35 citations · 93 across the 19 of their papers we have counts for
27 papers
Dynamical Canonical Heights and Finite Trees
Philipp Habegger, Harry Schmidt
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 f…
Degeneracy loci in the universal family of abelian varieties
Ziyang Gao, Philipp Habegger
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 Relative Manin-Mumford Conjecture
Ziyang Gao, Philipp Habegger
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…
Integer matrices with a given characteristic polynomial and multiplicative dependence of matrices
Philipp Habegger, Alina Ostafe, Igor E. Shparlinski
We consider the set of -matrices with integer elements of size at most and obtain a new upper bound on the number of matrices from $\ma…
Lower Bounds for the Canonical Height of a Unicritical Polynomial and Capacity
Philipp Habegger, Harry Schmidt
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…
A consequence of the relative Bogomolov conjecture
Vesselin Dimitrov, Ziyang Gao, Philipp Habegger
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 co…
Exponential periods and o-minimality
Johan Commelin, Philipp Habegger, Annette Huber
Let be an exponential period. We show that the real and imaginary part of are up to signs volumes of sets definable in the o-minimal structure generated by $\…
Uniformity in Mordell-Lang for curves
Vesselin Dimitrov, Ziyang Gao, Philipp Habegger
Consider a smooth, geometrically irreducible, projective curve of genus defined over a number field of degree . It has at most finitely many rational points by t…
Galois orbits of torsion points near atoral sets
Vesselin Dimitrov, Philipp Habegger
We prove that the Galois equidistribution of torsion points of the algebraic torus extends to the singular test functions of the form , where is a…
Uniform bound for the number of rational points on a pencil of curves
Vesselin Dimitrov, Ziyang Gao, Philipp Habegger
Consider a one-parameter family of smooth, irreducible, projective curves of genus defined over a number field. Each fiber contains at most finitely many rational points b…
The geometric Bogomolov conjecture
Serge Cantat, Ziyang Gao, Philipp Habegger +1
We prove the geometric Bogomolov conjecture over a function field of characteristic zero.
No singular modulus is a unit
Yu. Bilu, P. Habegger, L. Kühne
A result of the second-named author states that there are only finitely many CM-elliptic curves over whose -invariant is an algebraic unit. His proof depends on Duk…
Heights in families of abelian varieties and the Geometric Bogomolov Conjecture
Ziyang Gao, Philipp Habegger
On an abelian scheme over a smooth curve over a symmetric relatively ample line bundle defines a fiberwise Néon-Tate height. If the base curve is inside a pr…
The Norm of Gaussian Periods
P. Habegger
Gaussian periods are cyclotomic integers with a long history in number theory and connections to problems in combinatorics. We investigate the asymptotic behavior of the absolute n…
Diophantine Approximations on Definable Sets
P. Habegger
Consider the vanishing locus of a real analytic function on restricted to . We bound the number of rational points of bounded height that approximate this s…
Quasi-equivalence of Heights and Runge's Theorem
P. Habegger
Let be a polynomial that depends on two variables and and has algebraic coefficients. If and are algebraic numbers with , then by work of Néron $h(x)/…
Bad reduction of genus curves with CM jacobian varieties
Philipp Habegger, Fabien Pazuki
We show that a genus 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 Fa…
Teichmueller curves in genus three and just likely intersections in
Matt Bainbridge, Philipp Habegger, Martin Moeller
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 holomo…
O-minimality and certain atypical intersections
Philipp Habegger, Jonathan Pila
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…
Small points and free abelian groups
Robert Grizzard, Philipp Habegger, Lukas Pottmeyer
Let be an algebraic extension of the rational numbers and an elliptic curve defined over some number field contained in . The absolute logarithmic Weil height, respectiv…
Singular Moduli that are Algebraic Units
Philipp Habegger
We prove that only finitely many -invariants of elliptic curves with complex multiplication are algebraic units. A rephrased and generalized version of this result resembles Sie…
A Note on Divisible Points of Curves
Martin Bays, Philipp Habegger
Let 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…
The Tate-Voloch Conjecture in a Power of a Modular Curve
Philipp Habegger
Let be a prime. Tate and Voloch proved that a point of finite order in the algebraic torus cannot be -adically too close to a fixed subvariety without lying on it. The curre…
Effective Height Upper Bounds on Algebraic Tori
Philipp Habegger
The main emphasis will be on height upper bounds in the algebraic torus G^{n}_{m}. By height we will mean the absolute logarithmic Weil height. Section 3.2 contains a precise defin…