arXiv subjects

Diana Mocanu

Publications and source records attributed to Diana Mocanu.

7 recordsLinked to original sources

arXiv2025

Revisiting the equation $x^2+y^3=z^p$

Let $E/\mathbb Q$ be an elliptic curve and $p \geq 3$ a prime. The modular curve $X_E^-(p)$ parameterizes elliptic curves with $p$-torsion modules anti-symplectically isomorphic to $E[p]$. The work of Freitas--Naskręcki--Stoll uses the modular method to show that all primitive non-trivial solutions of the Fermat-type equation $x^2 + y^3 = z^p$ give rise to rational points on $X_E^-(p)$ with $E \in \{27a1,54a1,96a1,288a1,864a1,864b1,864c1 \}$. Using a criterion classifying the existence of local points due to the first two authors, we show that, for $E$ any of the curves with conductor 864 and certain primes $p \equiv 19 \pmod{24}$, we have $X_E^-(p)(\mathbb Q_\ell) = \emptyset$. Furthermore, for each $E$ in the list and any $p$, we prove that either $X_E^-(p)$ can be discarded using the same criterion, or it cannot be discarded using purely local information.

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arXiv2025

Local points on twists of $X(p)$ with applications

Let $E/\mathbb Q$ be an elliptic curve and $p \geq 3$ a prime. The modular curve $X_E^-(p)$ parametrizes elliptic curves with $p$-torsion modules anti-symplectically isomorphic to $E[p]$. We give a complete classification of when $X_E^-(p)(\mathbb Q_\ell)$ is non-empty, for all primes $\ell\neq p$; our result also includes $\ell=p$ in most cases when $E$ is semistable at $p$. We give two different applications. First, we classify CM curves $E/\mathbb Q$ where the modular curve $X_E^-(p)$ is a counterexample to the Hasse principle for infinitely many $p$. Assuming the Frey--Mazur conjecture, we prove that for at least $60\%$ of rational elliptic curves $E$, the modular curve $X_E^-(p)$ is a counterexample to the Hasse principle for at least $50\%$ of primes $p$. Secondly, we introduce a new technique to the elimination stage of the modular method and apply it to show that $x^3+y^3=5^αz^p$ has no non-trivial primitive solutions for various primes $p$ satisfying $(α/p)=-1$. Moreover, as a by-product of our work, we simplify the assumptions of several local symplectic criteria due to the first author and Alain Kraus.

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arXiv2025

Weil polynomials of abelian varieties over finite fields

In this paper, we investigate Weil polynomials and their relationship with isogeny classes of abelian varieties over finite fields. We give a necessary condition for a degree 12 polynomial with integer coefficients to be a Weil polynomial. Moreover, we provide explicit criteria that determine when a Weil polynomial of degree 14 occurs as the characteristic polynomial of a Frobenius endomorphism acting on an abelian variety.

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arXiv2024

Conductor exponents for families of hyperelliptic curves

We compute the conductor exponents at odd places using the machinery of cluster pictures of curves for three infinite families of hyperelliptic curves. These are families of Frey hyperelliptic curves constructed by Kraus and Darmon in the study of the generalised Fermat equations of signatures $(r,r,p)$ and $(p,p,r)$, respectively. Here, $r$ is a fixed prime number and $p$ is a prime that is allowed to vary. In the context of the modular method, Billerey-Chen-Dieulefait-Freitas computed all conductor exponents for the signature $(r,r,p)$. We recover their computations at odd places, providing an alternative approach. In a similar setup, Chen-Koutsianas computed all conductor exponents for the signature $(p,p,5)$. We extend their work to the general case of signature $(p,p,r)$ at odd places. Our work can also be used to compute local arithmetic data for the curves in these families.

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arXiv2024

Non-trivial Integer Solutions of $x^r+y^r=Dz^p$

In this paper, we use the modular method over totally real fields together with some standard conjectures (the Weak Frey--Mazur Conjecture and the Eichler--Shimura Conjecture) to prove that infinitely many equations of the type $x^r+y^r=Dz^p$ do not have any non-trivial primitive integer solutions, where $r \geq 5$ is a fixed prime, whenever $p$ is large enough. For $r \equiv 3 \pmod 4$, we get the same result with only assuming the Weak Frey--Mazur Conjecture.

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arXiv2022

Asymptotic Fermat for signatures $(r,r,p)$ using the modular approach

Let $K$ be a totally real field, and $r\geq 5$ a fixed rational prime. In this paper, we use the modular method as presented in the recent work of Freitas and Siksek to study non-trivial, primitive solutions $(x,y,z) \in \mathcal{O}_K^3$ of the signature $(r,r,p)$ equation $x^r+y^r=z^p$ (where $p$ is a prime that varies). An adaptation of the modular method is needed, and we follow the recent work of Freitas which constructs Frey curves over totally real subfields of $K(ζ_r)$. When $K=\mathbb{Q}$ we get that there are no non-trivial, primitive integer solutions $(x,y,z)$ with $2|z$ for signatures $(r,r,p)$ when $r \in \{5,7,11,13,19,23, 37,47,53,59,61,67,71,79,83,101,103,107,131,139,149\}$ and $p$ is sufficiently large. Similar results hold for quadratic fields, for example when $K=\mathbb{Q}(\sqrt{2})$ there are no non-trivial, primitive solutions $(x,y,z)\in \mathcal{O}_K^3$ with $\sqrt{2}|z$ for signatures $(5,5,p),(7,7,p)$, $(11,11,p),(13,13,p)$ and sufficiently large $p$.

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arXiv2022

Asymptotic Fermat for signatures $(p,p,2)$ and $(p,p,3)$ over totally real fields

Let $K$ be a totally real number field and consider a Fermat-type equation $Aa^p+Bb^q=Cc^r$ over $K$. We call the triple of exponents $(p,q,r)$ the signature of the equation. We prove various results concerning the solutions to the Fermat equation with signature $(p,p,2)$ and $(p,p,3)$ using a method involving modularity, level lowering and image of inertia comparison. These generalize and extend the recent work of Işik, Kara and Ozman. For example, consider $K$ a totally real field of degree $n$ with $2 \nmid h_K^+$ and $2$ inert. Moreover, suppose there is a prime $q\geq 5$ which totally ramifies in $K$ and satisfies $\gcd(n,q-1)=1$, then we know that the equation $a^p+b^p=c^2$ has no primitive, non-trivial solutions $(a,b,c) \in \mathcal{O}_K^3$ with $2 | b$ for $p$ sufficiently large.

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