arXiv subjects

Naina Praveen

Publications and source records attributed to Naina Praveen.

3 recordsLinked to original sources

arXiv2026

Distinguishing elliptic curves modulo $p$ and identifying images of product representations

Given two elliptic curves defined over $\mathbb{Q}$ and a rational prime $p$, we study the product of their residual Galois representations. Using Goursat's lemma, we explicitly enumerate and completely characterize all possible images of such product representations. We also define associated invariants to these image groups, which we call \textit{witness ratios}, and we explain their computational utility and their relationship to the well-known Sturm bound for testing congruences between modular forms.

math.NT↗

arXiv2026

Recovering Kodaira types from $\ell$-torsion on elliptic curves

The classical Néron-Ogg-Shafarevich criterion characterises good reduction of an elliptic curve $E$ over a $p$-adic field via the action of inertia on the $\ell$-adic Tate module. However, the action of inertia on $E[\ell]$ is not sufficient to distinguish between potentially good and multiplicative reduction, and the action on $T_{\ell}(E)$ is not sufficient to determine the Kodaira type. We remedy this situation by endowing $E[\ell]$ with a distance function that records the $p$-adic distances between the $x$-coordinates of the points. We show that, equipped with this additional structure, $E[\ell]$ determines the Kodaira type of the elliptic curve. In the case of residue characteristic $2$, we assume that $E$ does not have potentially good reduction of type $I_n^*$.

math.NT↗

arXiv2022

Restricted invertibility of continuous matrix functions

Motivated by an influential result of Bourgain and Tzafriri, we consider continuous matrix functions $A:\mathbb{R}\to M_{n\times n}$ and lower $\ell_2$-norm bounds associated with their restriction to certain subspaces. We prove that for any such $A$ with unit-length columns, there exists a continuous choice of subspaces $t\mapsto U(t)\subset \mathbb{R}^n$ such that for $v\in U(t)$, $\|A(t)v\|\geq c\|v\|$ where $c$ is some universal constant. Furthermore, the $U(t)$ are chosen so that their dimension satisfies a lower bound with optimal asymptotic dependence on $n$ and $\sup_{t\in \mathbb{R}}\|A(t)\|.$ We provide two methods. The first relies on an orthogonality argument, while the second is probabilistic and combinatorial in nature. The latter does not yield the optimal bound for $\dim(U(t))$ but the $U(t)$ obtained in this way are guaranteed to have a canonical representation as joined-together spaces spanned by subsets of the unit vector basis.

math.FA↗