Algebraic Curves and Finite Fields 2014
DOI: 10.1515/9783110317916.1
|View full text |Cite
|
Sign up to set email alerts
|

Generic Newton polygons for curves of given p-rank

Abstract: We survey results and open questions about the p-ranks and Newton polygons of Jacobians of curves in positive characteristic p. We prove some geometric results about the p-rank stratification of the moduli space of (hyperelliptic) curves. For example, if 0 ≤ f ≤ g − 1, we prove that every component of the p-rank f + 1 stratum of M g contains a component of the p-rank f stratum in its closure. We prove that the p-rank f stratum of M g is connected. For all primes p and all g ≥ 4, we demonstrate the existence of… Show more

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
2
1

Citation Types

0
35
0

Year Published

2014
2014
2024
2024

Publication Types

Select...
4
1
1

Relationship

3
3

Authors

Journals

citations
Cited by 9 publications
(35 citation statements)
references
References 19 publications
0
35
0
Order By: Relevance
“…A curve corresponding to a point in the pull back of a generic point in τ (∆ 1 ) ∩ Z [2,1] is formed from two smooth pointed curves X 1 and X 3 of genus 1 and 3 respectively. Then X 1 is ordinary and X 3 has Ekedahl-Oort type µ = [2, 1] or X 1 is supersingular and X 3 has Ekedahl-Oort type µ = [1]. In the first case, we consider the action of Verschiebung operator V on the de Rham cohomology.…”
Section: The Associated Final Type V Is the Increasing And Surjective...mentioning
confidence: 99%
See 1 more Smart Citation
“…A curve corresponding to a point in the pull back of a generic point in τ (∆ 1 ) ∩ Z [2,1] is formed from two smooth pointed curves X 1 and X 3 of genus 1 and 3 respectively. Then X 1 is ordinary and X 3 has Ekedahl-Oort type µ = [2, 1] or X 1 is supersingular and X 3 has Ekedahl-Oort type µ = [1]. In the first case, we consider the action of Verschiebung operator V on the de Rham cohomology.…”
Section: The Associated Final Type V Is the Increasing And Surjective...mentioning
confidence: 99%
“…) and the action of V . So the Ekedahl-Oort type is [2,1]. Note that the moduli space of pointed elliptic curves is irreducible of dimension 1 and Z [2,1] ⊂ M 3 is irreducible of dimension 3 by [7,Theorem 11.3].…”
Section: The Associated Final Type V Is the Increasing And Surjective...mentioning
confidence: 99%
“…, f (m − 1)). We denote by ord the Newton polygon of G 0,1 ⊕ G 1,0 which has slopes 0 and 1 with multiplicity 1 and by ss the Newton polygon of G 1,1 which has slope 1/2 with multiplicity 2. m = 3 p 1 mod 3 2 mod 3 prime orbits split (1, 2) a signature (1, 1, 1 1,1,0,0,0,0) ord 3 ord 2 ⊕ ss ord ⊕ ss 2 ss 3 (1,2,5) (1,1,0,0,1,0,0) ord 3 ss 3 ord 3 ss 3 (1,3,4) (1,0,1,0,0,0,0) ord 2 ord 2 ss 2 ss 2 m = 9 p 1 mod 9 2, 5 mod 9 4, 7 mod 9 8 mod 9 (1,2,4,8,7,5) (1, 4, 7), (2,8,5) (1, 8), (2, 7) prime orbits split (3,6) (3), (6) (4, 5), (3, 6) a signature (1,1,7) (1,1,1,1,0,0,0,0) ord 4 ss 4 (1/3, 2/3) ⊕ ord ss 4 (1,2,6) (1,1,0,0,1,0,0,0) ord 3 ss 3 (1/3, 2/3) ss 3 (1,3,5) (1,1,0,1,0,0,0,0) ord 3 ss 3 (1/3, 2/3) ss 3 m = 10 p 1 mod 10 3, 7 mod 10 9 mod 10 (1, 3,9,7) (1, 9), (2, 8) prime orbits split (2,6,8,4), (5) (3, 7), (4, 6), (5) a signature (1,1,8) (1,1,1,1,0,0,0,0,0) ord 4 ss 4 ss 4 (1,2,7) (1,1,1,0,0,1,0,0,0) ord 4 ss 4 ss 4 (1,4,5) (1,0,1,0,0,0,0,0,0) ord 2 ss 2 ss 2 m = 11 p 1 mod 11 2, 6, 7, 8 mod 11 3, 4, 5, 9 mod 11 10 mod 11 (1,3,4,5,9) (1, 10), (2,9) (3,9) (1, 11), (4, 8) (1, 5), (2, 10), (3) (2), (4), (5,11) (3,9), (2, 10) prime orbits split (4, 8), (6), (7, 11), (9) (6), (8), (10) (5, 7), (6) a signature (1,1,10) (1,1,1,1,1,0,0,0,0,0,0) ord 5 ord 3 ⊕ ss 2 ord 2 ⊕ ss 3 ss 5 (1,2,9) (1,1,1,0,0,0,1,0,0,0,0) ord 4 ord ⊕ ss 3 ord 3 ⊕ ss ss 4 (1,3,8) (1,1,0,0,1,0,0,0,0,0,0) ord 3 ord 2 ⊕ ss ord ⊕ ss 2 ss 3 (1,4,7) (1,1,0,1,0,0,1,0,0,0,0) ord 4 ss 4 ord 4 ss 4 (1...…”
Section: Tablesmentioning
confidence: 99%
“…genus Newton polygon congruence where 5 (1/5, 4/5) 3, 4, 5, 9 mod 11 m = 11, a = (1, 1, 9) 5 (2/5, 3/5) 3, 4, 5, 9 mod 11 m = 11, a = (1, 2, 8) 6 (1/3, 2/3) 2 3, 9 mod 13 m = 13, a = (1, 2, 10) 9, 11 mod 14 m = 13, a = (1, 1, 12) 7…”
mentioning
confidence: 99%
“…For physical p-power torsion, and thus for elements of order p in the class groups of (hyperelliptic) function fields, all the results surveyed in Sections 5.4 and 5.5 have immediate parallels for M f g , the moduli space of hyperelliptic curves of genus g and p-rank f , and for the hyperelliptic sublocus H However, it is not clear how to apply the methods of the present paper to strata M g,ξ or H g,ξ . Indeed, it is not even known which Newton strata in M g are nonempty -see [4] for a survey of what little is known -let alone finer strata M g,ξ , to say nothing of their p-adic monodromy. Thus, a version of Theorem 4.5 for hyperelliptic Jacobians remains a distant goal.…”
mentioning
confidence: 99%