# Barinder Singh Banwait's Homepage

B dot S dot Banwait at warwick dot ac dot uk

I am a fourth year PhD Student in Mathematics at the University of Warwick, U.K. My advisor is John Cremona. Here is my CV.

Research Interests, and What my PhD is about

I am interested in rational points on higher genus curves and higher dimensional varieties, as well as Diophantine Equations and the Generalised Fermat Problem.

A paper describing some of my thesis is available here. But here follows a short summary:

In my thesis I am studying a local-to-global principle for isogenies of prime degree on elliptic curves over number fields $E/K$, after Andrew Sutherland, who last year published a paper on this. If $E/K$ has a $K$-rational $l$-isogeny, then $E/K_\mathfrak{p}$ has a $K_\mathfrak{p}$-rational $l$-isogeny, for every prime $\mathfrak{p}$ of $K$. Sutherland asks the converse question: if all completions have $K_\mathfrak{p}$ rational $l$-isogeny, then must $E/K$ have a $K$-rational $l$-isogeny? (Well, technically, Sutherland only assumes that almost all completions have $K_\mathfrak{p}$ rational $l$-isogeny.) Sutherland shows that the answer is No, and shows that the obstruction is given by the image of the mod-$l$ representation associated to $E$ satisfying a certain property. Sutherland was not able to deal with all cases; he had to make a certain assumption on the pair $(K,l)$. The first thing I do is to deal with this case Sutherland does not deal with, and then proceed to classify and find all elliptic curves over quadratic fields which fail this local-to-global principle. An interesting part of this classification led me to ask if there are elliptic curves over $\mathbb{Q}(\sqrt{13})$ whose mod-13 image, when reduced mod scalar matrices, is isomorphic to the alternating group $A_4$ (as these would give failures of the local-to-global principle). This question took me over a year to answer, but finally, I found that the answer is YES; there are in fact elliptic curves over $\mathbb{Q}$ whose projective mod-13 image is isomorphic to $S_4$; thus, when basechanged to $\mathbb{Q}(\sqrt{13})$, they attain projective $A_4$ image at 13. I also have examples that are not base-changes from $\mathbb{Q}$. These results were obtained by explicitly studying the modular curve $X_{S_4}(13)$ parametrising these elliptic curves. I was able to write down an explicit model of this genus 3 curve over $\mathbb{Q}$, as a plane quartic in $\mathbb{P}^2$ (which was not easy). I then found points defined over $\mathbb{Q}$ and $\mathbb{Q}(\sqrt{13})$ on this curve by searching in Magma for points up to height 10 million, and evaluating the $j$-map at these points. I cannot prove that there are no other points on this curve; the problem is that the rank of the Jacobian is 3, so Chabauty's method does not apply.