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Elvira Lupoian

Publications and source records attributed to Elvira Lupoian.

9 recordsLinked to original sources
arXiv2026

Reduction Types of Genus 2 Curves

Tate produced a table for elliptic curves over local fields that beautifully summarises their arithmetic invariants in terms of the Kodaira type. We present an analogous set of tables for curves of genus 2. The invariants addressed include: reduction type of the minimal regular model, reduction type of the minimal regular model with normal crossings, Néron component group of the Jacobian, conductor exponent, valuation of the discriminant of a minimal Weierstrass model, and cluster pictures.

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arXiv2026

Intersections of the Ekedahl-Oort and Newton Strata of $\mathcal{A}_{5}$

The moduli space $\mathcal{A}_g$ of principally polarised abelian varieties of dimension $g$, defined over an algebraically closed field of characteristic $p >0$, is studied through various stratifications. The two most prominent ones are the Newton stratification, based on the isogeny class of the $p$-divisible group of an abelian variety, and the Ekedahl-Oort stratification, defined by the isomorphism class of its $p$-torsion group scheme. In general, it is not known how the strata of these two intersect. In this paper we completely determine which of these intersections are non-empty in dimension five. As a consequence, we give an explicit description of the induced Ekedahl-Oort stratification on the supersingular locus $\mathcal{S}_{5}$.

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arXiv2025

Two-Torsion Subgroups of some Modular Jacobians

We give a practical method to compute the 2-torsion subgroup of the Jacobian of a non-hyperelliptic curve of genus $3$, $4$ or $5$. The method is based on the correspondence between the 2-torsion subgroup and the theta hyperplanes to the curve. The correspondence is used to explicitly write down a zero-dimensional scheme whose points correspond to elements of the $2$-torsion subgroup. Using $p$-adic or complex approximations (obtained via Hensel lifting or homotopy continuation and Newton-Raphson) and lattice reduction we are then able to determine the points of our zero-dimensional scheme and hence the $2$-torsion points. We demonstrate the practicality of our method by computing the $2$-torsion of the modular Jacobians $J_{0}\left( N \right)$ for $N = 42, 55, 63, 72, 75$. As a result of this we are able to verify the generalised Ogg conjecture for these values.

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arXiv2025

Three-Torsion Subgroups and Wild Conductor Exponents of Plane Quartics

In this paper we give an algorithm to find the 3-torsion subgroup of the Jacobian of a smooth plane quartic curve with a marked rational point. We describe $3-$torsion points in terms of cubics which triply intersect the curve, and use this to define a system of equations whose solution set corresponds to the coefficients of these cubics. We compute the points of this zero-dimensional, degree $728$ scheme first by approximation, using homotopy continuation and Newton-Raphson, and then using continued fractions to obtain accurate expressions for these points. We describe how the Galois structure of the field of definition of the $3$-torsion subgroup can be used to compute local wild conductor exponents, including at $p=2$.

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arXiv2025

Three-Torsion Subgroups and Wild Conductors of Genus 3 Hyperelliptic Curves

We give a practical method for computing the 3-torsion subgroup of the Jacobian of a genus 3 hyperelliptic curve. We define a scheme for the 3-torsion points of the Jacobian and use complex approximations, homotopy continuation and lattice reduction to find precise expression for the 3-torsion. In the latter stages of the paper, we explain how the 3-torsion subgroup can be used to compute the wild part of the local exponent of the conductor at 2.

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arXiv2025

Ceresa Cycles of $X_{0}(N)$

The Ceresa cycle is an algebraic 1-cycle on the Jacobian of an algebraic curve. Although it is homologically trivial, Ceresa famously proved that for a very general complex curve of genus at least 3, it is non-trivial in the Chow group. In this paper we study the Ceresa cycle attached to the complete modular curve $X_{0}(N)$ modulo rational equivalence. For prime level $p$ we give a complete description, namely we prove that if $X_{0}(p)$ is not hyperelliptic, then its Ceresa cycle is non-torsion. For general level $N$, we prove that there are finitely many $X_{0}(N)$ with torsion Ceresa cycle. Our method relies on the relationship between the vanishing of the Ceresa cycle and Chow-Heegner points on the Jacobian. We use the geometry and arithmetic of modular Jacobians to prove that such points are of infinite order and therefore deduce non-vanishing of the Ceresa cycle.

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arXiv2025

Modular Units on $X_{1}( p)$ and Quotients of the Cuspidal Group

Modular units are functions on modular curves whose divisors are supported on the cusps. They form a free abelian group of rank at most one less than the number of cusps. In this paper we study the group of modular units on $X_{1}( p )$, with prime level $p \ge 5$. We give an explicit basis for this group and study certain rational subgroups of it. We use the basis to numerically investigate the structure of the cuspidal group of $X_{1}( p)$ and its rational subgroup. In the later stages of this paper we use our basis to determine a specific large quotient of the cuspidal group.

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arXiv2023

Computing the Cuspidal Subgroup of the Modular Jacobian $J_{H}\left( p \right)$

For a fixed prime $p$ congruent to $1$ modulo $4$ we may define the modular curve $X_{H}\left( p \right)$ associated to the subgroup of non-zero squares modulo $p$. This curve has four cusps and we consider the subgroup of the Jacobian $J_{H}\left( p \right)$ of $X_{H}\left( p \right)$ generated by these points, which we will call the cuspidal subgroup of $J_{H}\left( p \right)$. This is a finite subgroup by the results of Manin and Drinfeld, and lies inside the $\mathbb{Q} \left( \sqrt{ p} \right)$-rational torsion subgroup. In this paper we compute the cuspidal subgroup for all such curves of genus $g$, $2 \leq g \leq 10$, namely those with $p \in \{ 29, 37, 41, 53, 61, 73 \}$, and compare this with $J_{H}\left( \mathbb{Q} \left( \sqrt{p} \right) \right)_{\text{tors}}$.

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