## Publications

- Nils Bruin, E. Victor Flynn, Ari Shnidman, Genus two curves with full sqrt(3)-level structure and Tate-Shafarevich groups, ArXiv preprint arXiv:2102.04319 (2021).
- Nils Bruin, Daniel Lewis, Two-cover descent on plane quartics with rational bitangents,
*ANTS XIV: Proceedings of the Fourteenth Algorithmic Number Theory Symposium*, The Open Book Series 4-1 (2020), 73-89. See also Electronic resources. (or see ArXiv preprint arXiv:2003.00666). Conference Video Presentation (also on YouTube) and the slides. - Nils Bruin, Jordan Thomas, Anthony Várilly-Alvarado Explicit computation of symmetric differentials and its application to quasi-hyperbolicity, ArXiv preprint arXiv:1912.08908 (2019).
- Nils Bruin, Emre Can Sertöz, Prym varieties of genus four curves,
*Trans. AMS***373**(2020), 149-183. Electronically published on September 23, 2019. (or see ArXiv preprint arXiv:1808.07881, 2018). - Nils Bruin, Jeroen Sijsling, Alexandre Zotine,
Numerical computation of endomorphism rings,
*ANTS XIII: Proceedings of the Thirteenth Algorithmic Number Theory Symposium*, The Open Book Series 2-1 (2019), 155-171. See also Electronic Resources. (or see ArXiv preprint arXiv:1807.02605, 2018). - Nils Bruin, Brett Nasserden, Arithmetic aspects of the Burkhardt quartic threefold,
*J. London Math. Soc.***98**(2018), 536 - 556. See also Electronic Resources (or see ArXiv preprint arXiv:1705.09006) - Nils Bruin, Tom Fisher, Visibility of 4-covers of elliptic curves,
*Research in Number Theory***4**:11 (2018) (or see ArXiv preprint arXiv:1701.07528,, 2017). - Nils Bruin, Bjorn Poonen, Michael Stoll, Generalized explicit descent and its application to curves of genus 3, Forum Math. Sigma 4 (2016), e6, 80 pp.
- Nils Bruin, E.Victor Flynn, Damiano Testa,
Descent via (3,3)-isogeny on Jacobians of genus 2 curves,
*Acta Arithmetica***165**(2014), 201-223. (or see ArXiv preprint arXiv:1401.0580, 2014). - Nils Bruin,
Success and challenges in determining the rational points on curves,
*ANTS X: Proceedings of the Tenth Algorithmic Number Theory Symposium*, The Open Book Series 1-1 (2013), 187-212. - Nils Bruin, Brett Hemenway,
On congruent primes and class numbers of imaginary quadratic fields,
*Acta Arithmetica***159**(2013), 63-87. (or see ArXiv preprint arXiv:1110.5959, 2011). - Nils Bruin, Alexander Molnar,
Minimal models for rational functions in a dynamical setting,
*LMS Journal of Computation and Mathematics*,**15**(2012), 400-417. See also Electronic Resources (or see ArXiv preprint arXiv:1204.4967, 2012). - Jason P. Bell, Nils Bruin, Michael Coons, Transcendence of generating functions whose coefficients are multiplicative,
*Trans. Amer. Math. Soc.***364**(2012), 933-959 (or see ArXiv preprint arXiv:1003.2221, 2010). - Nils Bruin, Kevin Doerksen,
The arithmetic of genus two curves with (4,4)-split Jacobians,
*Canad. J. Math.***63**(2011), 992-1021. See also Electronic Resources (or see ArXiv preprint arXiv:0902.3480, 2009). - Nils Bruin, Michael Stoll, The Mordell-Weil sieve: Proving non-existence of rational points on curves,
*LMS Journal of Computation and Mathematics***13**(2010), 272-306. (or see ArXiv preprint arXiv:0906.1934, 2009). - Nils Bruin, Sander R. Dahmen, Visualizing elements of Sha[3] in genus 2 jacobians,
*Algorithmic Number Theory*, 9th International Symposium, ANTS-IX, Lecture Notes in Computer Science 6197, Springer 2010, pp. 110 - 125, (ArXiv preprint arXiv:1001.5302, 2010.) - Nils Bruin, Michael Stoll, Two-cover descent on hyperelliptic Curves,
*Math. Comp.*78 (2009), 2347-2370. See also Electronic Resources. (or see ArXiv preprint arXiv:0803.2052, 2008) -
Nils Bruin and Michael Stoll,
Deciding existence of rational points on curves: an experiment,
*Experiment. Math.*17 (2008), no. 2, 181-189. See Electronic data for additional resources. (or see arXiv preprint, 2006). - Nils Bruin, The arithmetic of Prym varieties in genus 3, Compositio Mathematica (2008), 144 : 317-338. See Electronic Data for supporting electronic resources. (or see arXiv preprint, 2004).
- Bright, M. J.; Bruin, N.; Flynn, E. V.; Logan, A. The Brauer-Manin obstruction and Sha[2],
*LMS J. Comput. Math.*10 (2007), 354--377. Electronic resources available in Appendix A or here. - Nils Bruin, Julio Fernández, Josep González, Joan-Carles
Lario,
Rational points on twists of X
_{0}(63),*Acta Arith.*126 (2007), no. 4, 361-385. Also available as Preprint at Universitat Politecnica Catalunya. - Nils Bruin, K. Györy, L. Hajdu, Sz. Tengely, Arithmetic progressions consisting of unlike powers,
*Indag. Mathem.*17 (4), 539-555 (2006). Also available as preprint. - Nils Bruin, E. Victor Flynn, Josep González, 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, Kálmán Györy, 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, 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 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), in
*Discovering Mathematics with Magma*, W. Bosma, J. Cannon (eds). Algorithms Comput. Math., 19, Springer, Berlin, 2006, pp. 63-91. See Electronic Data for supporting electronic resources. - Nils Bruin,
Chabauty methods using elliptic curves.
*J. Reine Angew. Math.*562 (2003), 27 - 49. (it seems the 2003 volumes of Crelle have fallen of the map, so as a stop-gap measure a cached copy. No copyright infringement intended) - 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. Also the accompanying*stellingen*(in dutch). (pdf version) -
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. (pdf version)

## Student Theses

- Filatov, Evgueni,
*Brauer-Severi varieties associated to twists of the Burkhardt quartic*, M.Sc., Summer 2020. - Lewis, Daniel,
*An implementation of two-cover descent on plane quartic curves*, M.Sc., Summer 2019. - Yadegarzadeh, Sepehr,
*On the density of parameterizations of generalized Fermat equations of signature (2,3,3) that produce locally primitive solutions*, M.Sc., Summer 2019. - Kulkarni, Avinash,
*Aspects of the arithmetic of uniquely trigonal genus four curves: arithmetic invariant theory and class groups of cubic number fields*, Ph.D., Fall 2018. - Brett Nasserden,
*Arithmetic aspects of the Burkhardt quartic threefold*, M.Sc., Summer 2016. - Navid Alaei,
*Symmetric Differential Forms on the Barth Sextic Surface*, M.Sc., Spring 2015. - Avinash Alexander Kulkarni,
*On Jacobians of dimension 2g that decompose into Jacobians of dimension g*, M.Sc., Summer 2014. - Patrick Ryan McMahon,
*Solvability of ternary equations of signature (3,3,2)*, M.Sc., Summer 2014. - Steven Kieffer,
*Computability in principle and in practice in algebraic number theory: Hensel to Zassenhaus*, M.Sc., Spring 2012. - Alexander Molnar,
*Fractional linear minimal models of rational functions*, M.Sc., Fall 2011. errata - Kevin Doerksen,
*On the arithmetic of genus 2 curves with (4,4)-split Jacobians*, Ph.D., Summer 2011. - Brett Hemenway,
*On recognizing congruent primes*, M.Sc., Fall 2006.