Timo Keller
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View article: Complete verification of strong BSD for many modular abelian surfaces over ${\mathbf {Q}}$
Complete verification of strong BSD for many modular abelian surfaces over ${\mathbf {Q}}$ Open
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…
View article: 17T7 is a Galois group over the rationals
17T7 is a Galois group over the rationals Open
We prove that the transitive permutation group 17T7, isomorphic to a split extension of $C_2$ by $\mathrm{PSL}_2(\mathbb{F}_{16})$, is a Galois group over the rationals. The group arises from the field of definition of the $2$-torsion on a…
View article: $p$-converse theorems for elliptic curves of potentially good ordinary reduction at Eisenstein primes
$p$-converse theorems for elliptic curves of potentially good ordinary reduction at Eisenstein primes Open
Let $E/\mathbf{Q}$ 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 (i.e., such that the residual representation $E[p]$ is red…
View article: On the anticyclotomic Iwasawa theory of newforms at Eisenstein primes of semistable reduction
On the anticyclotomic Iwasawa theory of newforms at Eisenstein primes of semistable reduction Open
Let $f$ be a newform of weight $k=2r$ and level $N$ with trivial nebentypus. Let $\mathfrak{p}\nmid 2N$ be a maximal ideal of the ring of integers of the coefficient field of $f$ such that the self-dual twist of the mod-$\mathfrak{p}$ Galo…
View article: Complete verification of strong BSD for many modular abelian surfaces over $\mathbf{Q}$
Complete verification of strong BSD for many modular abelian surfaces over $\mathbf{Q}$ Open
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 $…
View article: Towards a classification of isolated $j$-invariants
Towards a classification of isolated $j$-invariants Open
We develop an algorithm to test whether a non-CM elliptic curve $E/\mathbb{Q}$ gives rise to an isolated point of any degree on any modular curve of the form $X_1(N)$. This builds on prior work of Zywina which gives a method for computing …
View article: Specialization of Mordell–Weil ranks of abelian schemes over surfaces to curves
Specialization of Mordell–Weil ranks of abelian schemes over surfaces to curves Open
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…
View article: Computing quadratic points on modular curves $X_0(N)$
Computing quadratic points on modular curves $X_0(N)$ Open
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,…
View article: Quadratic Chabauty for Atkin–Lehner quotients ofmodular curves of prime level and genus 4, 5, 6
Quadratic Chabauty for Atkin–Lehner quotients ofmodular curves of prime level and genus 4, 5, 6 Open
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 four, five, and six. We fi…
View article: On the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math>-torsion of the Tate–Shafarevich group of abelian varieties over higher dimensional bases over finite fields
On the -torsion of the Tate–Shafarevich group of abelian varieties over higher dimensional bases over finite fields Open
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 prove the invariance of the finiteness of the Tate–Shafarevic…
View article: Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings
Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings Open
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 c…
View article: Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces
Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces Open
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 .
View article: On the $p$-torsion of the Tate-Shafarevich group of abelian varieties over higher dimensional bases over finite fields
On the $p$-torsion of the Tate-Shafarevich group of abelian varieties over higher dimensional bases over finite fields Open
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 prove the invariance of the finiteness of the Tate-Shafarevic…
View article: Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings
Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings Open
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 classic…
View article: Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces
Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces Open
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-Tate group of $J/\mathbb{Q}$ is trivial. This verifies the st…
View article: On an Analogue of the Conjecture of Birch and Swinnerton-Dyer for Abelian Schemes over Higher Dimensional Bases over Finite Fields
On an Analogue of the Conjecture of Birch and Swinnerton-Dyer for Abelian Schemes over Higher Dimensional Bases over Finite Fields Open
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 characteristic p . We prove the prime-to- p part conditionally o…