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J. Steffen Müller

Publications and source records attributed to J. Steffen Müller.

At least 19 recordsLinked to original sources
arXiv2026

Algorithms for hyperelliptic Mumford Curves $p$-adic Uniformization, $p$-adic integrals and $p$-adic heights

Mumford curves generalize the Tate uniformization of elliptic curves with split multiplicative reduction and provide p-adic analogues of the uniformization of Riemann surfaces. In this paper, we present several algorithms for hyperelliptic Mumford curves. For a given hyperelliptic Mumford curve $X$ defined over a finite extension of the field of p-adic numbers for some $p\neq 2$, we first describe how to compute a p-adic Schottky group W that uniformizes X; this is based on our extension to Kadziela's approximation theorem. As applications, we explain how to use this uniformization in order to compute p-adic Abelian integrals and $p$-adic Schneider heights on X; the latter uses Werner's formula expressing the p-part of the Schneider height in terms of theta functions. We illustrate our algorithms with numerical examples computed using the computer algebra system SageMath.

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arXiv2026

p-adic adelic metrics and Quadratic Chabauty I

We give a new construction of $p$-adic heights on varieties over number fields using $p$-adic Arakelov theory. In analogy with Zhang's construction of real-valued heights in terms of adelic metrics, these heights are given in terms of $p$-adic adelic metrics on line bundles. In particular, we describe a construction of canonical $p$-adic heights on abelian varieties and we show that we recover the canonical Mazur--Tate height and, for Jacobians, the height constructed by Coleman and Gross. Our main application is a new and simplified approach to the Quadratic Chabauty method for the computation of rational points on certain curves over the rationals, by pulling back the canonical height on the Jacobian with respect to a carefully chosen line bundle. We show that our construction allows us to reprove, without using $p$-adic Hodge theory or arithmetic fundamental groups, several results due to Balakrishnan and Dogra. Our method also extends to primes $p$ of bad reduction. One consequence of our work is that for any canonical height ($p$-adic or $\mathbb{R}$-valued) on an abelian variety (and hence on pull-backs to other varieties), the local contribution at a finite prime $q$ can be constructed using $q$-analytic methods.

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arXiv2026

Coleman-Gross Heights and $p$-adic Néron Functions on Jacobians of Genus $2$ Curves

We develop a theory of $p$-adic Néron functions on abelian varieties, depending on various auxiliary choices, and show that the global $p$-adic height functions constructed by Mazur and Tate can be decomposed into a sum of $p$-adic Néron functions if the same auxiliary choices are made. We also consider a decomposition of the $p$-adic height constructed by Coleman and Gross for good reduction, and extended to arbitrary reduction by Colmez and Besser, into a sum of certain local height functions for Jacobians of odd degree genus~$2$ curves. We show that this local height function is equal to the $p$-adic Néron function with the same auxiliary choices, regardless of the reduction type of the curve. This extends work of Balakrishnan and Besser for elliptic curves. When the curve has semistable reduction and the reduction of the Jacobian is ordinary, we also describe the $p$-adic Néron function that arises from the canonical Mazur--Tate splitting explicitly in terms of a generalisation of the $p$-adic sigma function constructed by Blakestad.

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arXiv2025

Linear quadratic Chabauty

We present a new quadratic Chabauty method to compute the integral points on certain even degree hyperelliptic curves. Our approach relies on a nontrivial degree zero divisor supported at the two points at infinity to restrict the $p$-adic height to a linear function; we can then express this restriction in terms of holomorphic Coleman integrals under the standard quadratic Chabauty assumption. Then we use this linear relation to extract the integral points on the curve. We also generalize our method to integral points over number fields. Our method is significantly simpler and faster than all other existing versions of the quadratic Chabauty method. We give examples over $\Q$ and $\Q(\sqrt{7})$.

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arXiv2025

Algorithms for $p$-adic Heights on Hyperelliptic Curves of Arbitrary Reduction

In this paper, we develop an algorithm for computing Coleman--Gross (and hence Nekovář) $p$-adic heights on hyperelliptic curves over number fields with arbitrary reduction type above $p$. This height is defined as a sum of local heights at each finite place and we use algorithms for Vologodsky integrals, developed by Katz and the second-named author, to compute the local heights above $p$. We also discuss an alternative method to compute these for odd degree genus 2 curves via $p$-adic sigma functions, via work of the first-named author. For both approaches one needs to choose a splitting of the Hodge filtration. A canonical choice for this is due to Blakestad in the case of an odd degree curve of genus $2$ that has semistable ordinary reduction at $p$. We provide an algorithm to compute Blakestad's splitting, which is conjecturally the unit root splitting for the action of Frobenius. We give several numerical examples, including the first worked quadratic Chabauty example in the literature for a curve with bad reduction.

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arXiv2025

Rational points on the non-split Cartan modular curve of level 27 and quadratic Chabauty over number fields

Thanks to work of Rouse, Sutherland, and Zureick-Brown, it is known exactly which subgroups of GL$_2(\mathbf{Z}_3)$ can occur as the image of the $3$-adic Galois representation attached to a non-CM elliptic curve over $\mathbf{Q}$, with a single exception: the normaliser of the non-split Cartan subgroup of level 27. In this paper, we complete the classification of 3-adic Galois images by showing that the normaliser of the non-split Cartan subgroup of level 27 cannot occur as a 3-adic Galois image of a non-CM elliptic curve. Our proof proceeds via computing the $\mathbf{Q}(ζ_3)$-rational points on a certain smooth plane quartic curve $X'_H$ (arising as a quotient of the modular curve $X_{ns}^+(27)$) defined over $\mathbf{Q}(ζ_3)$ whose Jacobian has Mordell--Weil rank 6. To this end, we describe how to carry out the quadratic Chabauty method for a modular curve $X$ defined over a number field $F$, which, when applicable, determines a finite subset of $X(F\otimes\mathbf{Q}_p)$ in certain situations of larger Mordell--Weil rank than previously considered. Together with an analysis of local heights above 3, we apply this quadratic Chabauty method to determine $X'_H(\mathbf{Q}(ζ_3))$. This allows us to compute the set $X_{ns}^+(27)(\mathbf{Q})$, finishing the classification of 3-adic images of Galois.

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arXiv2024

Computing p-adic heights on hyperelliptic curves

We describe an algorithm to compute the local Coleman-Gross p-adic height at p on a hyperelliptic curve. Previously, this was only possible using an algorithm due to Balakrishnan and Besser, which was limited to odd degree. While we follow their general strategy, our algorithm is significantly faster and simpler and works for both odd and even degree. We discuss a precision analysis and an implementation in SageMath. Our work has several applications, also discussed in this article. These include various versions of the quadratic Chabauty method, and numerical evidence for a p-adic version of the conjecture of Birch and Swinnerton-Dyer in cases where this was not previously possible.

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arXiv2023

Linear and quadratic Chabauty for affine hyperbolic curves

We give sufficient conditions for finiteness of linear and quadratic refined Chabauty-Kim loci of affine hyperbolic curves. We achieve this by constructing depth $\leq 2$ quotients of the fundamental group, following a construction of Balakrishnan-Dogra in the projective case. We also apply Betts' machinery of weight filtrations to give unconditional explicit upper bounds on the number of S-integral points when our conditions are satisfied.

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arXiv2023

Computing torsion subgroups of Jacobians of hyperelliptic curves of genus 3

We introduce an algorithm to compute the rational torsion subgroup of the Jacobian of a hyperelliptic curve of genus 3 over the rationals. We apply a Magma implementation of our algorithm to a database of curves with low discriminant due to Sutherland as well as a list of curves with small coefficients. In the process, we find several torsion structures not previously described in the literature. The algorithm is a generalisation of an algorithm for genus 2 due to Stoll, which we extend to abelian varieties satisfying certain conditions. The idea is to compute p-adic torsion lifts of points over finite fields using the Kummer variety and to check whether they are rational using heights. Both have been made explicit for Jacobians of hyperelliptic curves of genus 3 by Stoll. This article is partially based on the second-named author's Master thesis.

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arXiv2022

Rational points on $X^+_0(125)$

We compute the rational points on the Atkin-Lehner quotient $X^+_0(125)$ using the quadratic Chabauty method. Our work completes the study of exceptional rational points on the curves $X^+_0(N)$ of genus between 2 and 6. Together with the work of several authors, this completes the proof of a conjecture of Galbraith.

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arXiv2020

Explicit quadratic Chabauty over number fields

We generalize the explicit quadratic Chabauty techniques for integral points on odd degree hyperelliptic curves and for rational points on genus 2 bielliptic curves to arbitrary number fields using restriction of scalars. This is achieved by combining equations coming from Siksek's extension of classical Chabauty with equations defined in terms of p-adic heights attached to independent continuous idele class characters. We give several examples to show the practicality of our methods.

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arXiv2020

Two recent p-adic approaches towards the (effective) Mordell conjecture

We give an introductory account of two recent approaches towards an effective proof of the Mordell conjecture, due to Lawrence--Venkatesh and Kim. The latter method, which is usually called the method of Chabauty--Kim or non-abelian Chabauty in the literature, has the advantage that in some cases it has been turned into an effective method to determine the set of rational points on a curve, and we illustrate this by presenting three new examples of modular curves where this set can be determined.

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arXiv2019

Explicit arithmetic intersection theory and computation of Néron-Tate heights

We describe a general algorithm for computing intersection pairings on arithmetic surfaces. We have implemented our algorithm for curves over $\mathbb Q$, and we show how to use it to compute regulators for a number of Jacobians of smooth plane quartics, and to numerically verify the conjecture of Birch and Swinnerton-Dyer for the Jacobian of the split Cartan curve of level 13, up to squares.

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arXiv2018

Archimedean local height differences on elliptic curves

To compute generators for the Mordell-Weil group of an elliptic curve over a number field, one needs to bound the difference between the naive and the canonical height from above. We give an elementary and fast method to compute an upper bound for the local contribution to this difference at an archimedean place, which sometimes gives better results than previous algorithms.

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arXiv2017

Explicit Chabauty-Kim for the Split Cartan Modular Curve of Level 13

We extend the explicit quadratic Chabauty methods developed in previous work by the first two authors to the case of non-hyperelliptic curves. This results in an algorithm to compute the rational points on a curve of genus $g \ge 2$ over the rationals whose Jacobian has Mordell-Weil rank $g$ and Picard number greater than one, and which satisfies some additional conditions. This algorithm is then applied to the modular curve $X_{s}(13)$, completing the classification of non-CM elliptic curves over $\mathbf{Q}$ with split Cartan level structure due to Bilu-Parent and Bilu-Parent-Rebolledo.

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arXiv2016

Canonical Heights on Genus Two Jacobians

Let $K$ be a number field and let $C/K$ be a curve of genus 2 with Jacobian variety $J$. In this paper, we study the canonical height $\hat{h} \colon J(K) \to \mathbb R$. More specifically, we consider the following two problems, which are important in applications: (1) for a given $P \in J(K)$, compute $\hat{h}(P)$ efficiently; (2) for a given bound $B > 0$, find all $P \in J(K)$ with $\hat{h}(P) \le B$. We develop an algorithm running in polynomial time (and fast in practice) to deal with the first problem. Regarding the second problem, we show how one can tweak the naive height $h$ that is usually used to obtain significantly improved bounds for the difference $h - \hat{h}$, which allows a much faster enumeration of the desired set of points. Our approach is to use the standard decomposition of $h(P) - \hat{h}(P)$ as a sum of local `height correction functions'. We study these functions carefully, which leads to efficient ways of computing them and to essentially optimal bounds. To get our polynomial-time algorithm, we have to avoid the factorization step needed to find the finite set of places where the correction might be nonzero. The main innovation at this point is to replace factorization into primes by factorization into coprimes. Most of our results are valid for more general fields with a set of absolute values satisfying the product formula. An analogous approach to (1) above for elliptic curves was used in our recent paper arXiv:1509.08748.

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arXiv2015

Computing Canonical Heights on Elliptic Curves in Quasi-Linear Time

We introduce an algorithm that can be used to compute the canonical height of a point on an elliptic curve over the rationals in quasi-linear time. As in most previous algorithms, we decompose the difference between the canonical and the naive height into an archimedean and a non-archimedean term. Our main contribution is an algorithm for the computation of the non-archimedean term that requires no integer factorization and runs in quasi-linear time.

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