Picard Curves with Small Conductor

Börner, Michel; Bouw, Irene I.; Wewers, Stefan

doi:10.1007/978-3-319-70566-8_4

Cited by 4 publications

(27 citation statements)

References 22 publications

(55 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then

Y

has semistable reduction over

L = K

, and its stable model

scriptY

is smooth over

O_{K}

. Therefore, we are in Case (a) of [2, Lemma 4.1, Theorem 4.2]. It follows that

X = Y / G

is a smooth model of

P_{K}^{1}

such that the closure

\overset{D}{true}

scriptX

of the branch divisor

\overset{D}{true}

is étale over

Spec {scriptO}_{K}

.…”

Section: Nonspecial Picard Curves As Superelliptic Curvesmentioning

confidence: 97%

“…In the general case, one needs to add the number of loops of the dual graph of

{\bar{Y}}^{u}

. We refer to [2, Section 2.2] for a precise formula. Remark Let

Y / {double-struckQ}_{3}^{1false normalnr}

be a Picard curve given as superelliptic curve of exponent 3 ().…”

Section: Comparing the Conductor And The Discriminantmentioning

confidence: 99%

“…Remark This definition of a Picard curve is slightly more general than that in [15, Appendix I] and [2], in that we do not require the Galois cover

φ_{\bar{K}}

to be defined over

K

.…”

Section: Picard Curvesmentioning

confidence: 99%

“…Let

L

be a minimal Galois extension of

K

over which

Y

has semistable reduction. Then by [2, Section 4.1], the curve

Y_{L}

has a unique stable

O_{L}

‐model

scriptY

. Moreover, the action of

G

extends to

scriptY

and the quotient scheme

X = Y / G

is a semistable model of

Y_{L} / G = {double-struckP}_{L}^{1}

.…”

Section: Nonspecial Picard Curves As Superelliptic Curvesmentioning

confidence: 99%

“…The induced map

\bar{Y} \to \bar{X}

is called the stable reduction of the cover

Y \to {double-struckP}_{K}^{1}

. Lemma 4.1 and Theorem 4.2 of [2] give a precise description of this map, distinguishing five cases (a)–(e). Case (a) is the only case where

\bar{Y}

is smooth.…”

Section: Nonspecial Picard Curves As Superelliptic Curvesmentioning

confidence: 99%

See 4 more Smart Citations

Conductor and discriminant of Picard curves

Bouw

Koutsianas

Sijsling

et al. 2020

J. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

We describe normal forms and minimal models of Picard curves, discussing various arithmetic aspects of these. We determine all so-called special Picard curves over Q with good reduction outside 2 and 3, and use this to determine the smallest possible conductor a special Picard curve may have. We also collect a database of Picard curves over Q of small conductor.view of covers of the projective line in Propositions 3.2.1 and 4.3.2. This criterion is formulated in terms of the discriminant of the binary form f from ( * ).From Section 4 onward we concentrate on special Picard curves. Our main result here is Theorem 5.1.16, which states that the smallest possible value for the conductor of a special Picard curve over Q is 2 6 3 6 . This value is attained for the standard special Picard curve Y 0 defined by (1.16). To prove this, we apply results from [7] on the computation of the conductor of a superelliptic curves via stable reduction. In Section 5.1 we extend results from [2] for nonspecial Picard curves to special ones, and prove lower bounds on the local conductor f p at a prime of bad reduction. Our results illustrate how to analyze the effect of twisting on the conductor.The key ingredient in the proof of Theorem 5.1.16 is Theorem 4.4.54, which classifies all special Picard curves defined over Q with good reduction outside 2 and 3: there are precisely 800 different Q-isomorphism classes. The proof uses methods in Galois cohomology that generalize beyond this particular case.We expect N = 2 6 3 6 to be the smallest conductor for any Picard curve defined over Q. Following the strategy for studying this question outlined in [2, Section 5], we have constructed a large database of Picard curves over Q that have good reduction outside two small primes, and more precisely outside of the pairs {2, 3}, {3, 5}, and {3, 7}. These curves were obtained by methods described in the forthcoming work [4] (summarized in Appendix B) as well as an effective enumeration by Sutherland [38]. Our database gives equations for these Picard curves, as well as their invariants, discriminants, and conductors. Its construction is briefly discussed in the concluding Appendix A.The database provides evidence for the question, discussed in Section 5.2, whether the conductor of a Picard curve divides its minimal discriminant, as is the case for curves of genus 1 and 2.Notations and conventions. In this article, a curve is a separated scheme of dimension 1 over a field. Given an affine equation for a curve, we will identify it with the smooth projective curve with the same function field.

show abstract

“…Then

Y

has semistable reduction over

L = K

, and its stable model

scriptY

is smooth over

O_{K}

. Therefore, we are in Case (a) of [2, Lemma 4.1, Theorem 4.2]. It follows that

X = Y / G

is a smooth model of

P_{K}^{1}

such that the closure

\overset{D}{true}

scriptX

of the branch divisor

\overset{D}{true}

is étale over

Spec {scriptO}_{K}

.…”

Section: Nonspecial Picard Curves As Superelliptic Curvesmentioning

confidence: 97%

“…In the general case, one needs to add the number of loops of the dual graph of

{\bar{Y}}^{u}

. We refer to [2, Section 2.2] for a precise formula. Remark Let

Y / {double-struckQ}_{3}^{1false normalnr}

be a Picard curve given as superelliptic curve of exponent 3 ().…”

Section: Comparing the Conductor And The Discriminantmentioning

confidence: 99%

“…Remark This definition of a Picard curve is slightly more general than that in [15, Appendix I] and [2], in that we do not require the Galois cover

φ_{\bar{K}}

to be defined over

K

.…”

Section: Picard Curvesmentioning

confidence: 99%

“…Let

L

be a minimal Galois extension of

K

over which

Y

has semistable reduction. Then by [2, Section 4.1], the curve

Y_{L}

has a unique stable

O_{L}

‐model

scriptY

. Moreover, the action of

G

extends to

scriptY

and the quotient scheme

X = Y / G

is a semistable model of

Y_{L} / G = {double-struckP}_{L}^{1}

.…”

Section: Nonspecial Picard Curves As Superelliptic Curvesmentioning

confidence: 99%

“…The induced map

\bar{Y} \to \bar{X}

is called the stable reduction of the cover

Y \to {double-struckP}_{K}^{1}

. Lemma 4.1 and Theorem 4.2 of [2] give a precise description of this map, distinguishing five cases (a)–(e). Case (a) is the only case where

\bar{Y}

is smooth.…”

Section: Nonspecial Picard Curves As Superelliptic Curvesmentioning

confidence: 99%

See 3 more Smart Citations

Conductor and discriminant of Picard curves

Bouw

Koutsianas

Sijsling

et al. 2020

J. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

Reduction Types of Genus-3 Curves in a Special Stratum of their Moduli Space

Bouw

Coppola

Kılıçer

et al. 2021

Association for Women in Mathematics Series

View full text Add to dashboard Cite

We study a 3-dimensional stratum M 3,V of the moduli space M 3 of curves of genus 3 parameterizing curves Y that admit a certain action of V C 2 × C 2 . We determine the possible types of the stable reduction of these curves to characteristic different from 2. We define invariants for M 3,V and characterize the occurrence of each of the reduction types in terms of them. We also calculate the j-invariant (resp. the Igusa invariants) of the irreducible components of positive genus of the stable reduction Y in terms of the invariants.

show abstract

Chabauty–Coleman Computations on Rank 1 Picard Curves

Hashimoto

Morrison

2021

Arithmetic Geometry, Number Theory, and Computation

View full text Add to dashboard Cite

Picard Curves with Small Conductor

Cited by 4 publications

References 22 publications

Conductor and discriminant of Picard curves

Conductor and discriminant of Picard curves

Reduction Types of Genus-3 Curves in a Special Stratum of their Moduli Space

Chabauty–Coleman Computations on Rank 1 Picard Curves

Contact Info

Product

Resources

About