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Oana Padurariu

Publications and source records attributed to Oana Padurariu.

11 recordsLinked to original sources
arXiv2026

Point counts, automorphisms, and gonalities of Shimura curves

We implement an algorithm to compute the number of points over finite fields for the Shimura curves $X_0^D(N)$ over $\mathbb{Q}$ and their Atkin--Lehner quotients. Our computations identify $116$ such quotients over finite fields (out of $783514$ tested) that attain a number of rational points exceeding that of any previously known curve of the same genus over the same finite field. To illustrate the utility of our point counts algorithm in addressing arithmetic questions, we prove that all automorphisms are Atkin--Lehner for $9288$ of the $10609$ curves $X_0^D(N)$ of genus $g > 2$ with $D$ the discriminant of an indefinite quaternion algebra over $\mathbb{Q}$, $N$ a squarefree positive integer coprime to $D$, and $DN\leq 10000$, and we determine all tetragonal and geometrically tetragonal curves $X_0^D(N)$ up to a small number of possible exceptions.

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arXiv2026

Cyclic isogenies of elliptic curves over fixed quadratic fields

Building on Mazur's 1978 work on prime degree isogenies, Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over $\mathbb{Q}$. Although more than 40 years have passed, the determination of cyclic isogenies of elliptic curves over a single other number field has hitherto not been realised. In this paper we develop a procedure to assist in establishing such a determination for a given quadratic field. Executing this procedure on all quadratic fields $\mathbb{Q}(\sqrt{d})$ with $|d| < 10^4$ we obtain, conditional on the Generalised Riemann Hypothesis, the determination of cyclic isogenies of elliptic curves over $19$ quadratic fields, including $\mathbb{Q}(\sqrt{213})$ and $\mathbb{Q}(\sqrt{-2289})$. To make this procedure work, we determine all of the finitely many quadratic points on the modular curves $X_0(125)$ and $X_0(169)$, which may be of independent interest.

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arXiv2025

Shimura curve Atkin--Lehner quotients of genus at most two

We provide a complete enumeration of all quotients of genus $0, 1$ and $2$ of the Shimura curves $X_0^D(N)$ over $\mathbb{Q}$ by non-trivial subgroups of Atkin--Lehner involutions. For all $1270$ genus $1$ quotients $X$ with $N$ squarefree, we determine the isomorphism class of the Jacobian $X$. For $146$ non-elliptic genus $1$ curves $X$ and for $405$ curves genus $2$ quotients $X$, we provide a defining equation for $X$. A main tool for us is the theory of Čerednik--Drinfeld uniformizations of the curves $X_0^D(N)$, which we implement in wider generality than has previously been done in the literature.

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arXiv2025

Bielliptic Shimura curves $X_0^D(N)$ with nontrivial level

We work towards completely classifying all bielliptic Shimura curves $X_0^D(N)$ with nontrivial level $N$ coprime to $D$, extending a result of Rotger that provided such a classification for level one. Combined with prior work, this allows us to determine the list of all relatively prime pairs $(D,N)$ for which $X_0^D(N)$ has infinitely many degree $2$ points. As an application, we use these results to make progress on determining which curves $X_0^D(N)$ have sporadic points. Using tools similar to those that appear in this study, we also determine all of the geometrically trigonal Shimura curves $X_0^D(N)$ with $\gcd(D,N)=1$ (none of which are trigonal over $\mathbb{Q}$).

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arXiv2025

Counting 5-isogenies of elliptic curves over $\mathbb{Q}$

We show that the number of $5$-isogenies of elliptic curves defined over $\mathbb{Q}$ with naive height bounded by $H > 0$ is asymptotic to $C_5\cdot H^{1/6} (\log H)^2$ for some explicitly computable constant $C_5 > 0$. This settles the asymptotic count of rational points on the genus zero modular curves $X_0(m)$. We leverage an explicit $\mathbb{Q}$-isomorphism between the stack $\mathscr{X}_0(5)$ and the generalized Fermat equation $x^2 + y^2 = z^4$ with $\mathbb{G}_m$-action of weights $(4, 4, 2)$.

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arXiv2022

Rational points on rank 2 genus 2 bielliptic curves in the LMFDB

Building on work of Balakrishnan, Dogra, and of the first author, we provide some improvements to the explicit quadratic Chabauty method to compute rational points on genus $2$ bielliptic curves over $\mathbb{Q}$, whose Jacobians have Mordell-Weil rank equal to $2$. We complement this with a precision analysis to guarantee correct outputs. Together with the Mordell-Weil sieve, this bielliptic quadratic Chabauty method is then the main tool that we use to compute the rational points on the $411$ locally solvable curves from the LMFDB which satisfy the aforementioned conditions.

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arXiv2022

Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings

We complete the computation of all $\mathbb{Q}$-rational points on all the $64$ maximal Atkin-Lehner quotients $X_0(N)^*$ such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty--Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method combined with the Mordell-Weil sieve. Additionally, for square-free levels $N$, we classify all $\mathbb{Q}$-rational points as cusps, CM points (including their CM field and $j$-invariants) and exceptional ones. We further indicate how to use this to compute the $\mathbb{Q}$-rational points on all of their modular coverings.

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