arXiv subjects

Tommaso Giorgio Centeleghe

Publications and source records attributed to Tommaso Giorgio Centeleghe.

6 recordsLinked to original sources

arXiv2022

Categories of abelian varieties over finite fields II: Abelian varieties over finite fields and Morita equivalence

The category of abelian varieties over $\mathbb{F}_q$ is shown to be anti-equivalent to a category of $\mathbb{Z}$-lattices that are modules for a non-commutative pro-ring of endomorphisms of a suitably chosen direct system of abelian varieties over $\mathbb{F}_q$. On full subcategories cut out by a finite set $w$ of conjugacy classes of Weil $q$-numbers, the anti-equivalence is represented by what we call $w$-locally projective abelian varieties.

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arXiv2017

Integral Frobenius for Abelian Varieties with Real Multiplication

In this paper we introduce the concept of $\mathit{integral}$ $\mathit{Frobenius}$ to formulate an integral analogue of the classical compatibility condition linking the collection of rational Tate modules $V_λ(A)$ arising from abelian varieties over number fields with real multiplication. Our main result gives a recipe for constructing an integral Frobenius when the real multiplication field has class number one. By exploiting algorithms already existing in the literature, we investigate this construction for three modular abelian surfaces over $\mathbf{Q}$.

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arXiv2015

Integral Tate modules and splitting of primes in torsion fields of elliptic curves

Let $E$ be an elliptic curve over a finite field $k$, and $\ell$ a prime number different from the characteristic of $k$. In this paper we consider the problem of finding the structure of the Tate module $T_\ell(E)$ as an integral Galois representations of $k$. We indicate an explicit procedure to solve this problem starting from the characteristic polynomial $f_E(x)$ and the $j$-invariant $j_E$ of $E$. Hilbert Class Polynomials of imaginary quadratic orders play here an important role. We give a global application to the study of prime-splitting in torsion fields of elliptic curves over number fields.

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arXiv2015

Categories of abelian varieties over finite fields I. Abelian varieties over $\mathbb{F}_p$

We assign functorially a $\mathbb{Z}$-lattice with semisimple Frobenius action to each abelian variety over $\mathbb{F}_p$. This establishes an equivalence of categories that describes abelian varieties over $\mathbb{F}_p$ avoiding $\sqrt{p}$ as an eigenvalue of Frobenius in terms of simple commutative algebra. The result extends the isomorphism classification of Waterhouse and Deligne's equivalence for ordinary abelian varieties.

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arXiv2011

On certain cusp forms on a definite quaternion algebra

If $D$ is the definite quaternion algebra over $\qu$ of discriminant $p$, we compute, for any prime $p>3$, the number of infinite dimensional cusp forms on $D^*$ which are trivial at infinity, tamely ramified at $p$, and have given conductor $N$ away from $p$. We include a detail explanation of a Deuring--type correspondence between supersingular elliptic curves in characteristic $p$ and a certain double coset arising from the adelic points of $D^*$.

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arXiv2010

Computing the number of certain Galois representations mod $p$

Using the link between mod $p$ Galois representations of $\qu$ and mod $p$ modular forms established by Serre's Conjecture, we compute, for every prime $p\leq 1999$, a lower bound for the number of isomorphism classes of continuous Galois representation of $\qu$ on a two--dimensional vector space over $\fbar$ which are irreducible, odd, and unramified outside $p$.

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