Filip Najman
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Research Interests
Filip Najman has a total of 37 co-authors. Here are the top co-authors by number of shared publications:
- Peter Bruin
Mathematical institute, Faculty of Science, Leiden University, the Netherlands
7 shared publications - Enrique González-Jiménez
Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, Spain
4 shared publications - Florian Luca
Mathematics Division, Stellenbosch University
3 shared publications - Harris B. Daniels
Department of Mathematics and Statistics, Amherst College, Amherst, MA 01002, USA
3 shared publications - Philippe Michaud-Jacobs
Mathematics Institute, University of Warwick, CV4 7AL, United Kingdom
3 shared publications - Timo Keller
Leibniz Universität Hannover, Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Welfengarten 1, 30167 Hannover, Germany
3 shared publications
Latest Publications
- 25 June 2026
- Published by Springer Nature in Research in Number Theory
- Vol. 12 (3)
Correction to: Quadratic points on modular curves $X_0(N)$ for $N\le 100$- 25 June 2026
- Published by Springer Nature in Research in Number Theory
- Vol. 12 (3)
Correction to: Quadratic points on modular curves $X_0(N)$ for $N\le 100$- 3 March 2026
- Published by Springer Nature in Research in Number Theory
- Vol. 12 (2) , 23
Quadratic points on modular curves $X_0(N)$ for $N\le 100$- 3 March 2026
- Published by Springer Nature in Research in Number Theory
- Vol. 12 (2) , 23
Quadratic points on modular curves $X_0(N)$ for $N\le 100$- Open AccessSporadic points on $X_0(N)$We determine all integers $N$ for which the modular curve $X_0(N)$ admits a sporadic CM point (of any degree), as well as all $N$ for which $X_0(N)$ admits a sporadic point, whether CM or non-CM. In a sense, our results generalize the classification of isogenies of elliptic curves over $\Q$ due to Mazur and Kenku: their work determines the $X_0(N)$ with degree 1 sporadic points, whereas we classify all $X_0(N)$ that have a sporadic point of arbitrary degree.Open AccessSporadic points on $X_0(N)$We determine all integers $N$ for which the modular curve $X_0(N)$ admits a sporadic CM point (of any degree), as well as all $N$ for which $X_0(N)$ admits a sporadic point, whether CM or non-CM. In a sense, our results generalize the classification of isogenies of elliptic curves over $\Q$ due to Mazur and Kenku: their work determines the $X_0(N)$ with degree 1 sporadic points, whereas we classify all $X_0(N)$ that have a sporadic point of arbitrary degree.
- 11 October 2025
- Published by Walter de Gruyter GmbH in Journal für die reine und angewandte Mathematik (Crelles Journal)
Classification of torsion of elliptic curves over quartic fields
Let 𝐸 be an elliptic curve over a quartic field 𝐾. By the Mordell–Weil theorem, is a finitely generated group. We determine all the possibilities for the torsion group , where 𝐾 ranges over all quartic fields 𝐾 and 𝐸 ranges over all elliptic curves over 𝐾. We show that there are no sporadic torsion groups, or in other words, that all torsion groups either do not appear or they appear for infinitely many non-isomorphic elliptic curves 𝐸. Proving this requires showing that numerous modular curves have no non-cuspidal degree 4 points. We deal with almost all the curves using one of 3 methods: a method for the rank 0 cases requiring no computation; the Hecke sieve, a local method requiring computer-assisted computations; and the global method, an argument for the positive rank cases also requiring no computation. We deal with the handful of remaining cases using ad hoc methods.- 11 October 2025
- Published by Walter de Gruyter GmbH in Journal für die reine und angewandte Mathematik (Crelles Journal)
Classification of torsion of elliptic curves over quartic fields
Let 𝐸 be an elliptic curve over a quartic field 𝐾. By the Mordell–Weil theorem, is a finitely generated group. We determine all the possibilities for the torsion group , where 𝐾 ranges over all quartic fields 𝐾 and 𝐸 ranges over all elliptic curves over 𝐾. We show that there are no sporadic torsion groups, or in other words, that all torsion groups either do not appear or they appear for infinitely many non-isomorphic elliptic curves 𝐸. Proving this requires showing that numerous modular curves have no non-cuspidal degree 4 points. We deal with almost all the curves using one of 3 methods: a method for the rank 0 cases requiring no computation; the Hecke sieve, a local method requiring computer-assisted computations; and the global method, an argument for the positive rank cases also requiring no computation. We deal with the handful of remaining cases using ad hoc methods.- Open AccessQuadratic points on modular curves $X_0(N)$ for $N\leq 100$We determine the quadratic points on the modular curves $X_0(N)$ for $N\leq 100$ for which this has not been previously done, namely the cases $$N\in\{66,70,78,82,84,86,87,88,90,96,99\}.$$ We accomplish this by improving on the ``going down method," which uses the fact that we have a moduli description of all the (infinitely many) quadratic points on $X_0(n)$ for some divisor $n$ of $N$.Open AccessQuadratic points on modular curves $X_0(N)$ for $N\leq 100$We determine the quadratic points on the modular curves $X_0(N)$ for $N\leq 100$ for which this has not been previously done, namely the cases $$N\in\{66,70,78,82,84,86,87,88,90,96,99\}.$$ We accomplish this by improving on the ``going down method," which uses the fact that we have a moduli description of all the (infinitely many) quadratic points on $X_0(n)$ for some divisor $n$ of $N$.
- 4 December 2024
- Published by Elsevier in Journal of Algebra
- Vol. 666, 279-288
Quadratic points on X0(163)
- 4 December 2024
- Published by Elsevier in Journal of Algebra
- Vol. 666, 279-288
Quadratic points on X0(163)
- 29 November 2024
- Published by Oxford University Press (OUP) in The Quarterly Journal of Mathematics
- Vol. 76 (1) , 1-18
Hyperelliptic and Trigonal Modular Curves in Characteristic p
Let $X_\Delta(N)$ be an intermediate modular curve of level N, meaning that there exist (possibly trivial) morphisms $X_1(N)\rightarrow X_\Delta(N) \rightarrow X_0(N)$. For all such intermediate modular curves, we give an explicit description of all primes $p \nmid N$ such that $X_\Delta(N)_{\overline{\mathbb{F}}_p}$ is either hyperelliptic or trigonal. Furthermore we also determine all primes p such that $X_\Delta(N)_{\mathbb{F}_p}$ is trigonal. This is done by first using the Castelnuovo–Severi inequality to establish a bound N0 such that if $X_0(N)_{{\overline{\mathbb{F}}_p}}$ is hyperelliptic or trigonal, then $N \leq N_0$. To deal with the remaining small values of N, we develop a method based on the careful study of the canonical ideal to determine, for a fixed curve $X_\Delta(N)$, all the primes p such that $X_\Delta(N)_{{\overline{\mathbb{F}}_p}}$ is trigonal or hyperelliptic. Furthermore, as a byproduct of the developed methods we show that $X_\Delta(N)_{{\overline{\mathbb{F}}_p}}$ is not a smooth plane quintic, for any N and any p.- 29 November 2024
- Published by Oxford University Press (OUP) in The Quarterly Journal of Mathematics
- Vol. 76 (1) , 1-18
Hyperelliptic and Trigonal Modular Curves in Characteristic p
Let $X_\Delta(N)$ be an intermediate modular curve of level N, meaning that there exist (possibly trivial) morphisms $X_1(N)\rightarrow X_\Delta(N) \rightarrow X_0(N)$. For all such intermediate modular curves, we give an explicit description of all primes $p \nmid N$ such that $X_\Delta(N)_{\overline{\mathbb{F}}_p}$ is either hyperelliptic or trigonal. Furthermore we also determine all primes p such that $X_\Delta(N)_{\mathbb{F}_p}$ is trigonal. This is done by first using the Castelnuovo–Severi inequality to establish a bound N0 such that if $X_0(N)_{{\overline{\mathbb{F}}_p}}$ is hyperelliptic or trigonal, then $N \leq N_0$. To deal with the remaining small values of N, we develop a method based on the careful study of the canonical ideal to determine, for a fixed curve $X_\Delta(N)$, all the primes p such that $X_\Delta(N)_{{\overline{\mathbb{F}}_p}}$ is trigonal or hyperelliptic. Furthermore, as a byproduct of the developed methods we show that $X_\Delta(N)_{{\overline{\mathbb{F}}_p}}$ is not a smooth plane quintic, for any N and any p.- 25 April 2024
- Published by American Mathematical Society (AMS) in Mathematics of Computation
- Vol. 94 (351) , 447-473
Towards a classification of isolated 𝑗-invariants
We develop an algorithm to test whether a non-complex multiplication elliptic curve E / Q E/\mathbf {Q} gives rise to an isolated point of any degree on any modular curve of the form X 1 ( N ) X_1(N) . This builds on prior work of Zywina which gives a method for computing the image of the adelic Galois representation associated to E E . Running this algorithm on all elliptic curves presently in the L L -functions and Modular Forms Database and the Stein–Watkins Database gives strong evidence for the conjecture that E E gives rise to an isolated point on X 1 ( N ) X_1(N) if and only if j ( E ) = − 140625 / 8 , − 9317 j(E)=-140625/8, -9317 , 351 / 4 351/4 , or − 162677523113838677 -162677523113838677 .
- 25 April 2024
- Published by American Mathematical Society (AMS) in Mathematics of Computation
- Vol. 94 (351) , 447-473
Towards a classification of isolated 𝑗-invariants
We develop an algorithm to test whether a non-complex multiplication elliptic curve E / Q E/\mathbf {Q} gives rise to an isolated point of any degree on any modular curve of the form X 1 ( N ) X_1(N) . This builds on prior work of Zywina which gives a method for computing the image of the adelic Galois representation associated to E E . Running this algorithm on all elliptic curves presently in the L L -functions and Modular Forms Database and the Stein–Watkins Database gives strong evidence for the conjecture that E E gives rise to an isolated point on X 1 ( N ) X_1(N) if and only if j ( E ) = − 140625 / 8 , − 9317 j(E)=-140625/8, -9317 , 351 / 4 351/4 , or − 162677523113838677 -162677523113838677 .
- 7 November 2023
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 152 (02) , 591-598
Two results on 𝑥^{𝑟}+𝑦^{𝑟}=𝑑𝑧^{𝑝}
This note proves two theorems regarding Fermat-type equation x r + y r = d z p x^r + y^r = dz^p where r ≥ 5 r \geq 5 is a prime. Our main result shows that, for infinitely many integers d d , the previous equation has no non-trivial primitive solutions such that 2 ∣ x + y 2 \mid x+y or r ∣ x + y r \mid x+y , for a set of exponents p p of positive density. We use the modular method with a symplectic argument to prove this result.
- 7 November 2023
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 152 (02) , 591-598
Two results on 𝑥^{𝑟}+𝑦^{𝑟}=𝑑𝑧^{𝑝}
This note proves two theorems regarding Fermat-type equation x r + y r = d z p x^r + y^r = dz^p where r ≥ 5 r \geq 5 is a prime. Our main result shows that, for infinitely many integers d d , the previous equation has no non-trivial primitive solutions such that 2 ∣ x + y 2 \mid x+y or r ∣ x + y r \mid x+y , for a set of exponents p p of positive density. We use the modular method with a symplectic argument to prove this result.