Number Theory Research Group

The Number Theory Research Group works on the connections of number theory to geometry and analytic number theory. It is led by Prof. Habegger and Prof. Le Boudec.

One interesting question is how many points with integral or rational coordinates lie on a plane curve. Or can the sum of two cube numbers, such as 1, 8, 27, and so on, also be a cubic number? Such a problem can also be investigated with geometric tools.The group’s research especially focuses on elliptic curves. Using the construction of chords and tangents depicted on the right, two points can be “added” to a third point.
This “addition on the curve” connects geometry and arithmetic and is a powerful tool for studying elliptic curves. It plays an important role in applications in cryptography.

Members

Associated member and lecturer: Dr. Fabrizio Barroero Office 03.002, Spiegelgasse 1

Upcoming Events

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Publications

Gao, Z. and Habegger, P. (2019) ‘Heights in families of abelian varieties and the geometric Bogomolov conjecture’, Preprints Fachbereich Mathematik. Universität Basel.   edoc | Open Access
Bilu, Y., Habegger, P. and Kühne, L. (2018) ‘No singular modulus is a unit’, International mathematics research notices, p. 37ages. doi: 10.1093/imrn/rny274.   
Le Boudec, P. (2018) ‘Average rank in families of quadratic twists: a geometric point of view’, Math. Ann, 371(1-2), pp. 695-705. doi: 10.1007/s00208-017-1619-y.   
Le Boudec, P. (2018) ‘On the distribution of squarefree integers in arithmetic progressions’, Math. Z, 290(1-2), pp. 421-429. doi: 10.1007/s00209-017-2023-8.   
Gao, Z. and Habegger, P. (2018) ‘Heights in families of abelian varieties and the geometric Bogomolov Conjecture’, Annals of mathematics, p. 78ages.   
Corvaja, P., Masser, D. and Zannier, U. (2018) ‘Torsion curves on abelian schemes and Betti coordinates’, Math. Annalen, 371, pp. 1013-1045.   
Derksen, H. and Masser, D. (2018) ‘Linear equations over multiplicative groups, recurrences, and mixing III’, Ergodic Th. and Dynamical Systems, 38, pp. 2625-2643.   
Corvaja, P., Masser, D. and Zannier, U. (2017) ‘Torsion hypersurfaces on abelian schemes and Betti coordinates’, Preprints Fachbereich Mathematik. Universität Basel.   edoc | Open Access
Bilu, Y., Luca, F. and Masser, D. (2017) ‘Collinear CM-points’, Preprints Fachbereich Mathematik. Universität Basel.   edoc | Open Access
Masser, D. (2017) ‘Abcological anecdotes’, Preprints Fachbereich Mathematik. Universität Basel.   edoc | Open Access
Masser, D. and Brownawell, W. D. (2017) ‘Unlikely intersections for curves in additive groups over positive characteristic’, Proceedings of the American Mathematical Society. American Mathematical Society, 145(11), pp. 4617-4627. doi: 10.1090/proc/13617.   edoc
Dill, G. A. (2017) ‘Effective approximation and Diophantine applications’, Acta Arithmetica. Institute of Mathematics, Polish Academy of Sciences, 177(2), pp. 169-199. doi: 10.4064/aa8430-9-2016.   edoc
Canci, J. K. and Vishkautsan, S. (2017) ‘Quadratic maps with a periodic critical point of period 2’, International Journal of Number Theory. World Scientific Publishing Company, 13(6), pp. 1393-1417. doi: 10.1142/S1793042117500786.   edoc
Checcoli, S., Veneziano, F. and Viada, E. (2017) ‘On the explicit Torsion Anomalous Conjecture’, Transactions of the American Mathematical Society. American Mathematical Society, 369(9), pp. 6465 -6491. doi: 10.1090/tran/6893 .   edoc | Open Access
Habegger, P. (2017) ‘The Norm of Gaussian Periods’, The Quaterly Journal of Mathematics, p. 30.   
Masser, D., Habegger, P. and Jones. G. O. (2017) ‘Six unlikely intersection problems in search of effectivity’, Math. Proc. Cambridge Phil. Soc, 162(3), pp. 447-477.   
Masser, D. and Brownawell, W. D. (2017) ‘Zero estimates with moving targets’, J. London Math. Soc, 95(2), pp. 441-455.   
Habegger, P. and Pazuki, F. (2017) ‘Bad reduction of genus 2 curves with CM jacobian varieties’, Compositio mathematica, p. 42.   
Masser, D., Amoroso, F. and Zannier, U. (2017) ‘Bounded height in pencils of finitely generated subgroups’, Duke Math. J, 166(13), pp. 2599-2642.   
Masser, D. (2017) ‘Integration in elementary terms’, Newsletter London Math. Soc, 473, pp. 30-36.   
Canci, J. K., Solomon and Vishkautsan (2017) ‘Scarcity of cycles for rational functions over a number field’, Transaction of the american mathematical society. American Mathematical Soiety, p. 26. doi: 10.1090/tran/7217.   
Habegger, P. and Ih, S. (2017) ‘Distribution of integral division points on the algebraic torus’, Transactions of the American Mathematical Society, p. 30.   
Masser, D., Bilu, Y. and Luca, F. (2017) ‘Collinear CM-points’, Algebra and Number Theory, 11(5), pp. 1047-1087.   
Vávra, T. and Veneziano, F. (2017) ‘Pisot unit generators in number fields’, Journal of Symbolic Computation. Elsevier Ltd, p. 15. doi: 10.1016/j.jsc.2017.11.005.   
Habegger, P. (2017) ‘Diophantine Approximations on Definable Sets’, Selecta mathematica, p. 43.   
Checcoli, S., Veneziano, F. and Viada, E. (2016) ‘The explicit Mordell Conjecture for families of curves (with an appendix by M. Stoll)’. Cornell: Cornell University Library. Available at: https://arxiv.org/pdf/1602.04097.pdf.   edoc
Brownawell, W. D. and Masser, D. (2016) ‘Zero estimates with moving targets’, Preprints Fachbereich Mathematik. Universität Basel.   edoc | Open Access
Brownawell, W. D. and Masser, D. (2016) ‘Unlikely intersections for curves in additive groups over positive characteristic’, Preprints Fachbereich Mathematik. Universität Basel.   edoc | Open Access
Habegger, P. (2016) ‘The norm of a Gaussian periods’, Preprints Fachbereich Mathematik. Universität Basel.   edoc | Open Access
Bainbridge, M., Habegger, P. and Möller, M. (2016) ‘Teichmüller curves in genus three and just likely intersections in ${\bf G}_m^n\times{\bf G_a}^n$’, Preprints Fachbereich Mathematik. Universität Basel.   edoc | Open Access
Pillay, A., Bertrand, D., Masser, D. and Zannier, U. (2016) ‘Relative Manin-Mumford for semi-abelian surfaces’, Preprints Fachbereich Mathematik. Universität Basel.   edoc | Open Access
Canci, J. K. and Paladino, L. (2016) ‘On preperiodic points of rational functions defined over F_p(t)’, Preprints Fachbereich Mathematik. Universität Basel.   edoc | Open Access
Capuano, L., Masser, D., Pila, J. and Zannier, U. (2016) ‘Rational points on Grassmannians and unlikely intersections in tori’, Preprints Fachbereich Mathematik. Universität Basel.   edoc | Open Access
Derksen, H. and Masser, D. (2016) ‘Linear equations over multiplicative groups recurrences, and mixing III’, Preprints Fachbereich Mathematik. Universität Basel.   edoc | Open Access
Amoroso, F. and Masser, D. (2016) ‘Lower bounds for the height in Galois extensions’, Preprints Fachbereich Mathematik. Universität Basel.   edoc | Open Access
Dill, G. (2016) ‘Effective approximation and Diophantine applications’, Preprints Fachbereich Mathematik. Universität Basel.   edoc | Open Access
Habegger, P. (2016) ‘Diophantine approximations on definable sets’, Preprints Fachbereich Mathematik. Universität Basel.   edoc | Open Access
Habegger, P. (2016) ‘Effective Height Upper Bounds on Algebraic Tori’, in Habegger, P., Rémond, G., Scanlon, T., Emmanuel, U., and Yafaev, A. (eds) Panoramas et synthèses. Paris: Société mathématique de France (Around the Zilber-Pink conjecture), pp. 167-242.   edoc
Habegger, P. and Pila, J. (2016) ‘O-minimality and certain atypical intersections’, Annales Scientifiques de l’École Normale Supérieure. Société Mathématique de France, 49(4), pp. 813-858. Available at: http://smf4.emath.fr/Publications/AnnalesENS/4_49/html/ens_ann-sc_49_813-858.php.   edoc
Pottmeyer, L. (2016) ‘A Note on Extensions of ℚtr’, Journal de Théorie des Nombres de Bordeaux. Institut de Mathématiques de Bordeaux, 28(3), pp. 735-742.   edoc
Canci, J. K. and Paladino, L. (2016) ‘Preperiodic points for rational functions defined over a global field in terms of good reduction’, Proceedings of the American Mathematical Society. American Mathematical Society, 144(12), p. 18. doi: 10.1090/proc/13096.   edoc
Bainbridge, M., Habegger, P. and Möller, M. (2016) ‘Teichmüller curves in genus three and just likely intersections in \mathbf{G}_{m}^{n}\times\mathbf{G}_{a}^{n}’, Publications mathematiques de l’IHES. Springer, 124(1), pp. 1-98. doi: 10.1007/s10240-016-0084-6.   edoc
Canci, J. K. and Paladino, L. (2016) ‘On preperiodic points for rational functions defined over $mathbb{F}_p(t)$’, Rivista di Matematica della Università di Parma. Università di Parma, 7(1), p. 12. Available at: https://arxiv.org/pdf/1601.07293.pdf.   edoc
Habegger, P., Jones, G. and Masser, D. (2016) ‘Six unlikely intersection problems in search of effectivity’, Mathematical Proceedings of the Cambridge Philosophical Society. Cambridge University Press, 162(3), pp. 447-477. doi: 10.1017/S0305004116000682.   edoc
Habegger, P. (2016) ‘Quasi-equivalence of Heights and Runge’s Theorem’, in Elsholtz, C. and Grabner, P. (eds). Cham: Springer (Diophantine problems, uniform distribution and applications, Festschrift in honour of Robert F. Tichy’s 60th birthday), pp. 257-280.   edoc
Vávra, T. and Veneziano, F. (2015) ‘Pisot units in number fields’. ArXiv. Available at: https://arxiv.org/abs/1512.08786.   edoc
Masser, D. and Zannier, U. (2015) ‘Torsion points on families of simple abelian surfacesand Pell’s equation over polynomial rings’, Preprints Fachbereich Mathematik. Universität Basel.   edoc | Open Access
Habegger, P. (2015) ‘Singular moduli that are algebraic units’, Preprints Fachbereich Mathematik. Universität Basel.   edoc | Open Access
Habegger, P., Jones, G. and Masser, D. (2015) ‘Six unlikely intersection problems in search of effectivity’, Preprints Fachbereich Mathematik. Universität Basel.   edoc | Open Access
Canci, J. K. and Vishkautsan, S. (2015) ‘Quadratic maps with a periodic critical point of period 2’, Preprints Fachbereich Mathematik. Universität Basel.   edoc | Open Access
Canci, J. K. (2015) ‘Preperiodic points for rational functions defined over a rational function field of characteristic zero’, Preprints Fachbereich Mathematik. Universität Basel.   edoc | Open Access
Canci, J. K. and Paladino, L. (2015) ‘Preperiodic points for rational functions defined over a global field in terms of good reduction’, Preprints Fachbereich Mathematik. Universität Basel.   edoc | Open Access
Habegger, P. and Pila, J. (2015) ‘O-minimality and certain atypical intersections’, Preprints Fachbereich Mathematik. Universität Basel.   edoc | Open Access
Canci, J. K. (2015) ‘Good reduction for endomorphisms of the projective line in terms of the branch locus’, Preprints Fachbereich Mathematik. Universität Basel.   edoc | Open Access
Habegger, P. and Pazuki, F. (2015) ‘Bad reduction of genus 2 curves with CM jacobian varieties’, Preprints Fachbereich Mathematik. Universität Basel.   edoc | Open Access
Bays, M. and Habegger, P. (2015) ‘A note on divisible points of curves’, Transactions of the American Mathematical Society. American Mathematical Society, 367(2), pp. 1313-1328. doi: 10.1090/S0002-9947-2014-06494-5.   edoc
Habegger, P. (2015) ‘Singular Moduli that are Algebraic Units’, Algebra and Number Theory. Mathematical Sciences Publishers, 9(7), pp. 1515-1524. doi: 10.2140/ant.2015.9.1515.   edoc | Open Access
Masser, D. (2015) ‘Relative Manin-Mumford for abelian varieties ’, in Jones, G. O. and Wilkie, A. J. (eds) London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press (O-Minimality and Diophantine Geometry), pp. 193-203. doi: 10.1017/CBO9781316106839.008.   edoc
Habegger, P. (2015) ‘The Manin-Mumford conjecture, an elliptic curve, its torsion points & their Galois orbits’, in Jones, G. and Wilkie, A. (eds) London Mathematical Society lecture note series. Cambridge: Cambridge University Press (O-Minimality and Diophantine Geometry), pp. 1-40. doi: 10.1017/CBO9781316106839.002.   edoc
Ghioca, D., Masser, D. and Zannier, U. (2015) ‘Bounded height conjecture for function fields’, New York journal of mathematics. European Mathematical Society, 21, pp. 837-846.   edoc
Masser, D. and Zannier, U. (2015) ‘Torsion points on families of abelian surfaces and Pell’s equation over polynomial rings’, Journal of the European Mathematical Society. EMS, 17(9), pp. 2379-2416. doi: 10.4171/JEMS/560.   edoc
Derksen, H. and Masser, D. (2015) ‘Linear equations over multiplicative groups, recurrences, and mixing II’, Indagationes Mathematicae . Elsevier, 26(1), pp. 113-136. doi: 10.1016/j.indag.2014.08.002.   edoc
Pottmeyer, L. (2015) ‘Heights of points with bounded ramification’, Annali della Scuola Normale di Pisa - Classe di Scienze. Scuola Normale Superiore, Pisa, 14(3), pp. 965-981. doi: 10.2422/2036-2145.201302_002.   edoc
Grizzard, R., Habegger, P. and Pottmeyer, L. (2015) ‘Small points and free abelian groups’, International mathematics research notices. Hindawi Publ. Corp., 20, pp. 10657-10679. doi: 10.1093/imrn/rnv007.   edoc
Pottmeyer, L. (2015) ‘Heights and totally p-adic numbers’, Acta arithmetica. Seminarjum matematyczne, 171(3), pp. 277-291. doi: 10.4064/aa171-3-5.   edoc
Canci, J. K. (2015) ‘Preperiodic points for rational functions defined over a rational function field of characteristic zero’, New York journal of mathematics. European Mathematical Society, (21), pp. 1295-1310. Available at: http://nyjm.albany.edu/j/2015/21-59v.pdf.   edoc | Open Access
Schmidt, H. (2015) Multiplication polynomials and relative Manin-Mumford. doi: 10.5451/unibas-006466673.   edoc | Open Access
Masser, D. (2014) ‘Relative Manin-Mumford for Abelian Varieties’, Preprints Fachbereich Mathematik. Universität Basel.   edoc | Open Access
Masser, D. and Wüstholz, G. (2014) ‘Polarization Estimates for Abelian Varieties’, Preprints Fachbereich Mathematik. Universität Basel.   edoc | Open Access
Derksen, H. and Masser, D. (2014) ‘Linear Equations over Multiplicative Groups, Recurrences, and Mixing II’, Preprints Fachbereich Mathematik. Universität Basel.   edoc | Open Access
Habegger, P. (2014) ‘The Tate-Voloch conjecture in a power of a modular curve’, International Mathematics Research Notices. Oxford University Press, 2014(12), pp. 3303-3339. doi: 10.1093/imrn/rnt025.   edoc
Checcoli, S., Veneziano, F. and Viada, E. (2014) ‘On torsion anomalous intersections’, Rendiconti Lincei. Matematica e Applicazioni. Accademia Nazionale dei Lincei, 25(1), pp. 1-36. doi: 10.4171/RLM/666.   edoc
Masser, D. and Wuestholz, G. (2014) ‘Polarization estimates for abelian varieties’, Algebra & number theory. Mathematical Sciences Publ., 8(5), pp. 1045-1070.   edoc
Masser, D. and Zannier, U. (2014) ‘Torsion points of families of products of elliptic curves’, Advances in mathematics. Academic Press, Vol. 259, pp. 116-133.   edoc
Masser, D. (2014) ‘Unlikely intersections for curves in multiplicative groups over positive characteristic’, The quarterly journal of mathematics. Oxford University Press, 65(2), pp. 505-515.   edoc
Dvornicich, R., Veneziano, F. and Zannier, U. (2014) ‘On the integral values of a curious recurrence’. arXiv. Available at: http://arxiv.org/abs/1403.3470.   
Habegger, P. (2013) ‘Small height and infinite nonabelian extensions’, Duke Mathematical Journal. Duke University Press, 162(11), pp. 2027-2076. doi: 10.1215/00127094-2331342.   edoc
Habegger, P. (2013) ‘Torsion points on elliptic curves in Weierstrass form’, Annali della Scuola Normale di Pisa - Classe di Scienze. Scuola Normale Superiore, 12(3), pp. 687-715. doi: 10.2422/2036-2145.201105_001.   edoc
Habegger, P. (2013) ‘Special points on fibered powers of elliptic surfaces’, Journal für die reine und angewandte Mathematik. De Gruyter, 685, pp. 143-179. doi: 10.1515/crelle-2012-0007.   edoc
Pottmeyer, L. (2013) ‘Heights and totally real numbers’, Rendiconti Lincei. Matematica e applicazioni. Accademia nazionale dei Lincei, 24(4), pp. 471-483. doi: 10.4171/RLM/662.   edoc
Canci, J. K., Peruginelli Giulio and Tossici, D. (2013) ‘On some notions of good reduction for endomorphisms of the projective line’, Manuscripta mathematica. Springer, 141(1-2), p. 17. doi: 10.1007/s00229-012-0573-y.   edoc
Chatzidakis, Z., Ghioca, D., Masser, D. and Maurin, G. (2013) ‘Unlikely, likely and impossible intersections without algebraic groups’, Atti della Accademia nazionale dei Lincei. Classe di scienze fisiche, matematiche e naturali. Rendiconti lincei. Matematica e applicazioni. Accademia nazionale dei Lincei, 24(4), pp. 485-501.   edoc
Bilu, Y., Masser, D. and Zannier, U. (2013) ‘An effective “theorem of André” for CM-points on a plane curve’, Mathematical proceedings of the Cambridge Philosophical Society. Cambridge University Press, 154(1), pp. 145-152.   edoc
Leitner, D. (2013) Aspects of x + y - z = 1 in positive characteristic. doi: 10.5451/unibas-006116269.   edoc | Open Access
Habegger, P. and Pila, J. (2012) ‘Some unlikely intersections beyond André-Oort’, Compositio Mathematica. Cambridge University Press, 148(1), pp. 1-27. doi: 10.1112/S0010437X11005604.   edoc
Checcoli, S., Veneziano, F. and Viada, E. (2012) ‘A sharp Bogomolov-type bound’, New York journal of mathematics. European Mathematical Society, 18, pp. 891-910. Available at: http://nyjm.albany.edu:8000/j/2012/18_891.html.   edoc | Open Access
Zannier, U. (2012) Some problems of unlikely intersections in arithmetic and geometry, Annals of mathematics studies. Princeton: Princeton University Press.   edoc
Derksen, H. and Masser, D. (2012) ‘Linear equations over multiplicative groups, recurrences, and mixing I’, Proceedings of the London Mathematical Society. Oxford University Press, 104(5), pp. 1045-1083.   edoc
Masser, D. and Zannier, U. (2012) ‘Torsion points on families of squares of elliptic curves’, Mathematische Annalen. Springer, 352(2), pp. 453-484.   edoc
Veneziano, F. (2011) ‘Quadratic integral solutions to double Pell equations’, Rendiconti del Seminario Matematico della Università di Padova. Libreria Internazionale Cortina, 126, pp. 47-61. doi: 10.4171/RSMUP/126-3.   edoc
Bombieri, E., Habegger, P., Masser, D. and Zannier, U. (2010) ‘A note on Maurin’s theorem’, Rendiconti Lincei. Matematica e Applicazioni. European Mathematical Society, 21(3), pp. 251-260. doi: 10.4171/RLM/570.   edoc
Habegger, P. (2010) ‘Weakly bounded height on modular curves’, Acta Mathematica Vietnamica. Institute of Mathematics, 35(1), pp. 43-69. Available at: http://journals.math.ac.vn/acta/pdf/1001043.pdf.   edoc
Canci, J. K. (2010) ‘Rational periodic points for quadratic maps’, Annales de l’Institut Fourier. Institut Fourier, 60(3), p. 33. doi: 10.5802/aif.2544.   edoc
Habegger, P. (2009) ‘On the Bounded Height Conjecture’, International mathematics research notices. Hindawi Publ. Corp., 2009, no. 5, pp. 860-886.   edoc
Habegger, P. (2009) ‘A Bogomolov property modulo algebraic subgroups’, Bulletin de la Société mathématique de France. Société mathématique de France, 137(1), pp. 93-125.   edoc
Habegger, P. (2009) ‘Multiplicative dependence and isolation II’, The journal of the London Mathematical Society. Oxford University Press, 80(2), pp. 495-513.   edoc
Habegger, P. (2009) ‘Intersecting subvarieties of abelian varieties with algebraic subgroups of complementary dimension’, Inventiones mathematicae. Springer International, 176(2), pp. 405-447.   edoc
Masser, D. (2009) ‘Multiplicative dependence of values of algebraic functions’, in. Cambridge: Cambridge University Press (Analytic Number Theory : Essays in Honour of Klaus Roth), pp. 324-333.   edoc
Habegger, P. (2008) ‘Intersecting subvarieties of ${mathbf G}_m^n$ with algebraic subgroups’, Mathematische Annalen. Springer, 342(2), pp. 449-466.   edoc
Widmer, M. L. (2008) Asymptotically counting points of bounded height. doi: 10.5451/unibas-004414562.   edoc | Open Access
Canci, J. K. (2007) ‘Finite orbits for rational maps’, Indagationes Mathematicae . Elsevier, 18(2), p. 12. doi: 10.1016/S0019-3577(07)80017-6.   edoc
Habegger, P. (2007) Heights and multiplicative relations on algebraic varieties. doi: 10.5451/unibas-004256285.   edoc | Open Access
Habegger, P. (2007) ‘Multiplicative dependence and isolation I’, in Zannier, U. (ed.). Pisa: Publications of the Scuola Normale Superiore (Diophantine Geometry, Scuola Normale Superiore), p. ?.   
Canci, J. K. (2006) ‘Cycles for rational maps of good reduction outside a prescribed set’, Monatshefte für Mathematik. Springer, 149(4), pp. 265-287. doi: 10.1007/s00605-006-0387-7.   edoc
Habegger, P. (2005) ‘The equation $x+y=alpha$ in multiplicatively dependent unknowns’, Acta arithmetica. Seminarjum matematyczne, Vol. 119, pp. 349-372.   edoc