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Nils Bruin Department of Mathematics Simon Fraser University Burnaby, BC CANADA V5A 1S6 Tel : (778) 782 3794 Fax : (778) 782 4947 Email : nbruin@sfu.ca

• Research subjects
• Previous track
• Publications
• Software
• Teaching
• Workshops
• Various other (mathematical) subjects

Previous track

Awards

• Stieltjes Prize 1999 for best thesis within Thomas Stieltjes Institute for Mathematics.
• C.J. Kok Prize 1999 by Leiden University for research on covering techniques with respect to Chabauty's method and its applications.

Publications

• Nils Bruin, K. Gyory, L. Hajdu, Sz. Tengely, Arithmetic progressions consisting of unlike powers, Indag. Mathem. 17 (4), 539-555 (2006). Also available as preprint.
• Nils Bruin, Julio Fernandez, Josep Gonzalez, Joan-Carles Lario, Rational points on twists of X0(63). Available as Preprint at Universitat Politecnica Catalunya, 2006. Accepted for publication in Acta Arithmetica.
• Nils Bruin, E. Victor Flynn, Josep Gonzalez, Victor Rotger, On finiteness conjectures for endomorphism algebras of abelian surfaces , Math. Proc. Camb. Phil. Soc. (2006) 141, 383-408. (or see arXiv preprint). See Electronic Data for supporting electronic resources.
• Michael A. Bennett, Nils Bruin, Kalman Gyory, Lajos Hajdu, Powers from Products of Consecutive Terms in Arithmetic Progression, Proc. London Math. Soc. (3) 92 (2006), no. 2, 273 - 306.
• Nils Bruin, E. Victor Flynn, Exhibiting Sha[2] on Hyperelliptic Jacobians, Journal of Number Theory, 118 (2006), pp. 266-291.
  
• Nils Bruin and Michael Stoll, Deciding existence of rational points on curves: an experiment, 2006. See Electronic data for additional resources.
• Nils Bruin, The primitive solutions to x^3+y^9=z^2, Journal of Number Theory, 111 (2005), pp. 179-189. (Or see arXiv preprint, 2003). See Transcript of computations for supporting electronic resources.
  
• Nils Bruin and E. Victor Flynn, Towers of 2-covers of hyperelliptic curves, Trans. Amer. Math. Soc. 357 (2005), 4329-4347. (or see Towers of 2-covers of hyperelliptic curves, Preprint PIMS-01-12, 2001)
• Nils Bruin, The arithmetic of Prym varieties in genus 3, 2004. See Electronic Data for supporting electronic resources.
• Nils Bruin and E. Victor Flynn, Rational Divisors in Rational Divisor Classes, in Duncan Buell (Ed.): Algorithmic Number Theory, 6th International Symposium, ANTS-VI, Lecture Notes in Computer Science 3076 Springer 2004, pp. 132 - 139.
• Nils Bruin, Visualisation of Sha[2] in Abelian Surfaces, Math. Comp. 73 (2004), no. 247, 1459 - 1476. See Electronic Data for supporting electronic resources.(or see arXiv preprint, 2002).
• Nils Bruin, Some ternary Diophantine equations of signature (n,n,2), to appear in Discovering Mathematics with Magma, W. Bosma, J. Cannon (eds). (Springer), 2003. See Electronic Data for supporting electronic resources.
• Nils Bruin, Chabauty methods using elliptic curves. J. Reine Angew. Math. 562 (2003), 27 - 49.
• N.R. Bruin, Chabauty methods and covering techniques applied to generalized Fermat equations, CWI Tract 133, 77 pages, 2002.
• Nils Bruin and Noam D. Elkies, Trinomials ax^7+bx+c and ax^8+bx+c with Galois Groups of Order 168 and 8*168, in Claus Fieker, David R. Kohel (Eds.): Algorithmic Number Theory, 5th International Symposium, ANTS-V, Lecture Notes in Computer Science 2369 Springer 2002, pp. 172 - 188.
• Nils Bruin and E. Victor Flynn, N-Covers of hyperelliptic curves, Math. Proc. Cambridge Philos. Soc. 134 (2003), no. 3, 397--405.
• Nils Bruin, On powers as sums of two cubes, in Wieb Bosma (ed.),Algorithmic Number Theory 4th International Symposium, ANTS-IV Leiden, The Netherlands, July 2-7, 2000 Proceedings. Springer LNCS 1838. Associated files: prf335.g, tor334.mpl, prf335.mpl.
• Nils Bruin, Chabauty methods and covering techniques applied to generalised Fermat equations, PhD-thesis, University of Leiden, 1999. errata
• Nils Bruin, Chabauty methods using covers on curves of genus 2, Report no. W99-15, University of Leiden.
• Nils Bruin, Chabauty methods using elliptic curves, Report no. 1999-14, University of Leiden.
• Nils Bruin, The diophantine equations x^2+-y^4=+-z^6 and x^2+y^8=z^3, Compositio Math. 118 (1999) 305-321. Associated files
• H.J. Broersma, N. Bruin, J.L. Hurink, L.E. Meester, S.S. op de Beek, J.H. Westhuis, Throughput of ADSL modems, in Proceedings of the 33rd European Study Group with Industry, Syllabus 46: 11-26, CWI, Amsterdam (1999).
• Nils Bruin, Generalization of the ABC-conjecture, Master Thesis, Leiden University, 1995

Online Lecture

Slides and talks

Workshop Computational Arithmetic Geometry, 18-20 June 2003, Sydney, Prym varieties of curves of genus 3, pdf slides

Software

Algae

Algorithms for arithmetic on elliptic curves over general number fields, based on KASH. The main feature is an implementation of 2-descent and 2-isogeny-descent routines (taking care of even class numbers).

Current version: Algae 0.beta. You can download it as ell.shar. You may be interested in reading the README and the (at this time very crude and perhaps intimidating) documentation.

UPDATE March 27, 2001. Bug fix. The full 2-descent code did not compute properly at the infinite primes where ec.gamma is negative. Most people would probably not use this feature anyway, but it is fixed now.

WARNING. Selmer group computations also involve places at infinity. Precision settings may affect their outcome. The curve $y^2=x^3-1042011$ is an example of this phenomenon. With OrderPrec(50), you get a rank bound of 3, where with OrderPrec(300) the bound is 4, which is the rank of the curve. This is due to incorrect answers returned by EltCon. The command HEAP(O, "USE_ELT_PROD_REP", 50) has been recommended by the KASH team and seems to remedy the problem. It tells KASH to use another, safer, method for computing EltCon in O. If the package constructs an order automatically, then it sets this flag for that order.

NEW. A development version for computing 2 (isogeny) Selmer groups of elliptic curves over number fields in MAGMA is now available as m-Algae.

You might also be interested in TECC. It is another elliptic curve package, also based on KASH. It offers functionality mostly complimentary to ell.g. Unfortunately, the packages do not have compatible data structures for representing elliptic curves (yet?). I have not tested using both packages within the same session.

Denis Simon also has a program for computing 2-Selmer groups of elliptic curves over number fields, based on PARI/GP.

Software related to x^3+y^3=z^p

• prf335.g KASH program proving Lemmas 9 and 10. Needs Algae.
• tor334.mpl MAPLE program proving Lemma 6.
• prf335.mpl MAPLE program proving Lemma 7. Needs divcalc.sh.

Software related to my thesis

• thesis.sh shar-archive of KASH-scripts for computations concerning reports W99-14, W99-15 and my PhD-thesis. Also a version on this server.

Teaching

• Fall 2005: Calculus for Science students, first year undergraduate course. (MATH151)
• Summer 2005: Elementary Number Theory, third year undergraduate course. (MATH342)
• Fall 2004: Introduction to Algebraic Systems, third year undergraduate course. (MATH332)
• Fall 2004: Algebraic Curves, graduate course.
• Summer 2004: Linear Algebra, fourth year undergraduate course. (MATH438)
• Spring 2004: Algebraic Number Theory, graduate course.
• Spring 2002: Advanced Real Analysis. (MATH320)
• Fall 2001: Elementary Linear Algebra. (MATH232)

Conferences and Workshops

Various other (mathematical) subjects

Maple bug