Samuele Anni
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View article: On the computation of endomorphism rings of abelian surfaces over finite fields
On the computation of endomorphism rings of abelian surfaces over finite fields Open
We study endomorphism rings of principally polarized abelian surfaces over finite fields from a computational viewpoint with a focus on exhaustiveness. In particular, we address the cases of non-ordinary and non-simple varieties. For each …
View article: On smooth plane models for modular curves of Shimura type
On smooth plane models for modular curves of Shimura type Open
In this paper we prove that there are finitely many modular curves that admit a smooth plane model. Moreover, if the degree of the model is greater than or equal to 19, no such curve exists. For modular curves of Shimura type we show that …
View article: Deep congruences + the Brauer-Nesbitt theorem
Deep congruences + the Brauer-Nesbitt theorem Open
We prove that mod-$p$ congruences between polynomials in $\mathbb{Z}_p[X]$ are equivalent to deeper $p$-power congruences between power-sum functions of their roots. This result generalizes to torsion-free $\mathbb{Z}_{(p)}$-algebras modul…
View article: On smooth plane models for modular curves of Shimura type
On smooth plane models for modular curves of Shimura type Open
In this paper we prove that there are finitely many modular curves that admit a smooth plane model. Moreover, if the degree of the model is greater than or equal to 19, no such curve exists. For modular curves of Shimura type we show that …
View article: Computing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ℒ</mml:mi></mml:math>-invariants via the Greenberg–Stevens formula
Computing -invariants via the Greenberg–Stevens formula Open
In this article, we describe how to compute slopes of -adic -invariants of Hecke eigenforms of arbitrary weight and level by means of the Greenberg–Stevens formula. Our method is based on the work of Lauder and Vonk on computing the revers…
View article: Constructing hyperelliptic curves with surjective Galois representations
Constructing hyperelliptic curves with surjective Galois representations Open
In this paper we show how to explicitly write down equations of hyperelliptic curves over such that for all odd primes the image of the mod Galois representation is the general symplectic group. The proof relies on understanding the act…
View article: Automorphic Forms and Related Topics
Automorphic Forms and Related Topics Open
The Langlands Programme predicts that a weight 2 newform f over a number eld K with integer Hecke eigenvalues generally should have an associated elliptic curve Ef over K. In [GMS14], we associated, building on works of Darmon [Dar01] and …
View article: Computing $\mathcal{L}$-invariants via the Greenberg-Stevens formula
Computing $\mathcal{L}$-invariants via the Greenberg-Stevens formula Open
In this article, we describe how to compute slopes of $p$-adic $\mathcal{L}$-invariants of arbitrary weight and level by means of the Greenberg-Stevens formula. Our method is based on work of Lauder and Vonk on computing the reverse charac…
View article: Modular elliptic curves over real abelian fields and the generalized Fermat equation x2ℓ+ y2m= zp
Modular elliptic curves over real abelian fields and the generalized Fermat equation x2ℓ+ y2m= zp Open
Using a combination of several powerful modularity theorems and class field\ntheory we derive a new modularity theorem for semistable elliptic curves over\ncertain real abelian fields. We deduce that if $K$ is a real abelian field of\ncond…
View article: Residual representations of semistable principally polarized abelian varieties
Residual representations of semistable principally polarized abelian varieties Open
Let $A$ be a semistable principally polarized abelian variety of dimension $d$ defined over the rationals. Let $\ell$ be a prime and let $\bar{\rho}_{A,\ell} : G_{\mathbb{Q}} \rightarrow \mathrm{GSp}_{2d}(\mathbb{F}_\ell)$ be the represent…
View article: A note on the minimal level of realization for a mod $\ell$ eigenvalue system
A note on the minimal level of realization for a mod $\ell$ eigenvalue system Open
In this article we give a criterion for a mod $\ell$ eigenvalue system attached to a mod $\ell$ Katz cuspform to arise from lower level or weight. Namely, we prove the following: the eigenvalue system associated to a ring homomorphism $f:\…
View article: On the generalized Fermat equation $x^{2\ell}+y^{2m}=z^p$ for $3 \le p \le 13$
On the generalized Fermat equation $x^{2\ell}+y^{2m}=z^p$ for $3 \le p \le 13$ Open
We show that the generalized Fermat equation $x^{2\ell}+y^{2 m}=z^p$ has no non-trivial primitive solutions for primes $\ell$, $m \ge 5$, and $3 \le p \le 13$. This is achieved by relating a putative solution to a Frey curve over a real su…