. Ellenberg and David Zureick-Brown (2013). Random Dieudonné modules, random -divisible groups, and random curves over nite elds.Abstract We describe a probability distribution on isomorphism classes of principally quasi-polarized p-divisible groups over a finite field k of characteristic p which can reasonably be thought of as a 'uniform distribution', and we compute the distribution of various statistics (p-corank, a-number, etc.) of p-divisible groups drawn from this distribution. It is then natural to ask to what extent the p-divisible groups attached to a randomly chosen hyperelliptic curve (respectively, curve; respectively, abelian variety) over k are uniformly distributed in this sense. This heuristic is analogous to conjectures of Cohen-Lenstra type for char k = p, in which case the random p-divisible group is defined by a random matrix recording the action of Frobenius. Extensive numerical investigation reveals some cases of agreement with the heuristic and some interesting discrepancies. For example, plane curves over F 3 appear substantially less likely to be ordinary than hyperelliptic curves over F 3 .

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Section: Relation With Geometry Of Moduli Spacesmentioning

confidence: 97%

Random Dieudonné modules, random -divisible groups, and random curves over finite fields

Cais¹,

Ellenberg²,

Zureick-Brown

2013

J. Inst. Math. Jussieu

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“…Let X be a singular curve formed by intersecting two curves C 1 and C 2 (at points x 1 ∈ C 1 and x 2 ∈ C 2 ) in an ordinary double point. By (1),…”

Section: 3mentioning

confidence: 97%

“…There are 2g − 2i − 2(g−i) + 1 points of order which do not have this property. Since Q = Π −1 (S), without loss of generality, one can suppose that the point of order is one of these or, equivalently, that ξ is in ∆ i:g−i , completing part (1).…”

Section: Ordinary Pryms Of a Generic Curve Of Given P-rankmentioning

confidence: 99%

“…Section 6 contains results about non-ordinary Pryms in the low genus cases g = 2, 3, generalizing [9, Theorem 6.1]. Section 7 contains the proof of Theorem 1.2 (1). The proof is again an inductive argument which uses the boundary component ∆ i:g−i , but it relies on more refined information from each of Sections 3, 4, 5, and 6.…”

Section: Introductionmentioning

confidence: 99%

“…Application 1.3 (Corollary 7.3) follows from Theorems 1.1 and 1.2 using a 'straightforward' deformation argument for f ≤ g − 3: Suppose π s is an unramified double cover of a singular curve of genus g and p-rank f for which the Prym has p-rank f . Applying [1,Section 3], one can deform π s to an unramified double cover of a smooth curve of genus g whose p-rank is still f . However, it is possible that the p-rank f of the Prym increases in this deformation.…”

Section: Introductionmentioning

confidence: 99%

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Ordinary and almost ordinary Prym varieties

Ozman

Pries

2019

Asian Journal of Mathematics

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We study the p-rank stratification of the moduli space of Prym varieties in characteristic p > 0. For arbitrary primes p and with = p and integers g ≥ 3 and 0 ≤ f ≤ g, the first theorem generalizes a result of Nakajima by proving that the Prym varieties of all the unramified Z/ -covers of a generic curve X of genus g and p-rank f are ordinary. Furthermore, when p ≥ 5 and = 2, the second theorem implies that there exists a curve of genus g and p-rank f having an unramified double cover whose Prym has p-rank f for each g 2 − 1 ≤ f ≤ g − 2; (these Pryms are not ordinary). Using work of Raynaud, we use these two theorems to prove results about the (non)-intersection of the -torsion group scheme with the theta divisor of the Jacobian of a generic curve X of genus g and p-rank f .

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On topological and combinatorial structures of pointed stable curves over algebraically closed fields of positive characteristic

2023

Mathematische Nachrichten

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In this paper, we study anabelian geometry of curves over algebraically closed fields of positive characteristic. Let be a pointed stable curve over an algebraically closed field of characteristic and the admissible fundamental group of . We prove that there exists a group‐theoretical algorithm, whose input datum is the admissible fundamental group , and whose output data are the topological and the combinatorial structures associated with . This result can be regarded as a mono‐anabelian version of the combinatorial Grothendieck conjecture in the positive characteristic. Moreover, by applying this result, we construct clutching maps for moduli spaces of admissible fundamental groups.

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Monodromy of the p-Rank Strata of the Moduli Space of Curves

Cited by 20 publications

References 25 publications

Random Dieudonné modules, random -divisible groups, and random curves over finite fields

Random Dieudonné modules, random -divisible groups, and random curves over finite fields

Ordinary and almost ordinary Prym varieties

On topological and combinatorial structures of pointed stable curves over algebraically closed fields of positive characteristic

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