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Publications (16)

math.NT2022

Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings

Nikola AdÅ¾aga, Shiva Chidambaram, Timo Keller +1

We complete the computation of all $Q$ -rational points on all the $64$ maximal Atkin-Lehner quotients $X_{0} (N)^{*}$ such that the quotient is hyperelliptic. To achieve this,…

math.NT2021

Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces

Timo Keller, Michael Stoll

Let $X$ be one of the $28$ Atkin-Lehner quotients of a curve $X_{0} (N)$ such that $X$ has genus $2$ and its Jacobian variety $J$ is absolutely simple. We show that the Shafarevich-Ta…

math.NT2019

On an analogue of the conjecture of Birch and Swinnerton-Dyer for Abelian schemes over higher dimensional bases over finite fields

Timo Keller

We formulate an analogue of the conjecture of Birch and Swinnerton-Dyer for Abelian schemes with everywhere good reduction over higher dimensional bases over finite fields of chara…

math.NT2024

Complete verification of strong BSD for many modular abelian surfaces over $Q$

Timo Keller, Michael Stoll

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 $Q$ .…

math.NT2025

Towards a classification of isolated $j$ -invariants

Abbey Bourdon, Sachi Hashimoto, Timo Keller +5

We develop an algorithm to test whether a non-CM elliptic curve $E / Q$ gives rise to an isolated point of any degree on any modular curve of the form $X_{1} (N)$ . This builds…

math.NT2024

On the anticyclotomic Iwasawa theory of newforms at Eisenstein primes of semistable reduction

Timo Keller, Mulun Yin

Let $f$ be a newform of weight $k = 2 r$ and level $N$ with trivial nebentypus. Let $p ∤ 2 N$ be a maximal ideal of the ring of integers of the coefficient field of $f$ s…

math.NT2025

Rational points on $X_{0} (N)^{*}$ when $N$ is non-squarefree

Sachi Hashimoto, Timo Keller, Samuel Le Fourn

Let $N$ be a non-squarefree integer such that the quotient $X_{0} (N)^{*}$ of the modular curve $X_{0} (N)$ by the full group of Atkin-Lehner involutions has positive genus. Elkies conject…

math.NT2023

Specialization of Mordell-Weil ranks of abelian schemes over surfaces to curves

Timo Keller

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…

math.NT2021

Quadratic Chabauty for Atkin-Lehner Quotients of Modular Curves of Prime Level and Genus 4, 5, 6

Nikola AdÅ¾aga, Vishal Arul, Lea Beneish +4

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 curv…

math.NT2022

On the $p$ -torsion of the Tate-Shafarevich group of abelian varieties over higher dimensional bases over finite fields

Timo Keller

We prove a finiteness theorem for the first flat cohomology group of finite flat group schemes over integral normal proper varieties over finite fields. As a consequence, we can pr…

math.NT2026

The constructive inverse Galois problem via Hilbert modular forms: realizing the transitive group 17T7

Raymond van Bommel, Edgar Costa, Noam D. Elkies +3

We show how Hilbert modular forms can be used in the constructive inverse Galois problem over the rationals. In particular, we prove that the transitive permutation group 17T7, iso…

math.AG2025

Comparison of different Tate conjectures

Veronika Ertl, Timo Keller, Yanshuai Qin

For an abelian variety $A$ over a finitely generated field $K$ of characteristic $p > 0$ , we prove that the algebraic rank of $A$ is at most a suitably defined analytic rank. Moreo…

math.NT2016

On the Tate-Shafarevich group of Abelian schemes over higher dimensional bases over finite fields

Timo Keller

We study analogues for the Tate-Shafarevich group for Abelian schemes with everywhere good reduction over higher dimensional bases over finite fields.

math.NT2024

$p$ -converse theorems for elliptic curves of potentially good ordinary reduction at Eisenstein primes

Timo Keller, Mulun Yin

Let $E / Q$ be an elliptic curve and $p \geq 3$ be a prime. We prove the $p$ -converse theorems for elliptic curves of potentially good ordinary reduction at Eisenstein primes…

math.NT2026

Point counts of abelian varieties over finite fields determining their zeta function

Shiva Chidambaram, Timo Keller

The paper proves that for an abelian variety of dimension g over a sufficiently large finite field, the point counts over the first g extensions uniquely determine its zeta functio…

#abelian varieties#finite fields#zeta functions#point counts

math.NT2023

Computing quadratic points on modular curves $X_{0} (N)$

Nikola AdÅ¾aga, Timo Keller, Philippe Michaud-Jacobs +3

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…

Publications (16)

Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings

Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces

On an analogue of the conjecture of Birch and Swinnerton-Dyer for Abelian schemes over higher dimensional bases over finite fields

Complete verification of strong BSD for many modular abelian surfaces over Q

Towards a classification of isolated j-invariants

On the anticyclotomic Iwasawa theory of newforms at Eisenstein primes of semistable reduction

Rational points on X0​(N)∗ when N is non-squarefree

Specialization of Mordell-Weil ranks of abelian schemes over surfaces to curves

Quadratic Chabauty for Atkin-Lehner Quotients of Modular Curves of Prime Level and Genus 4, 5, 6

On the p-torsion of the Tate-Shafarevich group of abelian varieties over higher dimensional bases over finite fields

The constructive inverse Galois problem via Hilbert modular forms: realizing the transitive group 17T7

Comparison of different Tate conjectures

On the Tate-Shafarevich group of Abelian schemes over higher dimensional bases over finite fields

p-converse theorems for elliptic curves of potentially good ordinary reduction at Eisenstein primes

Point counts of abelian varieties over finite fields determining their zeta function

Computing quadratic points on modular curves X0​(N)

Complete verification of strong BSD for many modular abelian surfaces over $Q$

Towards a classification of isolated $j$ -invariants

Rational points on $X_{0} (N)^{*}$ when $N$ is non-squarefree

On the $p$ -torsion of the Tate-Shafarevich group of abelian varieties over higher dimensional bases over finite fields

$p$ -converse theorems for elliptic curves of potentially good ordinary reduction at Eisenstein primes

Computing quadratic points on modular curves $X_{0} (N)$