arXiv subjects

Anthony Kling

Publications and source records attributed to Anthony Kling.

3 recordsLinked to original sources

arXiv2024

A graph-theoretic approach to computing Selmer groups of elliptic curves $y^2 = x^3 + bx$ over $\mathbb{Q}(i)$

We develop a graph-theoretic algorithm to compute the $\varphi$-Selmer group of the elliptic curve $E_b: y^2 = x^3 + bx$ over $\mathbb{Q}(i)$, where $b \in \mathbb{Z}[i]$ and $\varphi$ is a degree 2 isogeny of $E_b$. We associate to $E_b$ a weighted graph $G_b$, whose vertices are the odd Gaussian primes dividing $b$, and whose edge weights are determined by the quartic residue symbol between pairs of these primes. By applying our algorithm, we explicitly compute the $\varphi$-Selmer group of $E_b$ when $b$ is a product of inert primes, and we construct several infinite families of elliptic curves over $\mathbb{Q}(i)$ with trivial Mordell-Weil rank.

math.NT↗

arXiv2023

Comparison of integral structures on the space of modular forms of full level N

Let $N\geq3$ and $r\geq1$ be integers and $p\geq2$ be a prime such that $p\nmid N$. One can consider two different integral structures on the space of modular forms over $\mathbb{Q}$, one coming from arithmetic via $q$-expansions, the other coming from geometry via integral models of modular curves. Both structures are stable under the Hecke operators; furthermore, their quotient is finite torsion. Our goal is to investigate the exponent of the annihilator of the quotient. We will apply methods due to Brian Conrad to the situation of modular forms of even weight and level $\Gamma(Np^{r})$ over $\mathbb{Q}_{p}(\zeta_{Np^{r}})$ to obtain an upper bound for the exponent. We also use Klein forms to construct explicit modular forms of level $p^{r}$ whenever $p^{r}>3$, allowing us to compute a lower bound which agrees with the upper bound. Hence we are able to compute the exponent precisely.

math.NT↗

arXiv2021

Perfect congruences on bisimple $\omega$-semigroups

A congruence $\varepsilon$ on a semigroup $S$ is perfect if for any congruence classes $x\varepsilon$ and $y\varepsilon$ their product as subsets of $S$ coincides (as a set) with the congruence class $(xy)\varepsilon$. Perfect congruences on the bicyclic semigroup were found in \cite{key7}. Using the structure of bisimple $\omega$-semigroups determined in \cite{key25} and the description of congruences on these semigroups found in \cite{key20} and \cite{key1}, we obtain a complete characterization of perfect congruences on all bisimple $\omega$-semigroups, substantially generalizing the above mentioned result of \cite{key7}.

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