Heights and unlikely intersections

• Applicant: Professor Dr. Philipp Habegger
• Subject Area: Mathematics
• Term: from 2012 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 223746744

• Final Report Year: 2017

Final Report Abstract

Problems in unlikely intersections, introduced by Bombieri-Masser-Zannier, Pink, and Zilber, are intertwined with the theory of height functions. In collaboration with Bays and Pila we increased our understanding on unlikely intersections for curves and some surfaces using methods from model theory, specifically o-minimality, and other fields. The theory of heights was very helpful in solving a problem on Teichmüller curves in a collaboration of the PI with Bainbridge and Möller. Apart from theoretical work, this called for the efficient implementation of methods from unlikely intersections in the computer algebra system Sage. Points of small height appear naturally when investigating unlikely intersections in (semi-)abelian varieties. Studying small height goes back at least to Lehmer's work on primes. We shed new light on infinite algebraic extensions of the rationals that have a gap at the bottom of the height spectrum and explored the interplay between the structure of the multiplicative group of a field and the existence of a height gap. Connections between small height in the context of arithmetic dynamics led to the resolution of a conjecture of Narkiewicz by Pottmeyer, a young researcher supported by the grant. In a surprise he also gave the first example of a pseudo algebraically closed field that admits a height gap. Height lower bounds in connection with abelian surfaces with complex multiplication enabled Pazuki and the PI to investigate integrality properties of special points and prove a finiteness result.

Publications

• Bad reduction of genus 2 curves with cm jacobian varieties
  P. Habegger and F. Pazuki
  
• Quantitative Height Bounds under Splitting Conditions
  P. Fili and L. Pottmeyer
  
• A note on divisible points of curves, Trans. Amer. Math. Soc. 367 (2015), no. 2, 1313–1328
  M. Bays and P. Habegger
  
• Heights and totally p-adic numbers, Acta Arith. 171 (2015), no. 3, 277–291
  L. Pottmeyer
  
• Small points and free abelian groups, Int. Math. Res. Not. IMRN (2015), no. 20, 10657–10679
  R. Grizzard, P. Habegger, and L. Pottmeyer
  
• A note on extensions of Qtr , J. Théor. Nombres Bordeaux 28 (2016), no. 3, 735–742
  L. Pottmeyer
  
• O-minimality and certain atypical intersections, Ann. Sci. Ec. Norm. Supér. (4) 49 (2016), no. 4, 813–858
  P. Habegger and J. Pila
  
• Teichmüller curves in genus three and just likely intersections in Gm^n × Gn^a , Publ. Math. Inst. Hautes Etudes Sci. 124 (2016), 1–98
  M. Bainbridge, P. Habegger, and M. Möller