"Non UBC"@en . "DSpace"@en . "van Luijk, Ronald"@en . "2019-03-14T02:06:32Z"@en . "2018-05-30T09:00"@en . "Let $S$ be a del Pezzo surface of degree $1$ over a number field $k$. The main goal of this talk is to give easily verifiable sufficient conditions under which its set $S(k)$ of rational points is Zariski dense. It is well known that almost all fibers of the anticanonical map $\varphi \colon S \dashrightarrow \mathbb{P}^1$ are elliptic curves with the unique base point of $\varphi$ as zero. Suppose that $P \in S(k)$ is a point of finite order $n>1$ on its fiber. Then there is another elliptic fibration on the blow-up of $S$ at $P$. We will see where it comes from and how it can be used to define a proper closed subset $Z \subset S$ such that (1) it is easy to verify for any point on $S$ whether it lies in $Z$, and (2) the set $S(k)$ contains a point outside $Z$ if and only if $S(k)$ is Zariski dense. In other words, if $S(k)$ is Zariski dense, then we can prove this by exhibiting a rational point outside $Z$. We will also compare this to previous work. This is joint work with Jelle Bulthuis inspired by an example of Noam Elkies."@en . "https://circle.library.ubc.ca/rest/handle/2429/68702?expand=metadata"@en . "32.0"@en . "video/mp4"@en . ""@en . "Author affiliation: Universiteit Leiden"@en . "10.14288/1.0376868"@en . "eng"@en . "Unreviewed"@en . "Vancouver : University of British Columbia Library"@en . "Banff International Research Station for Mathematical Innovation and Discovery"@en . "Attribution-NonCommercial-NoDerivatives 4.0 International"@en . "http://creativecommons.org/licenses/by-nc-nd/4.0/"@en . "Researcher"@en . "BIRS Workshop Lecture Videos (Oaxaca de Ju\u00E1rez (Mexico))"@en . "Mathematics"@en . "Number theory"@en . "Algebraic geometry"@en . "Arithmetic number theory"@en . "Verifying Zariski density of rational points on del Pezzo surfaces of degree 1"@en . "Moving Image"@en . "http://hdl.handle.net/2429/68702"@en .