Timo Keller
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Timo Keller has a total of 18 co-authors. Here are the top co-authors by number of shared publications:
- Nikola Adžaga
Faculty of Civil Engineering, University of Zagreb, Zagreb, Croatia
5 shared publications - Filip Najman
Faculty of Science, Department of Mathematics, University of Zagreb, Zagreb, Croatia
3 shared publications - Shiva Chidambaram
Massachusetts Institute of Technology
3 shared publications - Borna Vukorepa
Department of Mathematics, Faculty of Science, University of Zagreb, Bijenicka cesta 30, 10000 Zagreb, Croatia
2 shared publications - Ekin Ozman
Faculty of Arts and Sciences, Mathematics Department, Bogazici University, Bebek, Istanbul, Turkey
2 shared publications - Michael Stoll
Universität Bayreuth
2 shared publications
Latest Publications
- Open Access
- 30 January 2025
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Sigma
- Vol. 13
Complete verification of strong BSD for many modular abelian surfaces over ${\mathbf {Q}}$We develop the theory and algorithms necessary to be able to verify the strong Birch–Swinnerton-Dyer Conjecture for absolutely simple modular abelian varieties over ${\mathbf {Q}}$ . We apply our methods to all 28 Atkin–Lehner quotients of $X_0(N)$ of genus $2$ , all 97 genus $2$ curves from the LMFDB whose Jacobian is of this type and six further curves originally found by Wang. We are able to verify the strong BSD Conjecture unconditionally and exactly in all these cases; this is the first time that strong BSD has been confirmed for absolutely simple abelian varieties of dimension at least $2$ . We also give an example where we verify that the order of the Tate–Shafarevich group is $7^2$ and agrees with the order predicted by the BSD Conjecture.- 30 January 2025
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Sigma
- Vol. 13
Open AccessComplete verification of strong BSD for many modular abelian surfaces over ${\mathbf {Q}}$We develop the theory and algorithms necessary to be able to verify the strong Birch–Swinnerton-Dyer Conjecture for absolutely simple modular abelian varieties over ${\mathbf {Q}}$ . We apply our methods to all 28 Atkin–Lehner quotients of $X_0(N)$ of genus $2$ , all 97 genus $2$ curves from the LMFDB whose Jacobian is of this type and six further curves originally found by Wang. We are able to verify the strong BSD Conjecture unconditionally and exactly in all these cases; this is the first time that strong BSD has been confirmed for absolutely simple abelian varieties of dimension at least $2$ . We also give an example where we verify that the order of the Tate–Shafarevich group is $7^2$ and agrees with the order predicted by the BSD Conjecture. - 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 .
- 3 October 2023
- Published by American Mathematical Society (AMS) in Mathematics of Computation
- Vol. 93 (347) , 1371-1397
Computing quadratic points on modular curves 𝑋₀(𝑁)
In this paper we improve on existing methods to compute quadratic points on modular curves and apply them to successfully find all the quadratic points on all modular curvesof genus up to , and genus up to with prime, for which they were previously unknown. The values of we consider are contained in the set We obtain that all the non-cuspidal quadratic points on for are complex multiplication (CM) points, except for one pair of Galois conjugate points on defined over . We also compute the -invariants of the elliptic curves parametrised by these points, and for the CM points determine their geometric endomorphism rings. - 3 October 2023
- Published by American Mathematical Society (AMS) in Mathematics of Computation
- Vol. 93 (347) , 1371-1397
Computing quadratic points on modular curves 𝑋₀(𝑁)
In this paper we improve on existing methods to compute quadratic points on modular curves and apply them to successfully find all the quadratic points on all modular curvesof genus up to , and genus up to with prime, for which they were previously unknown. The values of we consider are contained in the set We obtain that all the non-cuspidal quadratic points on for are complex multiplication (CM) points, except for one pair of Galois conjugate points on defined over . We also compute the -invariants of the elliptic curves parametrised by these points, and for the CM points determine their geometric endomorphism rings. - Open AccessComputing quadratic points on modular curves $X_0(N)$In this paper we improve on existing methods to compute quadratic points on modular curves and apply them to successfully find all the quadratic points on all modular curves $X_0(N)$ of genus up to $8$, and genus up to $10$ with $N$ prime, for which they were previously unknown. The values of $N$ we consider are contained in the set \[ \mathcal{L}=\{58, 68, 74, 76, 80, 85, 97, 98, 100, 103, 107, 109, 113, 121, 127 \}.\] We obtain that all the non-cuspidal quadratic points on $X_0(N)$ for $N\in \mathcal{L}$ are CM points, except for one pair of Galois conjugate points on $X_0(103)$ defined over $\mathbb{Q}(\sqrt{2885})$. We also compute the $j$-invariants of the elliptic curves parametrised by these points, and for the CM points determine their geometric endomorphism rings.Open AccessComputing quadratic points on modular curves $X_0(N)$In this paper we improve on existing methods to compute quadratic points on modular curves and apply them to successfully find all the quadratic points on all modular curves $X_0(N)$ of genus up to $8$, and genus up to $10$ with $N$ prime, for which they were previously unknown. The values of $N$ we consider are contained in the set \[ \mathcal{L}=\{58, 68, 74, 76, 80, 85, 97, 98, 100, 103, 107, 109, 113, 121, 127 \}.\] We obtain that all the non-cuspidal quadratic points on $X_0(N)$ for $N\in \mathcal{L}$ are CM points, except for one pair of Galois conjugate points on $X_0(103)$ defined over $\mathbb{Q}(\sqrt{2885})$. We also compute the $j$-invariants of the elliptic curves parametrised by these points, and for the CM points determine their geometric endomorphism rings.
- 27 March 2023
- Published by World Scientific Pub Co Pte Ltd in International Journal of Number Theory
- Vol. 19 (07) , 1671-1680
Specialization of Mordell–Weil ranks of abelian schemes over surfaces to curves
Using the Shioda–Tate theorem and an adaptation of Silverman’s specialization theorem, we reduce the specialization of Mordell–Weil ranks for abelian varieties over fields finitely generated over infinite finitely generated fields [ ] to the specialization theorem for Néron–Severi ranks recently proved by Ambrosi in positive characteristic. More precisely, we prove that after a blow-up of the base surface [ ], for all vertical curves [ ] of a fibration [ ] with [ ] from the complement of a sparse subset of [ ], the Mordell–Weil rank of an abelian scheme over [ ] stays the same when restricted to [ ].
- 27 March 2023
- Published by World Scientific Pub Co Pte Ltd in International Journal of Number Theory
- Vol. 19 (07) , 1671-1680
Specialization of Mordell–Weil ranks of abelian schemes over surfaces to curves
Using the Shioda–Tate theorem and an adaptation of Silverman’s specialization theorem, we reduce the specialization of Mordell–Weil ranks for abelian varieties over fields finitely generated over infinite finitely generated fields [ ] to the specialization theorem for Néron–Severi ranks recently proved by Ambrosi in positive characteristic. More precisely, we prove that after a blow-up of the base surface [ ], for all vertical curves [ ] of a fibration [ ] with [ ] from the complement of a sparse subset of [ ], the Mordell–Weil rank of an abelian scheme over [ ] stays the same when restricted to [ ].
- Open Access
Specialization of Mordell-Weil ranks of abelian schemes over surfaces to curves
Using the Shioda-Tate theorem and an adaptation of Silverman's specialization theorem, we reduce the specialization of Mordell-Weil ranks for abelian varieties over fields finitely generated over infinite finitely generated fields $k$ to the the specialization theorem for Néron-Severi ranks recently proved by Ambrosi in positive characteristic. More precisely, we prove that after a blow-up of the base surface $S$, for all vertical curves $S_x$ of a fibration $S \to U \subseteq \mathbf{P}^1_k$ with $x$ from the complement of a sparse subset of $|U|$, the Mordell-Weil rank of an abelian scheme over $S$ stays the same when restricted to $S_x$.Open AccessSpecialization of Mordell-Weil ranks of abelian schemes over surfaces to curves
Using the Shioda-Tate theorem and an adaptation of Silverman's specialization theorem, we reduce the specialization of Mordell-Weil ranks for abelian varieties over fields finitely generated over infinite finitely generated fields $k$ to the the specialization theorem for Néron-Severi ranks recently proved by Ambrosi in positive characteristic. More precisely, we prove that after a blow-up of the base surface $S$, for all vertical curves $S_x$ of a fibration $S \to U \subseteq \mathbf{P}^1_k$ with $x$ from the complement of a sparse subset of $|U|$, the Mordell-Weil rank of an abelian scheme over $S$ stays the same when restricted to $S_x$. - 1 January 2023
- Published by Institute of Mathematics, Polish Academy of Sciences in Acta Arithmetica
- Vol. 208 (1) , 15-49
Quadratic Chabauty for Atkin–Lehner quotients of modular curves of prime level and genus 4, 5, 6
We use the method of quadratic Chabauty on the quotients $X_0^+(N)$ of modular curves $X_0(N)$ by their Fricke involutions to provably compute all the rational points of these curves for prime levels $N$ of genus 4, 5, and 6. We find that the only such cu- 1 January 2023
- Published by Institute of Mathematics, Polish Academy of Sciences in Acta Arithmetica
- Vol. 208 (1) , 15-49
Quadratic Chabauty for Atkin–Lehner quotients of modular curves of prime level and genus 4, 5, 6
We use the method of quadratic Chabauty on the quotients $X_0^+(N)$ of modular curves $X_0(N)$ by their Fricke involutions to provably compute all the rational points of these curves for prime levels $N$ of genus 4, 5, and 6. We find that the only such cu- Open Access
- 12 October 2022
- Published by Springer Nature in Research in Number Theory
- Vol. 8 (4) , 1-24
Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings
We complete the computation of all $$\mathbb {Q}$$ -rational points on all the 64 maximal Atkin-Lehner quotients $$X_0(N)^*$$ such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty–Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method (from a forthcoming preprint of the fourth-named author) combined with the Mordell-Weil sieve. Additionally, for square-free levelsN, we classify all $$\mathbb {Q}$$ -rational points as cusps, CM points (including their CM field andj-invariants) and exceptional ones. We further indicate how to use this to compute the $$\mathbb {Q}$$ -rational points on all of their modular coverings.- 12 October 2022
- Published by Springer Nature in Research in Number Theory
- Vol. 8 (4) , 1-24
Open AccessRational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings
We complete the computation of all $$\mathbb {Q}$$ -rational points on all the 64 maximal Atkin-Lehner quotients $$X_0(N)^*$$ such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty–Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method (from a forthcoming preprint of the fourth-named author) combined with the Mordell-Weil sieve. Additionally, for square-free levelsN, we classify all $$\mathbb {Q}$$ -rational points as cusps, CM points (including their CM field andj-invariants) and exceptional ones. We further indicate how to use this to compute the $$\mathbb {Q}$$ -rational points on all of their modular coverings. - Open Access
Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings
We complete the computation of all $\mathbb{Q}$-rational points on all the $64$ maximal Atkin-Lehner quotients $X_0(N)^*$ such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty--Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method combined with the Mordell-Weil sieve. Additionally, for square-free levels $N$, we classify all $\mathbb{Q}$-rational points as cusps, CM points (including their CM field and $j$-invariants) and exceptional ones. We further indicate how to use this to compute the $\mathbb{Q}$-rational points on all of their modular coverings.Open AccessRational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings
We complete the computation of all $\mathbb{Q}$-rational points on all the $64$ maximal Atkin-Lehner quotients $X_0(N)^*$ such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty--Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method combined with the Mordell-Weil sieve. Additionally, for square-free levels $N$, we classify all $\mathbb{Q}$-rational points as cusps, CM points (including their CM field and $j$-invariants) and exceptional ones. We further indicate how to use this to compute the $\mathbb{Q}$-rational points on all of their modular coverings. - 23 May 2022
- Published by MathDoc/Centre Mersenne in Comptes Rendus Mathematique
- Vol. 360 (G5) , 483-489
Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces
Let be one of the Atkin–Lehner quotients of a curve such that has genus and its Jacobian variety is absolutely simple. We show that the Shafarevich–Tate group is trivial. This verifies the strong BSD conjecture for .- 23 May 2022
- Published by MathDoc/Centre Mersenne in Comptes Rendus Mathematique
- Vol. 360 (G5) , 483-489
Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces
Let be one of the Atkin–Lehner quotients of a curve such that has genus and its Jacobian variety is absolutely simple. We show that the Shafarevich–Tate group is trivial. This verifies the strong BSD conjecture for .