arXiv subjects

Stefan Wewers

Publications and source records attributed to Stefan Wewers.

At least 19 recordsLinked to original sources

arXiv2026

Semistable reduction of smooth quartics

We develop a method for computing stable reduction of smooth plane quartics over discretely valued fields, including residue characteristic p=2. The method uses the GIT-semistable plane models constructed in an earlier part of this project, together with an intrinsic description of hyperelliptic stable curves, to characterize when the stable model is obtained from a GIT-stable plane model by resolving its cusps. More precisely, for a smooth non-hyperelliptic curve of genus 3 with semistable reduction, we show that it admits a GIT-stable plane model if and only if its stable reduction is non-hyperelliptic. In that case, the stable model is obtained from the GIT-stable plane model by replacing each cusp by a 1-tail. Together with the companion paper on explicit local stable resolution of cusps, this gives an effective approach to computing stable reduction of smooth plane quartics. The resulting algorithms are implemented in the SageMath package "StabilityFunction".

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arXiv2026

Explicit local stable resolution of cusps

This article gives an explicit local stable resolution of cusps on GIT-stable plane quartic models. More generally, we consider a smoothing of an ordinary plane cusp over a complete discretely valued field. We show that, after a finite separable extension and a suitable choice of coordinates, a single weighted blow-up with weights ((1,2,3)) gives the stable resolution. The exceptional component is an explicit semistable Weierstrass cubic. The construction is effective and is implemented in the Sage package "StabilityFunction".

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arXiv2026

Models of hypersurfaces and Bruhat-Tits buildings

We propose a new approach to constructing semistable integral models of hypersurfaces over a discretely valued complete field K. For each stable hypersurface X over K we define a continuous stability function on the Bruhat-Tits building of PGL_{n+1}(K); its global minima control semistable hypersurface models after finite extensions of K. In particular, in residue characteristic zero the problem reduces to minimizing this function on the original building and then passing to a finite extension that turns a rational minimizer into a vertex. This extends work of Kollar and of Elsenhans-Stoll on minimal hypersurface models. We implement the resulting strategy for plane curves over p-adic number fields. In a follow-up article we use our results to compute the semistable reduction of smooth plane quartics.

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arXiv2025

Integral differential forms for superelliptic curves

Given a superelliptic curve $Y_K : y^n = f(x)$ over a local field $K$, we describe the theoretical background and an implementation of a new algorithm for computing the $\mathcal{O}_K$-lattice of integral differential forms on $Y_K$. We build on the results of Obus and the second author, which describe arbitrary regular models of the projective line using only valuations. One novelty of our approach is that we construct an $\mathcal{O}_K$-model of $Y_K$ with only rational singularities, but which may not be regular.

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arXiv2023

Computing the Weil representation of a superelliptic curve

We study the Weil representation $ρ$ of a curve over a $p$-adic field with potential reduction of compact type. We show that $ρ$ can be reconstructed from its stable reduction. For superelliptic curves of the form $y^n=f(x)$ at primes $p$ whose residue characteristic is prime to the exponent $n$ we make this explicit.

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arXiv2019

Explicit resolution of weak wild quotient singularities on arithmetic surfaces

A weak wild arithmetic quotient singularity arises from the quotient of a smooth arithmetic surface by a finite group action, where the inertia group of a point on a closed characteristic p fiber is a p-group acting with smallest possible ramification jump. In this paper, we give complete explicit resolutions of these singularities using deformation theory and valuation theory, taking a more local perspective than previous work has taken. Our descriptions answer several questions of Lorenzini. Along the way, we give a valuation-theoretic criterion for a normal snc-model of P^1 over a discretely valued field to be regular.

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arXiv2019

Conductor and discriminant of Picard curves

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 $\mathbb{Q}$ 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 $\mathbb{Q}$ of small conductor.

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arXiv2017

A non-abelian conjecture of Tate-Shafarevich type for hyperbolic curves

We state a conjectural criterion for identifying global integral points on a hyperbolic curve over $\mathbb{Z}$ in terms of Selmer schemes inside non-abelian cohomology functors with coefficients in $\mathbb{Q}_p$-unipotent fundamental groups. For $\mathbb{P}^1\setminus \{0,1,\infty\}$ and the complement of the origin in semi-stable elliptic curves of rank 0, we compute the local image of global Selmer schemes, which then allows us to numerically confirm our conjecture in a wide range of cases.

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arXiv2017

Picard curves with small conductor

We study the conductor of Picard curves over $\mathbb{Q}$, 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 $\mathbb{Q}$ always have bad reduction at $p=3$, with $f_3\geq 4$. As an application we discuss the question of finding Picard curves with small conductor.

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arXiv2017

Desingularization of arithmetic surfaces: algorithmic aspects

The quest for regular models of arithmetic surfaces allows different viewpoints and approaches: using valuations or a covering by charts. In this article, we sketch both approaches and then show in a concrete example, how surprisingly beneficial it can be to exploit properties and techniques from both worlds simultaneously.

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arXiv2016

Wild ramification kinks

Given a branched cover $f:Y\to X$ between smooth projective curves over a non-archimedian mixed-characteristic local field and an open rigid disk $D\subset X$, we study the question under which conditions the inverse image $f^{-1}(D)$ is again an open disk. More generally, if the cover $f$ varies in an analytic family, is this true at least for some member of the family? Our main result gives a criterion for this to happen.

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arXiv2015

The functional equation for L-functions of hyperelliptic curves

We compute the $L$-functions of a large class of algebraic curves, and verify the expected functional equation numerically. Our computations are based on our previous results on stable reduction to calculate the local $L$-factor and the conductor exponent at the primes of bad reduction. Most of our examples are hyperelliptic curves of genus $g\geq 2$ defined over $\mathbb{Q}$ which have semistable reduction at every prime $p$. We also treat a few more general examples of superelliptic curves.

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arXiv2015

Mixed Tate motives and the unit equation

This is the second installment in a sequence of articles devoted to "explicit Chabauty-Kim theory" for the thrice punctured line. Its ultimate goal is to construct an algorithmic solution to the unit equation whose halting will be conditional on Goncharov's conjecture about exhaustion of mixed Tate motives by motivic iterated integrals (refined somewhat with respect to ramification), and on Kim's conjecture about the determination of integral points via $p$-adic iterated integrals. In this installment we explain what this means while developing basic tools for the construction of the algorithm. We also work out an elaborate example, which goes beyond the cases that were understood before, and allows us to verify Kim's conjecture in a range of new cases.

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arXiv2014

Computing $L$-functions and semistable reduction of superelliptic curves

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 of the curve and the exponent of conductor at $\p$.

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arXiv2014

The Heisenberg coboundary equation: appendix to Explicit Chabauty-Kim theory

Let p be a regular prime number, let Gp denote the Galois group of the maximal unramified away from p extension of Q, and let H_et denote the Heisenberg group over Qp with Gp-action given by H_et = Qp(1)^2 \oplus Qp(2). Although Soulé vanishing guarantees that the map H^1(Gp, H_et) ---> H^1(Gp, Qp(1)^2) is bijective, the problem of constructing an explicit lifting of an arbitrary cocycle in H^1(Gp, Qp(1)^2) proves to be a challenge. We explain how we believe this problem should be analyzed, following an unpublished note by Romyar Sharifi, hereby making the original appendix to Explicit Chabauty-Kim theory available online in an arXiv-only note.

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arXiv2014

Explicit Chabauty-Kim theory for the thrice punctured line in depth two

Let $X= \mathbb{P}^1 \setminus \{0,1,\infty\}$, and let $S$ denote a finite set of prime numbers. In an article of 2005, Minhyong Kim gave a new proof of Siegel's theorem for $X$: the set $X(\mathbb{Z}[S^{-1}])$ of $S$-integral points of $X$ is finite. The proof relies on a `nonabelian' version of the classical Chabauty method. At its heart is a modular interpretation of unipotent $p$-adic Hodge theory, given by a tower of morphisms $h_n$ between certain $\mathbb{Q}_p$-varieties. We set out to obtain a better understanding of $h_2$. Its mysterious piece is a polynomial in $2|S|$ variables. Our main theorem states that this polynomial is quadratic, and gives a procedure for writing its coefficients in terms of $p$-adic logarithms and dilogarithms.

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arXiv2012

Fiercely ramified cyclic extensions of p-adic fields with imperfect residue field

We study the ramification of fierce cyclic Galois extensions of a local field $K$ of characteristic zero with a one-dimensional residue field of characteristic $p>0$. Using Kato's theory of the refined Swan conductor, we associate to such an extension a ramification datum, consisting of a sequence of pairs $(δ_i,ω_i)$, where $δ_i$ is a positive rational number and $ω_i$ a differential form on the residue field of $K$. Our main result gives necessary and sufficient conditions on such sequences to occur as a ramification datum of a fierce cyclic extension of $K$.

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arXiv2012

Another proof of the Semistable Reduction Theorem

We give a new proof of the Semistable Reduction Theorem for curves. The main idea is to present a curve $Y$ over a local field $K$ as a finite cover of the projective line $X=\PP^1_K$. By successive blowups (and after replacing $K$ by a suitable finite extension) we construct a semistable model of $X$ whose normalization with respect to the cover is a semistable model of $Y$.

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