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


 
[AG57] with B. Naskręcki and N. Freitas:
The generalized Fermat equation with exponents 2, 3, n (pdf, 620 kB)
Preprint (2016).
 
[AG56] with J.S. Müller:
Canonical heights on genus two Jacobians (pdf, 700 kB)  arXiv
Preprint (2016), submitted.
 
[AG55] with J.S. Müller:
Computing canonical heights on elliptic curves in quasi-linear time (pdf, 360 kB) (new version, 2015-12-15)  arXiv
Preprint (2015), submitted.
 
[AG54] Chabauty without the Mordell-Weil group (pdf, 555 kB)  arXiv
Preprint (2015)
 
[AG53] with I. Bauer:
Rational points on primary Burniat surfaces (pdf, 420 kB)  arXiv
Preprint (2015), submitted.
 
[AG52] Simultaneous torsion in the Legendre family (version from 2015-10-04; pdf, 514 kB)  arXiv
Preprint (2014-2015), submitted.
Files containing the data from the computations can be found here.
 
[AG51] An Explicit Theory of Heights for Hyperelliptic Jacobians of Genus Three (pdf, 471 kB)
Preprint (2014).
 
[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) [new version 2015-11-25]  arXiv
Preprint (2013-2015), submitted.
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 (Manuscript: 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 (Manuscript: 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 (Manuscript: 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 (Manuscript: pdf, 560 kB) arXiv
Duke Math. J. 137, 103-158 (2007)
 
[AG16] Finite descent obstructions and rational points on curves (Manuscript: pdf, 399 kB)  arXiv
Algebra & Number Theory 1, 349-391 (2007).
“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 (Manuscript: 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 (Preprint version: 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 (Preprint version: 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 (Manuscript: 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 (Manuscript: 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.), Advacnes 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


 
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)

 
Uniform bounds on the number of rational points on hyperelliptic curves of low Mordell-Weil rank
(Slides of a talk in Cetraro, 2014-07-21)  pdf (139 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,  May 17, 2016