Forschungsgruppe Zahlentheorie
Die Forschungsgruppe Zahlentheorie beschäftigt sich mit Verbindungen der Zahlentheorie zur Geometrie und mit der analytischen Zahlentheorie. Sie wird von Prof. Habegger und Prof. Le Boudec geleitet.
Eine interessante Frage ist, wie viele Punkte mit ganzzahligen oder rationalen Koordinaten auf einer ebenen Kurve liegen. Oder kann die Summe zweier Kubikzahlen, wie z.B. 1, 8, 27 usw., wieder eine Kubikzahl sein? Auch ein solches Problem lässt sich mit geometrischen Werkzeugen untersuchen.
Besonders elliptische Kurven sind im Fokus der Forschung. Mittels der rechts abgebildeten Sehnen- und Tangentenkonstruktion lassen sich zwei Punkte zu einem dritten Punkt «addieren». Diese «Addition auf der Kurve» verbindet Geometrie und Arithmetik und ist ein mächtiges Werkzeug, um die elliptische Kurve zu studieren. Sie spielt in Anwendungen der Kryptographie eine wichtige Rolle.
Mitglieder
Assoziertes Mitglied und Lehrbeauftragter: Dr. Fabrizio Barroero Office 03.002, Spiegelgasse 1
Publications
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, 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, 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, pp. 6465 -6491. doi: 10.1090/tran/6893 . edoc | Open Access
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.
Habegger, P. (2017) “The Norm of Gaussian Periods”, The Quaterly Journal of Mathematics, p. 30.
Habegger, P. and Pazuki, F. (2017) “Bad reduction of genus 2 curves with CM jacobian varieties”, Compositio mathematica, p. 42.
Masser, D. and Brownawell, W. D. (2017) “Zero estimates with moving targets”, J. London Math. Soc, 95(2), pp. 441-455.
Masser, D., Amoroso, F. and Zannier, U. (2017) “Bounded height in pencils of finitely generated subgroups”, Duke Math. J, 166(13), pp. 2599-2642.
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.
Masser, D., Bilu, Y. and Luca, F. (2017) “Collinear CM-points”, Algebra and Number Theory, 11(5), pp. 1047-1087.
Habegger, P. and Ih, S. (2017) “Distribution of integral division points on the algebraic torus”, Transactions of the American Mathematical Society, p. 30.
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
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
Habegger, P. (2016) “Effective Height Upper Bounds on Algebraic Tori”, in Hindry, M. (ed.) Panoramas et synthèses. to appear: Société Mathématique de France (La conjecture de Zilber-Pink), p. 68ages.
Vávra, T. and Veneziano, F. (2015) “Pisot units in number fields”. ArXiv. Available at: https://arxiv.org/abs/1512.08786. edoc
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
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., Vol. 8, H. 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, Vol. 65, H. 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, Vol. 141, H. 1, 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, Vol. 24, H. 4 , S. 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, Vol. 154, H. 1 , S. 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, Vol. 104, H. 5, pp. 1045-1083. edoc
Masser, D. and Zannier, U. (2012) “Torsion points on families of squares of elliptic curves”, Mathematische Annalen. Springer, Vol. 352, H. 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, Vol. 137, fasc. 1, pp. 93-125. edoc
Habegger, P. (2009) “Multiplicative dependence and isolation II”, The journal of the London Mathematical Society. Oxford University Press, Vol. 80, H. 2, pp. 495-513. edoc
Habegger, P. (2009) “Intersecting subvarieties of abelian varieties with algebraic subgroups of complementary dimension”, Inventiones mathematicae. Springer International, Vol. 176, H. 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, Vol. 342, H. 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