arXiv subjects

Aashraya Jha

Publications and source records attributed to Aashraya Jha.

4 recordsLinked to original sources

arXiv2025

Rational points on the non-split Cartan modular curve of level 27 and quadratic Chabauty over number fields

Thanks to work of Rouse, Sutherland, and Zureick-Brown, it is known exactly which subgroups of GL$_2(\mathbf{Z}_3)$ can occur as the image of the $3$-adic Galois representation attached to a non-CM elliptic curve over $\mathbf{Q}$, with a single exception: the normaliser of the non-split Cartan subgroup of level 27. In this paper, we complete the classification of 3-adic Galois images by showing that the normaliser of the non-split Cartan subgroup of level 27 cannot occur as a 3-adic Galois image of a non-CM elliptic curve. Our proof proceeds via computing the $\mathbf{Q}(ζ_3)$-rational points on a certain smooth plane quartic curve $X'_H$ (arising as a quotient of the modular curve $X_{ns}^+(27)$) defined over $\mathbf{Q}(ζ_3)$ whose Jacobian has Mordell--Weil rank 6. To this end, we describe how to carry out the quadratic Chabauty method for a modular curve $X$ defined over a number field $F$, which, when applicable, determines a finite subset of $X(F\otimes\mathbf{Q}_p)$ in certain situations of larger Mordell--Weil rank than previously considered. Together with an analysis of local heights above 3, we apply this quadratic Chabauty method to determine $X'_H(\mathbf{Q}(ζ_3))$. This allows us to compute the set $X_{ns}^+(27)(\mathbf{Q})$, finishing the classification of 3-adic images of Galois.

math.NT↗

arXiv2024

Finding Integral Points of Elliptic Curves over Imaginary Quadratic Fields

We determine the quadratic Chabauty set for integral points on elliptic curves of rank $2$ defined over imaginary quadratic fields using quadratic Chabauty. This builds on the work of Bianchi and Balakrishnan et al. We give the first instance of the implementation of anticyclotomic heights for curves which are not base changes, along with an implementation of a certain sieve for elliptic curves introduced by Balakrishnan et al. and used by Bianchi to determine integral points of rank $2$. We give the first example of the determination of the integral points of an elliptic curve of rank $2$ defined over an imaginary quadratic field, which is not a base change via quadratic Chabauty.

math.NT↗

arXiv2023

The Brauer Group of $\mathscr{Y}_0(2)$

We determine the Brauer group of the Deligne-Mumford stack $\mathscr{Y}_0(2)$, the moduli space of elliptic curves with a marked $2$-torsion subgroup over bases of arithmetic interest. Antieau and Meier determine the Brauer group for $\mathscr{M}_{1,1}$, the moduli stack of elliptic curves by exploiting the fact it is covered by the Legendre family and using the Hochschild-Serre spectral sequence. Over an algebraically closed field, Shin uses the coarse space map to determine the Brauer group of $\mathscr{M}_{1,1}$. We combine techniques from both papers to determine the Brauer group of $\mathscr{Y}_0(2)$.

math.AG↗

arXiv2017

Non-optimality of the Greedy Algorithm for subspace orderings in the method of alternating projections

The method of alternating projections involves projecting an element of a Hilbert space cyclically onto a collection of closed subspaces. It is known that the resulting sequence always converges in norm and that one can obtain estimates for the rate of convergence in terms of quantities describing the geometric relationship between the subspaces in question, namely their pairwise Friedrichs numbers. We consider the question of how best to order a given collection of subspaces so as to obtain the best estimate on the rate of convergence. We prove, by relating the ordering problem to a variant of the famous Travelling Salesman Problem, that correctness of a natural form of the Greedy Algorithm would imply that $\mathrm{P}=\mathrm{NP}$, before presenting a simple example which shows that, contrary to a claim made in the influential paper [Kayalar-Weinert, Math. Control Signals Systems, vol. 1(1), 1988], the result of the Greedy Algorithm is not in general optimal. We go on to establish sharp estimates on the degree to which the result of the Greedy Algorithm can differ from the optimal result. Underlying all of these results is a construction which shows that for any matrix whose entries satisfy certain natural assumptions it is possible to construct a Hilbert space and a collection of closed subspaces such that the pairwise Friedrichs numbers between the subspaces are given precisely by the entries of that matrix.

math.NA↗