COMPUTING<i>L</i>-FUNCTIONS AND SEMISTABLE REDUCTION OF SUPERELLIPTIC CURVES

Bouw, Irene I.; Wewers, Stefan

doi:10.1017/s0017089516000057

Cited by 37 publications

(113 citation statements)

References 31 publications

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“…It follows that Γ acts on the componentX 2 to which ∞ specializes. (This is similar to the argument in the proof of [2], Lemma 5.4.) Since there is exactly one other branch point specializing toX 2 , this point is fixed by Γ , as well.…”

Section: The Conductor Exponent In the Tame Casesupporting

confidence: 79%

“…Since Γ fixes at least 3 points on the genus-0 curveX 2 , it acts trivially onX 2 . Equation (2) implies that the action of Γ onȲ descends toX. It follows that Γ acts on W 2 via a subgroup of G. We conclude that Γ either fixes the three singularities ofȲ or cyclically permutes them.…”

Section: The Conductor Exponent In the Tame Casementioning

confidence: 81%

“…The theoretical background for this are the admissible covers, see [7], [12], § 10.4.3, or [26]. In the case of superelliptic curves the computation of f p has already been described in detail in [2], hence we can be much briefer than in the previous section.…”

Section: The Tame Case: P =mentioning

confidence: 99%

“…Let Y denote the normalization of X with respect to the cover Y L → X L . Theorem 3.4 from [2] shows that Y is a quasi-stable model of Y L . A priory, it is not clear whether Y is the stable model of Y .…”

Section: The Stable Modelmentioning

confidence: 99%

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Picard Curves with Small Conductor

Börner

Bouw

Wewers

2017

Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory

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We study the conductor of Picard curves over Q, which is a product of local factors. Our results are based on previous results on stable reduction of superelliptic curves that allow to compute the conductor exponent f p at the primes p of bad reduction. A careful analysis of the possibilities of the stable reduction at p yields restrictions on the conductor exponent f p . We prove that Picard curves over Q always have bad reduction at p = 3, with f 3 ≥ 4. As an application we discuss the question of finding Picard curves with small conductor.

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Section: The Conductor Exponent In the Tame Casesupporting

confidence: 79%

Section: The Conductor Exponent In the Tame Casementioning

confidence: 81%

Section: The Tame Case: P =mentioning

confidence: 99%

Section: The Stable Modelmentioning

confidence: 99%

See 2 more Smart Citations

Picard Curves with Small Conductor

Börner

Bouw

Wewers

2017

Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory

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“…is the modular curve X 0 (39). It has ∆ = 2 28 · 3 8 · 13 4 . The odd primes of bad reduction of C are 3 and 13.…”

Section: 1mentioning

confidence: 99%

Modular invariants for genus 3 hyperelliptic curves

et al. 2019

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In this article we prove an analogue of a theorem of Lachaud, Ritzenthaler, and Zykin, which allows us to connect invariants of binary octics to Siegel modular forms of genus 3. We use this connection to show that certain modular functions, when restricted to the hyperelliptic locus, assume values whose denominators are products of powers of primes of bad reduction for the associated hyperelliptic curves. We illustrate our theorem with explicit computations. This work is motivated by the study of the values of these modular functions at CM points of the Siegel upper-half space, which, if their denominators are known, can be used to effectively compute models of (hyperelliptic, in our case) curves with CM.

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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

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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.

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COMPUTINGL-FUNCTIONS AND SEMISTABLE REDUCTION OF SUPERELLIPTIC CURVES

Abstract: Abstract. We give an explicit description of the stable reduction of superelliptic curves of the form y n = f (x) at primes p whose residue characteristic is prime to the exponent n. We then use this description to compute the local L-factor and the exponent of conductor at p of the curve.

Cited by 37 publications

References 31 publications

Picard Curves with Small Conductor

Picard Curves with Small Conductor

Modular invariants for genus 3 hyperelliptic curves

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

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