Dr. Nicolas Mascot
Ussher Assistant Professor, Pure & Applied Mathematics
Publications and Further Research Outputs
Peer-Reviewed Publications
Nicolas Mascot, Explicit computation of Galois representations occurring in families of curves, 2023
Nicolas Mascot, Denis Simon, Computing the trace of an algebraic point on an elliptic curve, Expositiones Mathematicae, 2023
Nicolas Mascot, 'Package to compute with plane algebraic curves', Pari/GP, 2022, -
Nicolas Mascot, Explicit computation of a Galois representation attached to an eigenform over SL(3) from the H2 étale of a surface, Foundations of Computational Mathematics, 2022
Nicolas Mascot, Moduli-friendly Eisenstein series over the p-adics and the computation of modular Galois representations, Research in Number Theory, (8), 2022
A Prym variety with everywhere good reduction over Q(√61) in, editor(s)Balakrishnan, J.S., Elkies, N., Hassett, B., Poonen, B., Sutherland, A., Voight, J. , Arithmetic Geometry, Number Theory, and Computation, Springer, 2022, pp561 - 584, [Nicolas Mascot, Jeroen SIjsling, John Voight]
Nicolas Mascot, 'elltrace', Pari/GP, 2022, -
Nicolas Mascot, A method to prove that a modular Galois representation has large image, 2022
Nicolas Mascot, 'Package to compute Galois representations occurring in the torsion of Jacobians of curves', Pari/GP, 2021, -
Nicolas Mascot, Hensel-lifting torsion points on Jacobians and Galois representations, Mathematics of computation, (89), 2020, p1417 - 1455
Nicolas Mascot, Rigorous computation of the endomorphism ring of a Jacobian, Mathematics of computation, (88), 2019, p1303 - 1339
Nicolas Mascot, Certification of modular Galois representations, Mathematics of computation, (87), 2018, p381 - 423
Nicolas Mascot, Companion forms and explicit computation of PGL2 number fields with very little ramification, Journal of algebra, 509, 2018, p476 - 506
Nicolas Mascot, Computing modular Galois representations, Rendiconti del Circolo Matematico di Palermo, (62), 2013, p451 - 476
Non-Peer-Reviewed Publications
Nicolas Mascot, Explicit computations with étale cohomology of curves and surfaces, Geometry seminar, Univ. Tor Vergata, Rome, 9 May 2023, 2023, Giulio Codogni, Guido Maria Lido, Claudio Onorati
Nicolas Mascot, Computations with plane algebraic curves, UCD Algebra and Number Theory Seminar, University College Dublin, 9 Feb 2023, 2023, Kazim Buyukboduk, Robert Osburn
Nicolas Mascot, Plane algebraic curves in Pari/GP, Pari/GP workshop, Besancon, France, 10-14 January, 2022, Karim Belabas, Bill Allombert, Christophe Delaunay, Valentin Petit, Marine Rougnant
Nicolas Mascot, p-adic computation of mod l (modular) Galois representations, PARI day, Bordeaux, France (online), June 02, 2021, Bill Allombert, Karim Belabas
Nicolas Mascot, Modular Galois representations p-adically using Makdisi's moduli-friendly forms, LFANT seminar, Bordeaux, France (online), September 22, 2020, Aurel Page
Nicolas Mascot, Jeroen Sijsling, Workshop on Arithmetic Geometry, Number Theory, and Computation, June 1-5, In:Project: Explicit arithmetic of Jacobians, 2020, ICERM, Providence, RI, USA (online)
Nicolas Mascot, Hensel-lifting torsion points on Jacobians, and computation of Galois representations from higher etale cohomology, UCD Algebra and Number Theory Seminar, Univervisty College Dublin, January 21, 2020, Kazim Buyukboduk
Nicolas Mascot, Hensel-lifiting torsion points on Jacobians, and computation of Galois representations from higher etale cohomology, International Congress on Mathematical Software 2020, Braunschweig, Germany (online), 13 - 17 July, 2020, Emre Sertöz
Nicolas Mascot, Hensel-lifiting torsion points on Jacobians, and computation of Galois representations from higher etale cohomology, Number theory in flat lands, KU Leuven, Belgium, December 16, 2019, Wouter Castryck
Nicolas Mascot, Hensel-lifting torsion points on Jacobians and calculation of Galois representations in higher etale cohomology spaces, p-adic Langlands correspondence: a constructive and algorithmic approach, Rennes, France, 2-6 September, 2019
Nicolas Mascot, Modular Galois representations data, N/A, 2016
Research Expertise
Description
I am a computational number theorist. In a nutshell, I invent and implement algorithms to compute with mathematical objects which occur in algebra, number theory, and arithmetic geometry. I then publish these articles in peer-reviewed journals, and also make these algorithms accessible to the mathematics community by integrating them in number theory computer packages - mostly PARI/GP, a major open-source computer algebra system which received the 2021 Richard D. Jenks Memorial Prize for Excellence in Computer Engineering Applied to Computer Algebra, and to which I have contributed ca. 20,000 lines of code so far. My research is driven by the fact that central objects in modern number theory, such as Galois representations and etale cohomology, remain mysterious in spite of their major theoretical role, because they are never explicitly visualised, and are thus shrouded in a reputation of abstractness and difficulty. Yet I firmly believe that profound understanding of mathematics requires a crystal clear representation of the objects at play, which is best achieved by the study of explicit examples. My research therefore focuses on computing explicitly such objects, so as to lift the veil of abstract mystery off them, to get better acquainted with them, thus helping the community to progress by clarifying the big theoretical picture (Langlands programme) in which they play an essential role. This is essential, as the field is largely driven by phenomenology, so that computational tools are critical. More specifically, my work has mostly focused so far on explicit methods to handle Jacobian varieties of curves. These Jacobians are the instance of etale cohomology corresponding to the specific case of curves. Although they are directly attached to curves, they are immensely less accessible and more difficult to handle algorithmically than the curves themselves. My research has succeeded in developing efficient algorithms to overcome this. Recently, I have even managed to generalise my algorithms to the etale cohomology of higher-dimensional objects (such as surfaces instead of curves).Projects
- Title
- Explicit methods for Jacobians and etale cohomology
- Summary
- During the last 4 years I have developed algorithms to compute Galois representations, which are central objects in modern number theory but which are seldom seen explicitly. With this project, I plan to generalise my methods to more general and harder to access kinds of Galois representations, and to use my algorithms to solve Diophantine equations by performing descent on Jacobians of curves.
- Date From
- 2022
- Date To
- Ongoing
- Title
- Pari/GP package to handle Galois representations
- Summary
- Computation of subrepresentations and of quotient representations, efficient determination of the image of Frobenius elements
- Date From
- 2021
- Date To
- Ongoing
Recognition
Representations
Contribution of ca. 20,000 lines of code implementing new algorithms and opening up new area of mathematics to computational exploration
School of mathematics liaison with State Examinations Board to review and provide feedback on Leaving Cert maths questions.
Reporting and helping fix 15 bugs in the computer algebra package to which I contribute code
Awards and Honours
Teaching Hero (National Forum for the Enhancement of Teaching and Learning & Union of Students in Ireland)