Michael Stoll
Mathematisches Institut
Universität Bayreuth
95440 Bayreuth, Germany

Papers, Preprints and Lecture Notes

Arithmetic Geometry |  Combinatorics |  Mathematical Biology |  Group Theory |  Galois Theory
Jacobs Course Notes |  Other Course Notes |  German Lecture Notes |  Talk Notes
arXiv | MathSciNet | Zentralblatt MATH


Arithmetic Geometry


 
[AG58] with M. Derickx, S. Kamienny and W. Stein:
Torsion points on elliptic curves over number fields of small degree  arXiv
Preprint (2017).
 
[AG57] with B. Naskręcki and N. Freitas:
The generalized Fermat equation with exponents 2, 3, n (pdf, 650 kB; new version 2017-03-13)  arXiv
Preprint (2016-2017).
 
[AG56] with J.S. Müller:
Canonical heights on genus two Jacobians (pdf, 710 kB) (preprint version)  arXiv
Algebra & Number Theory 10, No. 10, 2153-2234 (2016). DOI: 10.2140/ant.2016.10.2153.
 
[AG55] with J.S. Müller:
Computing canonical heights on elliptic curves in quasi-linear time (pdf, 360 kB) (preprint version)  arXiv
LMS J. Comput. Math. 19 (Special issue A – Algorithmic Number Theory Symposium XII), 391-405 (2016).
This paper was awarded the Selfridge Prize 2016 at ANTS XII in Kaiserslautern.
 
[AG54] Chabauty without the Mordell-Weil group (pdf, 591 kB)  arXiv
Preprint (2015-2016), submitted.
 
[AG53] with I. Bauer:
Geometry and arithmetic of primary Burniat surfaces (pdf, 430 kB)  arXiv
Math. Nachr. (2017). DOI: 10.1002/mana.201600282
 
[AG52] Simultaneous torsion in the Legendre family (version from 2015-10-04; pdf, 514 kB)  arXiv
Experiment. Math. 26:4, 446-459 (2017)
Files containing the data from the computations can be found here.
 
[AG51] An Explicit Theory of Heights for Hyperelliptic Jacobians of Genus Three (pdf, 507 kB)  arXiv
Preprint (2014-2017), submitted. Related Magma files see here.
 
[AG50] Rational points on hyperelliptic curves: Recent developments (pdf, 26 kB)
Computeralgebra-Rundbrief 54, Fachgruppe Computeralgebra (2014).
 
[AG49] with S. Siksek:
The Generalised Fermat Equation x2 + y3 = z15 (pdf, 302 kB)  arXiv
Arch. Math. (Basel) 102:5, 411-421 (2014). DOI: 10.1007/s00013-014-0639-z
 
[AG48] Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell-Weil rank (pdf, 560 kB)  arXiv
to appear in J. Eur. Math. Soc.
See also the talk in Berkeley.
 
[AG47] with B. Poonen:
Most odd degree hyperelliptic curves have only one rational point (pdf, 570 kB)  arXiv
Ann. Math. 180:3, 1137-1166, (2014). DOI: 10.4007/annals.2014.180.3.7
See also the talk in Bad Boll.
 
[AG46] with N. Bruin and B. Poonen:
Generalized explicit descent and its application to curves of genus 3 (pdf, 676 kB)  arXiv
Forum of Mathematics, Sigma 4 (2016), e6, 80 pages. DOI: 10.1017/fms.2016.1
 
[AG45] with J.E. Cremona, T.A. Fisher, C. O'Neil and D. Simon:
Explicit n-descent on elliptic curves. III. Algorithms (pdf, 540 kB)  arXiv
Math. Comp. 84:292, 895-922 (2015), published online July 29, 2014.
 
[AG44] with R. Greenberg, K. Rubin and A. Silverberg:
On elliptic curves with an isogeny of degree 7 (pdf, 445 kB)  arXiv
Amer. J. Math., 136:1, 77-109 (2014). DOI: 10.1353/ajm.2014.0005
 
[AG43] with R. van Luijk:
Explicit Selmer groups for cyclic covers of P1 (pdf, 247 kB)  arXiv
Acta Arith. 159, 133-148 (2013). DOI: 10.4064/aa159-2-4
 
[AG42] with S. Siksek:
Partial descent on hyperelliptic curves and the generalized Fermat equation x3 + y4 + z5 = 0 (pdf, 284 kB)  arXiv
Bull. London Math. Soc. 44, 151-166 (2012), DOI: 10.1112/blms/bdr086
 
[AG41] with R.L. Miller:
Explicit isogeny descent on elliptic curves (pdf, 267 kB)  arXiv
Math. Comp. 82, 513-529 (2013) (published online June 11, 2012).
 
[AG40] with X. Faber and B. Hutz:
On the number of rational iterated pre-images of the origin under quadratic dynamical systems (pdf, 275 kB)  arXiv
Int. J. Number Theory 7:7, 1781-1806 (2011). DOI: 10.1142/S179304211100416.
MAGMA script for the computations
 
[AG39] with D. Testa:
The surface parametrizing cuboids (pdf, 192 kB)  arXiv
Preprint (2010).
 
[AG38] Rational points on curves (pdf, 307 kB)  arXiv
Extended notes of a talk given at the Journées Arithmétiques 2009.
Journal de Théorie des Nombres de Bordeaux 23, 257-277 (2011).
 
[AG37] How to solve a Diophantine equation (pdf, 208 kB)  arXiv
(Extended notes of a talk given at the IMO 2009.)
In: D. Schleicher, M. Lackmann (Eds): An invitation to mathematics, Springer (2011).
 
[AG36] with S. Siksek:
On a problem of Hajdu and Tengely (pdf, 244 kB)  arXiv
in: G. Hanrot, F. Morain, and E. Thomé (Eds.): ANTS-IX 2010, LNCS 6197, pp. 316-330. Springer, Heidelberg (2010).
See also the talk at ANTS-IX.
 
[AG35] with P.G. Walsh and Pingzhi Yuan:
On the Diophantine equation X2 - (22m+1) Y4 = -22m
Acta Arith. 139, 57-63 (2009). DOI: 10.4064/aa139-1-5
 
[AG34] Reduction theory of point clusters in projective space (pdf, 206 kB)  arXiv
Groups Geom. Dyn. 5, 553-565 (2011)
 
[AG33] with J.E. Cremona and T.A. Fisher:
Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves (pdf, 446 kB)  arXiv
Algebra & Number Theory 4, No. 6, 763-820 (2010). DOI: 10.2140/ant.2010.4.763
 
[AG32] Rational Points on Curves of Genus 2: Experiments and Speculations (pdf, 135 kB)
(Extended abstract of a talk in Dagstuhl, 2009), Dagstuhl Proceedings 09221
 
[AG31] with N. Bruin: The Mordell-Weil sieve: Proving non-existence of rational points on curves (pdf, 447 kB)  arXiv
LMS J. Comput. Math. 13, 272-306 (2010). DOI: 10.1112/S1461157009000187
 
[AG30] On the average number of rational points on curves of genus 2 (pdf, 629 kB)  arXiv
Preprint (2009).
 
[AG29] Documentation for the ratpoints program (pdf, 188 kB)  arXiv
Manuscript (2009).
 
[AG28] with N. Bruin:
Two-cover descent on hyperelliptic curves (pdf, 310 kB)  arXiv
Math. Comp. 78, 2347-2370 (2009). DOI: 10.1090/S0025-5718-09-02255-8. Published online March 11, 2009.
 
[AG27] Rational 6-cycles under iteration of quadratic polynomials (version 2008-06-11, pdf, 260 kB)  arXiv
London Math. Soc. J. Comput. Math. 11, 367-380 (2008).
MAGMA script for the computations
 
[AG26] Applications of the Mordell-Weil sieve (pdf, 113 kB)
Oberwolfach Report 34/2007, Oberwolfach Reports 4, no. 3, 1967-1970 (2007).
 
[AG25] with Y. Bugeaud, M. Mignotte, S. Siksek, and Sz. Tengely:
Integral points on hyperelliptic curves (pdf, 300 kB)  arXiv
Algebra & Number Theory 2, No. 8, 859-885 (2008).
 
[AG24] How to obtain global information from computations over finite fields (pdf, 23 kB)
In: Higher-dimensional geometry over finite fields, NATO Science for Peace and Security Series: Information and Communication Security 16, (D. Kaledin, Y. Tschinkel, eds.), p. 189-196, IOS Press (2008).
 
[AG23] with T.A. Fisher and E.F. Schaefer:
The yoga of the Cassels-Tate pairing (pdf, 220 kB) arXiv
LMS J. Comput. Math. 13, 451-460 (2010). DOI: 10.1112/S1461157010000185
 
[AG22] with J.E. Cremona, T.A. Fisher, C. O'Neil and D. Simon:
Explicit n-descent on elliptic curves. II. Geometry (pdf, 288 kB)  arXiv
J. reine angew. Math. 632, 63-84 (2009). DOI: 10.1515/CRELLE.2009.050. Published online June 16, 2009.
 
[AG21] with N. Bruin:
Deciding existence of rational points on curves: an experiment (pdf, 210 kB) arXiv
Experiment. Math. 17, 181-189 (2008)
.
See also my talk in Göttingen
 
[AG20] with J.E. Cremona, T.A. Fisher, C. O'Neil and D. Simon:
Explicit n-descent on elliptic curves. I. Algebra (pdf, 350 kB)  arXiv
J. reine angew. Math. 615, 121-155 (2008)
 
[AG19] On the number of rational squares at fixed distance from a fifth power (pdf, 160 kB)  arXiv
Acta Arith. 125, 79-88 (2006)
 
[AG18] Finite coverings and rational points (pdf, 94 kB)
Oberwolfach Report 32/2005, Oberwolfach Reports 2, no. 3, 1824-1827 (2005).
 
[AG17]with B. Poonen and E.F. Schaefer:
Twists of X(7) and primitive solutions to x2 + y3 = z7 (pdf, 560 kB) arXiv
Duke Math. J. 137, 103-158 (2007)
 
[AG16] Finite descent obstructions and rational points on curves (pdf, 399 kB)  arXiv
Algebra & Number Theory 1, 349-391 (2007).
This Errata note corrects some mistakes in the paper above.
“Draft Version 8” (pdf, 520 kB) with some discussion of conjectures around “Effective Mordell”.
See also my talk at MSRI, March 30, 2006.
 
[AG15] Independence of rational points on twists of a given curve (pdf, 240 kB)  arXiv
Compositio Math. 142, 1201-1214 (2006); DOI: 10.1112/S0010437X06002168, published online 25 Sep 2006
 
[AG14]with T. Yang:
On the L-function of the curves y2 = x5 + A (dvi, 66 kB  pdf, 250 kB)
J. London Math. Soc. (2) 68, 273-287 (2003)
 
[AG13] with J.E. Cremona:
Minimal models for 2-coverings of elliptic curves (pdf, 211 kB)
London Math. Soc. J. Comput. Math. 5, 220-243 (2002)
 
[AG12] with J.E. Cremona:
On the reduction theory of binary forms (pdf, 266 kB)
J. reine angew. Math. 565, 79-99 (2003)
See also my talk in Leiden, 2000 (dvi, 15 kB)
 
[AG11] with E.F. Schaefer:
How to do a p-descent on an elliptic curve
Trans. Amer. Math. Soc. 356, 1209-1231 (2004)
Long version:  dvi (163 kB)  pdf (360 kB)
 
[AG10] On the height constant for curves of genus two, II
Acta Arith. 104, 165-182 (2002)
See also my talk in Leiden, 1999 (dvi, 24 kB)
 
[AG9] with E.V. Flynn, F. Leprévost, E.F. Schaefer, W.A. Stein and J.L. Wetherell:
Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves
Math. Comp. 70, 1675-1697 (2001)
 
[AG8] Implementing 2-descent for Jacobians of hyperelliptic curves
Acta Arith. 98, 245-277 (2001)
 
[AG7] On the arithmetic of the curves  y2 = xl + A, II
J. Number Theory 93, 183-206 (2002).
 
[AG6] On the height constant for curves of genus two,
Acta Arith. 90, 183-201 (1999).
 
[AG5] with B. Poonen:
A local-global principle for densities
In: Scott D. Ahlgren (ed.) et al.: Topics in number theory. In honor of B. Gordon and S. Chowla.
Kluwer Academic Publishers, Dordrecht. Math. Appl., Dordr. 467, 241-244 (1999).
 
[AG4] with B. Poonen:
The Cassels-Tate pairing on principally polarized abelian varieties (pdf, slight update 2014-08-23, 509 kB) arXiv
Ann. of Math. (2) 150, 1109-1149 (1999).
 
[AG3] On the arithmetic of the curves  y2 = xl + A and their Jacobians
J. reine angew. Math. 501, 171-189 (1998).
 
[AG2] An example of a simple 2-dimensional abelian variety defined over Q with Mordell-Weil group of rank at least 20
C. R. Acad. Sci Paris 322. Série I, 849-851 (1996).
 
[AG1] Two simple 2-dimensional abelian varieties defined over Q with Mordell-Weil group of rank at least 19
C. R. Acad. Sci. Paris 321, Série I, 1341-1345 (1995).
 

Combinatorics


 
[CO2] with D. Schleicher:
An introduction to Conway's games and numbers (pdf, 300 kB) arXiv
Moscow Mathematical Journal 6, 359-388 (2006).
 
[CO1] Bounds for the length of recurrence relations for convolutions of P-recursive sequences
Europ. J. Comb. 18, 707-712 (1997).
 

Mathematical Biology


 
[MB3] with E. Geigant:
Stability of peak solutions of a non-linear transport equation on the circle (pdf, 959 kB)  arXiv
Electron. J. Diff. Equ. 2012, No. 157, 1-41 (2012).
 
[MB1] with E. Geigant:
Bifurcation analysis of an orientational aggregation model
J. Math. Biol. 46, 537-563 (2003), DOI: 10.1007/s00285-002-0187-1.
 
[MB2] with S. Schuster:
A simple method to analyze resonators in fish hearing
In: Elsner N., Wehner, R. (eds.): Proceedings of the 26th Göttingen Neurobiology Conference, Vol II, p. 303. Thieme, Stuttgart (1998).
 

Group Theory


 
[GR6] Coarse length can be unbounded in 3-step nilpotent Lie groups (pdf, 14 kB)
Preprint (2010). This gives an explicit example proving the claim made in the remark following Lemma 3.3 in [GR4] below.
 
[GR5] Some group presentations with rational growth (dvi, 29 kB)
Preprint (1995).
 
[GR4] Asymptotics of the growth of 2-step nilpotent groups
J. London Math. Soc. 58, 38-48 (1998).
See [GR6] above for an explicit example related to the remark following Lemma 3.3.
 
[GR3] Rational and transcendental growth series for the higher Heisenberg groups
Invent. math. 126, 85-109 (1996).
 
[GR2] Asymptotics of some number theoretic functions and an application to the growth of nilpotent groups (dvi, 370 kB)
Bonner Mathematische Schriften 266 (1994) (doctoral thesis).
 
[GR1] Regular geodesic languages for 2-step nilpotent groups
In: A.J. Duncan, N.D. Gilbert, J. Howie (eds): Combinatorial and Geometric Group Theory, Edinburgh 1993,
Cambridge University Press, Cambridge, LMS Lecture Note Series 204, 294-299 (1995).
 

Galois Theory


 
[GA2] Construction of semiabelian Galois extensions
Glasgow Math. J. 37, 99-104 (1995).
 
[GA1] Galois groups over Q of some iterated polynomials
Arch. Math. 59, 239-244 (1992).
 

Lecture Notes of Courses Taught at Jacobs University


 
[JN1] Linear Algebra I (pdf, 530 kB)
Notes from Fall 2006.
 
[JN2] Linear Algebra II (pdf, 500 kB)
Notes from Spring 2007.
 
[JN3] Introductory Algebra (pdf, 530 kB)
Notes from Fall 2005.
 
[JN4] Introductory Number Theory (pdf, 530 kB)
Notes from Spring 2006.
 
[JN5] Introductory Complex Analysis (pdf, 765 kB)
Notes from Spring 2007, with corrections.
 
[JN6] Introductory Geometry: Second Half (Algebraic Geometry) (pdf, 255 kB)
Notes from Fall 2005.
 
[JN7] Integration and Manifolds (pdf, 549 kB)
Notes from Fall 2007.
 

List of Jacobs University Courses


Other Course Notes


 
[CN4] Rational points on curves (pdf, 361 kB)
Lecture notes from a summer school in Hay-on-Wye, Wales, UK, 2015.
 
[CN3] Descent and covering collections (pdf, 290 kB)
Lecture notes from a summer school in Ohrid, Macedonia.
In: L. Beshaj, T. Shaska, E. Zhupa (Eds.), Advances on superelliptic curves and their applications, IOS Press, 2015, p. 176–193.
 
[CN1] Descent on Elliptic Curves (new version 2010/12/01, pdf, 329 kB)  arXiv
Short Course taught at IHP in Paris, October 2004.
In: Panoramas et Synthèses 36, Société Math. de France, 2012.
 
[CN2] Vector Bundles on Curves (pdf, 590 kB)  
Notes of a course taught by G. Faltings at the MPI in Bonn, 1995.
 

Lecture Notes (in German) of Courses Taught in Düsseldorf


 
[LN1] Abzählende Kombinatorik (pdf, 554 kB)
Vorlesungsskript vom Wintersemester 1999/2000.
 
[LN2] Elliptische Kurven I (pdf, 405 kB)
Vorlesungsskript vom Sommersemester 2000.
 
[LN3] Elliptische Kurven II (pdf, 495 kB)
Vorlesungsskript vom Wintersemester 2000/2001.
 
[LN4] Algebraische Kurven (pdf, 371 kB)
Vorlesungsskript vom Wintersemester 2001/2002.
 

Talk Notes


 
Rational Diophantine quintuples and diagonal genus 5 curves
(Slides of a talk at Diophantine Problems, University of Manchester, 2017-09-15)  pdf (155 kB)
 
Rational points on curves in practice
(Slides of a talk at the Journées Algophantiennes Bordelaises, Université de Bordeaux, 2017-06-08)  pdf (138 kB)
 
Simultaneous torsion in the Legendre family of elliptic curves
(Slides of a talk at the workshop on Heights and applications to unlikely intersections, Fields Institute, Toronto, 2017-02-17)  pdf (474 kB)
 
The generalized Fermat equation x2 + y3 = z11
(Slides of a talk at the conference on Computational Aspects of Diophantine Equations, Salzburg, 2016-02-15)  pdf (155 kB)
 
Chabauty without the Mordell-Weil group
(Slides of a talk at Rational Points 2015, Schney, 2015-06-29)  pdf (169 kB)
 
Uniform bounds for the number of rational points on curves of low Mordell-Weil rank
(Slides of a talk at the conference on p-adic Methods in Number Theory, Berkeley, 2015-05-27)  pdf (164 kB)
 
Descent and covering collections
(Slides of a lecture series in Ohrid, 2014-09-01/03/05)
Frey-Kurven und die verallgemeinerte Fermatsche Gleichung
(Slides of a talk (in German) in Saarbrücken at the conferral of an honorary doctorate to Gerhard Frey, 2014-07-25)  pdf (615 kB)
 
Most odd degree hyperelliptic curves have only one rational point
(Slides of a talk in Bad Boll, 2014-03-04)  pdf (183 kB)
 
Many curves with few rational points
(Slides of a talk at the University of Warwick, 2012-09-25)  pdf (127 kB)
 
Wie man eine diophantische Gleichung löst
(Vortrag bei der regionalen Lehrerfortbildung, 2012-06-27)  pdf (170 kB)
 
Explicit Kummer Varieties for Hyperelliptic Curves of Genus 3
(Slides of a talk in Luminy, 2012-01-17)  pdf (93 kB)
 
On a Problem of Hajdu and Tengely
(Slides of a contributed talk at ANTS-IX, 2010-07-23)  pdf (126 kB)
 
Torsion Points on Elliptic Curves over Quartic Number Fields
(Slides of the scientific part of my invited talk at ANTS-IX, 2010-07-22)  pdf (128 kB)
 
Diophantische Gleichungen und wie man manche von ihnen lösen kann
(mathematischer Teil meiner Antrittsvorlesung am 29. Oktober 2009 in Bayreuth)  pdf (130 kB)
 
Rational Six-Cycles Under Quadratic Iteration
(Slides of a talk at the GTEM 3rd annual meeting at Warwick, 2009-09-08)  pdf (80 kB)
 
Rational Points on Curves
(Slides of a talk at the Journées Arithmétiques, 2009-07-09)  pdf (104 kB)
 
Rational Points on Curves of Genus 2: Experiments and Speculations
(Slides of a talk in Dagstuhl, 2009-05-25)  pdf (123 kB)
 
Die Vermutung von Birch und Swinnerton-Dyer
(Slides (in German) of a talk in Bayreuth, 2009-01-08)  pdf (410 kB)
 
Searching for Rational Points on Genus 2 Jacobians
(Slides of a talk in Banff, 2008-12-05)  pdf (90 kB)
 
Die Vermutung von Birch und Swinnerton-Dyer
(Slides (in German) of a talk in Bremen, 2008-11-22)  pdf (230 kB)
 
Harmlose Gleichungen — Schwierige Lösung
(Slides (in German) of a talk in Rostock, 2008-09-10)  pdf (70 kB)
 
Rational Points on Curves of Genus 2
(Slides of a colloquium talk at the University of Warwick, 2008-01-25)  pdf (209 kB)
 
How to Obtain Global Information From Computations Over Finite Fields
(Slides of a talk at a Summer School in Göttingen, 2007-07-05)  pdf (90 kB)
 
How to Determine the Set of Rational Points on a Curve
(Slides of a talk at the MEGA 2007 Conference in Strobl, 2007-06-28)  pdf (143 kB)
 
How to Find the Rational Points on a Rank 1 Genus 2 Curve
(Slides of a talk at a workshop in Leiden, 2007-05-14)  pdf (170 kB)
 
Coverings and Mordell-Weil Sieve
(Slides of a talk at a Banff workshop, 2007-02-06)  pdf (91 kB)
 
Deciding Existence of Rational Points on Genus 2 Curves: A Computational Experiment
(Slides of a talk at the Clay Summer School in Göttingen, 2006-07-21)  pdf (95 kB)
 
Finite Descent Obstructions and Rational Points
(Slides of a talk at MSRI, 2006-03-30)  pdf (86 kB)
 
Finite Coverings and Rational Points
(Notes of a talk in Oberwolfach, 2005-07-19)  pdf (120 kB)
 
x2 + y3 = z7
(Slides of a talk in Dagstuhl, 2004-05)  pdf (118 kB)
 
Die Gleichung x2 + y3 = z7
(Slides (in German) of a talk in Göttingen, 2004-01-15)  pdf (89 kB)
 
How many rational points can a curve have?
(Slides of a talk in Bordeaux, 2001-11-23)  dvi (12 kB)
 
Computation of canonical heights on genus 2 Jacobians
(Slides of a talk in Bordeaux, 2001-11-22)  dvi (15 kB)
 
Algorithmischer Beweis kombinatorischer Identitäten
(German, Habilitationsvortrag, 1999-06-30)  pdf (148 kB)
 

Michael Stoll,  September 15, 2017