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Philipp Habegger

Publications and source records attributed to Philipp Habegger.

At least 19 recordsLinked to original sources
arXiv2026

Dynamical Canonical Heights and Finite Trees

We establish lower bounds for the canonical height of a wandering point that decays like the square of the field degree. Our methods and results apply to centered, postcritically finite, hyperbolic polynomials of prime power degree whose coefficients are algebraic integers. Our approach is ultimately inspired by an idea of Dimitrov that led to his proof of the Schinzel--Zassenhaus Conjecture. The height lower bound is derived from the lower bound of the local canonical height at an archimedean place. In previous work, the authors used a result of Dubinin on the transfinite diameter of a star-shaped tree. In this paper, we develop tools to bound the transfinite diameter of more general finite trees in the complex plane. We construct these trees using the Hubbard tree of a postcritically finite polynomial. Thurston's notion of core entropy helps us analyze combinatorial properties of the Hubbard tree.

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arXiv2023

The Relative Manin-Mumford Conjecture

We prove the Relative Manin-Mumford Conjecture for families of abelian varieties in characteristic 0. We follow the Pila-Zannier method to study special point problems, and we use the Betti map which goes back to work of Masser and Zannier in the case of curves. The key new ingredients compared to previous applications of this approach are a height inequality proved by both authors of the current paper and Dimitrov, and the first-named author's study of certain degeneracy loci in subvarieties of abelian schemes. We also strengthen this result and prove a criterion for torsion points to be dense in a subvariety of an abelian scheme over $\mathbb{C}$. The Uniform Manin-Mumford Conjecture for curves embedded in their Jacobians was first proved by K\"{u}hne. We give a new proof, as a corollary to our main theorem, that does not use equidistribution.

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arXiv2023

Degeneracy loci in the universal family of abelian varieties

Recent developments on the uniformity of the number of rational points on curves and subvarieties in a moving abelian variety rely on the geometric concept of the degeneracy locus. The first-named author investigated the degeneracy locus in certain mixed Shimura varieties. In this expository note we revisit some of these results while minimizing the use of mixed Shimura varieties while working in a family of principally polarized abelian varieties. We also explain their relevance for applications in diophantine geometry.

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arXiv2022

Integer matrices with a given characteristic polynomial and multiplicative dependence of matrices

We consider the set $\mathcal{M}_n(\mathbb Z; H)$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain a new upper bound on the number of matrices from $\mathcal{M}_n(\mathbb Z; H)$ with a given characteristic polynomial $f \in \mathbb Z[X]$, which is uniform with respect to $f$. This complements the asymptotic formula of A. Eskin, S. Mozes and N. Shah (1996) in which $f$ has to be fixed and irreducible. Using this result, among others, we obtain upper and lower bounds on the number of $s$-tuples of matrices from $\mathcal{M}_n(\mathbb Z; H)$, satisfying various multiplicative relations, including multiplicative dependence and bounded generation of a subgroup of $\mathrm{GL}_n(\mathbb Q)$. These problems generalise those studied in the scalar case $n=1$ by F. Pappalardi, M. Sha, I. E. Shparlinski and C. L. Stewart (2018) with an obvious distinction due to the non-commutativity of matrices. Motivated by these problems, we also prove various properties of the variety of complex matrices with fixed characteristic polynomial, including computing the degree of this variety.

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arXiv2021

Lower Bounds for the Canonical Height of a Unicritical Polynomial and Capacity

In a recent breakthrough, Dimitrov solved the Schinzel-Zassenhaus Conjecture. We follow his approach and adapt it to certain dynamical systems arising from polynomials of the form $T^p+c$ where $p$ is a prime number and where the orbit of $0$ is finite. For example, if $p=2$, and $0$ is periodic under $T^2+c$ with $c\in\mathbb{R}\smallsetminus\{-2\}$, we prove a lower bound for the local canonical height of a wandering algebraic integer that is inversely proportional to the field degree. From this we are able to deduce a lower bound for the canonical height of a wandering point that decays like the inverse square of the field degree.

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arXiv2020

A consequence of the relative Bogomolov conjecture

We propose a formulation of the relative Bogomolov conjecture and show that it gives an affirmative answer to a question of Mazur's concerning the uniformity of the Mordell-Lang conjecture for curves. In particular we show that the relative Bogomolov conjecture implies the uniform Manin-Mumford conjecture for curves. The proof is built up on our previous work "Uniformity in Mordell-Lang for curves".

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arXiv2020

Exponential periods and o-minimality

Let $\alpha \in \mathbb{C}$ be an exponential period. We show that the real and imaginary part of $\alpha$ are up to signs volumes of sets definable in the o-minimal structure generated by $\mathbb{Q}$, the real exponential function and ${\sin}|_{[0,1]}$. This is a weaker analogue of the precise characterisation of ordinary periods as numbers whose real and imaginary part are up to signs volumes of $\mathbb{Q}$-semi-algebraic sets. Furthermore, we define a notion of naive exponential periods and compare it to the existing notions using cohomological methods. This points to a relation between the theory of periods and o-minimal structures.

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arXiv2020

Uniformity in Mordell-Lang for curves

Consider a smooth, geometrically irreducible, projective curve of genus $g \ge 2$ defined over a number field of degree $d \ge 1$. It has at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of $g$, $d$, and the Mordell-Weil rank of the curve's Jacobian, thereby answering in the affirmative a question of Mazur. In addition we obtain uniform bounded, in $g$ and $d$, for the number of geometric torsion points of the Jacobian which lie in the image of an Abel-Jacobi map. Both estimates generalize our previous work for $1$-parameter families. Our proof uses Vojta's approach to the Mordell Conjecture, and the key new ingredient is the generalization of a height inequality due to the second- and third-named authors.

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arXiv2019

Galois orbits of torsion points near atoral sets

We prove that the Galois equidistribution of torsion points of the algebraic torus $\mathbb{G}_{m}^d$ extends to the singular test functions of the form $\log{|P|}$, where $P$ is a Laurent polynomial having algebraic coefficients that vanishes on the unit real $d$-torus in a set whose Zariski closure in $\mathbb{G}_m^d$ has codimension at least $2$. Our result includes a power saving quantitative estimate of the decay rate of the equidistribution. It refines an ergodic theorem of Lind, Schmidt, and Verbitskiy, of which it also supplies a purely Diophantine proof. As an application, we confirm Ih's integrality finiteness conjecture on torsion points for a class of atoral divisors of $\mathbb{G}_m^d$.

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arXiv2019

Uniform bound for the number of rational points on a pencil of curves

Consider a one-parameter family of smooth, irreducible, projective curves of genus $g\ge 2$ defined over a number field. Each fiber contains at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of the family and the Mordell--Weil rank of the fiber's Jacobian. Our proof uses Vojta's approach to the Mordell Conjecture furnished with a height inequality due to the second- and third-named authors. In addition we obtain uniform bounds for the number of torsion in the Jacobian that lie each fiber of the family.

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arXiv2019

Heights in families of abelian varieties and the Geometric Bogomolov Conjecture

On an abelian scheme over a smooth curve over $\overline{\mathbb Q}$ a symmetric relatively ample line bundle defines a fiberwise Néon-Tate height. If the base curve is inside a projective space, we also have a height on its $\overline{\mathbb Q}$-points that serves as a measure of each fiber, an abelian variety. Silverman proved an asymptotic equality between these two heights on a curve in the abelian scheme. In this paper we prove an inequality between these heights on a subvariety of any dimension of the abelian scheme. As an application we prove the Geometric Bogomolov Conjecture for the function field of a curve defined over $\overline{\mathbb Q}$. Using Moriwaki's height we sketch how to extend our result when the base field of the curve has characteristic 0.

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arXiv2015

Bad reduction of genus $2$ curves with CM jacobian varieties

We show that a genus $2$ curve over a number field whose jacobian has complex multiplication will usually have stable bad reduction at some prime. We prove this by computing the Faltings height of the jacobian in two different ways. First, we use a formula by Colmez and Obus specific to the CM case and valid when the CM field is an abelian extension of the rationals. This formula links the height and the logarithmic derivatives of an $L$-function. The second formula involves a decomposition of the height into local terms based on a hyperelliptic model. We use results of Igusa, Liu, and Saito to show that the contribution at the finite places in our decomposition measures the stable bad reduction of the curve and subconvexity bounds by Michel and Venkatesh together with an equidistribution result of Zhang to handle the infinite places.

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arXiv2015

A Note on Divisible Points of Curves

Let $C$ be an irreducible algebraic curve defined over a number field and inside an algebraic torus of dimension at least 3. We partially answer a question posed by Levin on points on $C$ for which a non-trivial power lies again on $C$. Our results have connections to Zilber's Conjecture on Intersections with Tori and yield to methods arising in transcendence theory and the theory of o-minimal structures.

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arXiv2014

Teichmueller curves in genus three and just likely intersections in $G_m^n x G_a^n$

We prove that the moduli space of compact genus three Riemann surfaces contains only finitely many algebraically primitive Teichmueller curves. For the stratum consisting of holomorphic one-forms in genus three with a single zero, our approach to finiteness uses the Harder-Narasimhan filtration of the Hodge bundle over a Teichmueller curve to obtain new information on the locations of the zeros of eigenforms. By passing to the boundary of moduli space, this gives explicit constraints on the cusps of Teichmueller curves in terms of cross-ratios of six points on a projective line. These constraints are akin to those that appear in Zilber and Pink's conjectures on unlikely intersections in diophantine geometry. However, in our case one is lead naturally to the intersection of a surface with a family of codimension two algebraic subgroups of $G_m^n \times G_a^n$ (rather than the more standard $G_m^n$). The ambient algebraic group lies outside the scope of Zilber's Conjecture but we are nonetheless able to prove a sufficiently strong height bound. For the generic stratum in genus three, we obtain global torsion order bounds through a computer search for subtori of a codimension-two subvariety of $G_m^9$. These torsion bounds together with new bounds for the moduli of horizontal cylinders in terms of torsion orders yields finiteness in this stratum. The intermediate strata are handled with a mix of these techniques.

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arXiv2014

O-minimality and certain atypical intersections

We show that the strategy of point counting in o-minimal structures can be applied to various problems on unlikely intersections that go beyond the conjectures of Manin-Mumford and André-Oort. We verify the so-called Zilber-Pink Conjecture in a product of modular curves on assuming a lower bound for Galois orbits and a sufficiently strong modular Ax-Schanuel Conjecture. In the context of abelian varieties we obtain the Zilber-Pink Conjecture for curves unconditionally when everything is defined over a number field. For higher dimensional subvarieties of abelian varieties we obtain some weaker results and some conditional results.

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arXiv2014

Small points and free abelian groups

Let $F$ be an algebraic extension of the rational numbers and $E$ an elliptic curve defined over some number field contained in $F$. The absolute logarithmic Weil height, respectively the Néron-Tate height, induces a norm on $F^*$ modulo torsion, respectively on $E(F)$ modulo torsion. The groups $F^*$ and $E(F)$ are free abelian modulo torsion if the height function does not attain arbitrarily small positive values. In this paper we prove the failure of the converse to this statement by explicitly constructing counterexamples.

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arXiv2014

Singular Moduli that are Algebraic Units

We prove that only finitely many $j$-invariants of elliptic curves with complex multiplication are algebraic units. A rephrased and generalized version of this result resembles Siegel's Theorem on integral points of algebraic curves.

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