Peter Bruin

Universitair docent (assistant professor), Mathematisch Instituut, Universiteit Leiden

I am affiliated with the Quantum Software Consortium, an NWO Gravitation project.

Postadres:
Universiteit Leiden
Mathematisch Instituut
Postbus 9512
2300 RA Leiden

Bezoekadres:
Mathematisch Instituut
Gorlaeusgebouw, kamer BW.2.19
Einsteinweg 55
2333 CC Leiden

E-mail: P.J.Bruin@math.leidenuniv.nl
Telephone: (+31) 71 527 7142

Onderwijs | Teaching

• Seminarium Presenteren en Communiceren (met Hans Finkelnberg, Onno van Gaans en Mark Roelands), academisch jaar 2024–2025
• Lineaire algebra 1 voor natuur- en sterrenkunde (ook op Brightspace, najaar 2024
• Mathematics for Quantum Physics (QIST programme; responsible lecturer), autumn 2024
• Modular Forms (with Sander Dahmen), spring 2024
• Seminarium Presenteren en Communiceren (met Hans Finkelnberg, Onno van Gaans en Mark Roelands), academisch jaar 2023–2024
• Lineaire algebra 1 voor natuur- en sterrenkunde (ook op Brightspace), najaar 2023
• Mathematics for Quantum Physics (QIST programme; responsible lecturer), autumn 2023
• Seminarium Presenteren en Communiceren (met Hans Finkelnberg, Onno van Gaans en Mark Roelands), academisch jaar 2022–2023
• Lineaire algebra 1 voor natuur- en sterrenkunde (ook op Brightspace), najaar 2022
• Modular Forms (with Sander Dahmen), spring 2022
• Seminarium Presenteren en Communiceren (met Hans Finkelnberg en Mark Roelands), academisch jaar 2021–2022
• Topologie, najaar 2021
• Seminarium Presenteren en Communiceren (met Hans Finkelnberg en Mark Roelands), academisch jaar 2020–2021
• Topologie, najaar 2020
• Modular Forms (with Sander Dahmen), spring 2020
• Seminarium Presenteren en Communiceren (met Hans Finkelnberg), najaar 2019
• Topologie, najaar 2019
• Representation Theory of Finite Groups, spring 2019
• Calculus 2 voor MST, najaar 2018, blok 2
• Topologie, najaar 2018
• Modular Forms (with Sander Dahmen), spring 2018
• Calculus 2 voor MST, najaar 2017, blok 2
• Topologie, najaar 2017
• Topologie, voorjaar 2017
• Calculus 2 voor MST, najaar 2016, blok 2
• Galois Representations and Automorphic Forms (with Arno Kret), autumn 2016
• Modular Forms (with Sander Dahmen), spring 2016
• Topologie, voorjaar 2016
• Calculus 2 voor MST, najaar 2015, blok 2
• Topologie, voorjaar 2015
• Calculus 2 voor MST, najaar 2014, blok 2

Publications

• With H. Basold and D. R. Lawson: The Directed Van Kampen Theorem in Lean. In: 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs) 309, 8:1–8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024). Publication, arXiv preprint.
• Extensions and torsors for finite group schemes. Expositiones Mathematicae 41 (2023), no. 3 (special issue in memory of Bas Edixhoven), 514–530. Journal page, arXiv preprint.
• With L. Ducas and S. Gibbons: Genus distribution of random q-ary lattices. In: M. Grześkowiak, J. Pieprzyk and J. Pomykała (editors), Number-Theoretic Methods in Cryptology – NuTMiC 2021. Banach Center Publications 126 (2023), 137–159. Journal page, IACR ePrint.
• With F. Najman: Counting elliptic curves with prescribed level structures over number fields. Journal of the London Mathematical Society 105 (2022), no. 4, 2415–2435. Journal page, arXiv preprint.
• With M. Derickx and M. Stoll: Elliptic curves with a point of order 13 defined over cyclic cubic fields. Functiones et Approximatio 65 (2021), no. 2, 191–197. Journal page, arXiv preprint.
• With A. Perucca: Reductions of points on algebraic groups, II. Glasgow Mathematical Journal 63 (2021), no. 2, 484–502. Journal page, arXiv preprint.
• With A. Ferraguti: Strongly modular models of Q-curves. International Journal of Number Theory 15 (2019), no. 3, 505–526. Journal page, arXiv preprint.
• With F. Najman: Fields of definition of elliptic curves with prescribed torsion. Acta Arithmetica 181 (2017), no. 1, 85–95. Journal page, arXiv preprint.
• With A. Ferraguti: On L-functions of quadratic Q-curves. Mathematics of Computation 87 (2018), 459–499. Journal page, arXiv preprint.
• With F. Najman: A criterion to rule out torsion groups for elliptic curves over number fields. Research in Number Theory 2 (2016), no. 3, 1–13. Journal page, arXiv preprint.
• With F. Najman: Hyperelliptic modular curves X₀(n) and isogenies of elliptic curves over quadratic fields. LMS Journal of Computation and Mathematics 18 (2015), no. 1, 578–602. Journal page, arXiv preprint.
• With F. Najman: The growth of the rank of Abelian varieties upon extensions. The Ramanujan Journal 39 (2016). no. 2, 259–269. Journal page, arXiv preprint.
• Bornes optimales pour la différence entre la hauteur de Weil et la hauteur de Néron–Tate sur les courbes elliptiques sur Q. Acta Arithmetica 160 (2013), no. 4, 385–397. Journal page, arXiv preprint, PARI/GP script (last updated 4 April 2018).
• Explicit bounds on automorphic and canonical Green functions of Fuchsian groups. Mathematika 60 (2014), no. 2, 257–306. Journal page, PDF (version of 4 February 2013).
• Merkl's method of bounding Green functions. Appendix to: A. Javanpeykar, Polynomial bounds for Arakelov invariants of Belyi curves. Algebra & Number Theory 8 (2014), no. 1, 89–140. Journal page, arXiv preprint.
• With J. G. Bosman, A. Dujella and F. Najman: Ranks of elliptic curves with prescribed torsion over number fields. International Mathematical Research Notices 2014 (2014), no. 11, 2885–2923. Journal page, arXiv preprint.
• Computing in Picard groups of projective curves over finite fields. Mathematics of Computation 82 (2013), 1711–1756. Journal page, PDF (version of 15 October 2011).
• The Tate pairing for Abelian varieties over finite fields. Journal de théorie des nombres de Bordeaux 23 (2011), no. 2, 323–328. Journal page, PDF.
• Computing coefficients of modular forms. Publications mathématiques de Besançon, année 2011 (actes de la conférence Théorie des nombres et applications, CIRM, Luminy, 30 novembre–4 décembre 2009), 19–36. PDF (published version).

(Published versions differ slightly from preprints.)

Prepublications/Protopublications

• With I. Manterola Ayala: Counting rational points on weighted projective spaces over number fields. Submitted. arXiv preprint.
• On quantum computation of Kloosterman sums. arXiv preprint.
• Dual pairs of algebras and finite commutative group schemes. arXiv preprint.
• With S. Adonsou and M. Ozols: Hidden shift problem for complex functions. In preparation.
• With S. Anni: Checking isomorphisms of residual modular Galois representations. In preparation.

Mathematical software

For my research, I often use the free* mathematical software systems PARI/GP and SageMath. I regularly contribute to SageMath by adding new features, fixing bugs and improving existing code. I have also made a number of contributions to PARI/GP.

* Free as in "free speech", not only as in "free beer"; see the GNU project.

The abgalrep, ellgalrep and modgalrep packages

I have written three software packages for computing Galois representations (or finite commutative group schemes over Q) attached to arithmetic objects:

• abgalrep computes one-dimensional Galois representations attached to Dirichlet characters over finite fields;
• ellgalrep computes torsion subschemes of elliptic curves over Q;
• modgalrep computes two-dimensional Galois representations attached to modular forms over finite fields.

Note: modgalrep is by far the largest of these packages; it consists of a C program together with several libraries and GP scripts. The other two are smaller and just consist of GP scripts.

The dual-pairs package

This is a SageMath package for computing with finite commutative group schemes and Galois representations in the form of dual pairs of algebras, as described in the preprint Dual pairs of algebras and finite commutative group schemes.

• Source code
• Documentation

Agda packages

These are formalisations of small parts of mathematics in dependent type theory using the the Agda programming language.

• agda-real: a formalisation of the real numbers as two-sided Dedekind cuts.
• cubical-algebra: some algebraic constructions in cubical type theory.

Organisation

An Expedition into Arithmetic Geometry (2023)

Together with Samuele Anni, Hélène Esnault, Ronald van Luijk, Pierre Parent, Gabor Wiese and Andrei Yafaev, I organised the workshop An Expedition into Arithmetic Geometry, held in memory of Bas Edixhoven at the Lorentz Center (Leiden) from 30 May to 2 June 2023. The concluding event was the Bas Edixhoven Memorial Symposium on the afternoon of Friday 2 June.

COUNT workshop (2023)

I was one of the organisers of the workshop COUNT (COmputations and their Uses in Number Theory) at CIRM (Marseille) from 27 February to 3 March 2023.

6th QSC General Assembly (2021)

Wolfgang Löffler and I organised the 6th General Assembly of the Quantum Software Consortium at the Universiteit Leiden on Friday 5 November 2021.

Algorithms in Number Theory and Arithmetic Geometry (2017)

I organised the workshop Algorithms in Number Theory and Arithmetic Geometry at the Universiteit Leiden from 31 July to 4 August 2017.

Sage Days 51: Algorithms in Arithmetic Geometry (2013)

Together with Maarten Derickx and Michiel Kosters, I organised an edition of the Sage Days, titled Algorithms in Arithmetic Geometry, in the Lorentz Center in Leiden, 22–26 July 2013. See also the wiki.

Previous positions

Proefschrift | Ph. D. thesis

Modular curves, Arakelov theory, algorithmic applications
Stellingen (propositions, in Dutch)
Universiteit Leiden, 1 September 2010
Promotor: Prof. dr. S. J. Edixhoven
Copromotor: Dr. R. S. de Jong

Doctoraalscriptie | Master's thesis

Green functions on Riemann surfaces and an application to Arakelov theory
Universiteit Leiden, 28 June 2006
Supervised by Prof. dr. S. J. Edixhoven and Dr. R. S. de Jong

Notes of talks

• Goss L-functions, seminar on Drinfeld modules, 14 and 28 October 2015
• What is… an L-function? Zurich Graduate Colloquium, 30 October 2012
• Optimal bounds for the difference between the Néron–Tate height and the Weil height on elliptic curves over Q, Selmer Groups, Descent and the Distribution of Ranks, University of Warwick, 28 September 2012
• Modular curves and Galois representations, mini-course given at the workshop Modular Forms and Related Topics II, CAMS, American University of Beirut, 6–10 February 2012
• Groupes de Chow supérieurs, applications classe de cycle et régulateurs étales, groupe de travail Zéro-cycles sur les variétés p-adiques, ENS Paris, 9 février 2011
• Calcul de coefficients de formes modulaires, séminaire d'arithmétique et de géometrie algébrique, Orsay, 25 janvier 2011
• Computing coefficients of modular forms, Théorie des nombres et applications, CIRM, Luminy (Marseille), 30 November 2009
• Arakelov theory and height bounds, Forschungsseminar Arithmetische Geometrie, Humboldt-Universität zu Berlin, 17 November 2009
• Computing in Picard groups of projective curves over finite fields, Arbeitsgemeinschaft Zahlentheorie, Institut für Experimentelle Mathematik, Universität Duisburg–Essen, 10 November 2009
• Endomorphism rings of Abelian varieties and their representations, seminar on complex multiplication for Abelian varieties, 14 October and 11 November 2009
• Global class field theory, seminar on adelic methods, 6 October 2008
• Explicit towers of curves with many points, seminar on rational points on curves over finite fields, 25 April 2008, and DIAMANT/EIDMA symposium, Oegstgeest, 29 May 2008
• Moduli of elliptic curves, seminar on moduli spaces and modular forms, 12 September 2007
• Sketch of a construction of the Néron model of an Abelian variety, seminar on Néron models, 4 April 2007
• Smooth morphisms, seminar on Néron models, 21 February 2007
• Formal groups, seminar on complex multiplication, 2 March 2006
• Curves in P¹ × P¹, seminar on cohomology of quasi-coherent sheaves, 16 November 2005
• Kolmogorov–Arnold–Moser theory, Classical Mechanics and Dynamical Systems, 16 November 2004
• De kubus van Rubik, studentenseminarium, 11 maart 2004

Laatste wijziging: 2 september 2024.