arXiv subjects
Publications and source records attributed to Gabriele Ranieri.
We give upper and lower bounds on the Chevalley-Bass number of a field of characteristic zero, whenever this quantity is well-defined. We also describe an algorithm which computes the Chevalley-Bass number of a field, provided its maximal abelian subextension is known. As a primary application, we improve the value of a constant related to exponential diophantine equations.
We give a positive answer to a Conjecture by Manjul Bhargava, Daniel M. Kane, Hendrik W. Lenstra Jr., Bjorn Poonen and Eric Rains, concerning the cohomology of torsion subgroups of elliptic curves over global fields. This implies that, given a global field $k$ and an integer $n$, for $100\%$ of elliptic curves $E$ defined over $k$, the $n$-th Selmer group of $E$ is the intersection of two direct summands of the adelic cohomology group $H^1(\mathbf{A},E[n])$. We also give examples of elliptic curves for which the conclusion of this conjecture does not hold.
We partially answer to a question of Vidaux and Videla by constructing an infinite family of rings of algebraic integers of totally real subfields of Q whose Julia Robinson's Number is distinct from 4 and +$\infty$. Moreover the set of the Julia Robinson's Number that we construct is unbounded.
Let $p \geq 5$ be a prime number. We find all the possible subgroups $G$ of ${\rm GL}_2 ( \mathbb{Z} / p \mathbb{Z} )$ such that there exists a number field $k$ and an elliptic curve ${\mathcal{E}}$ defined over $k$ such that the ${\rm Gal} ( k ( {\mathcal{E}}[p] ) / k )$-module ${\mathcal{E}}[p]$ is isomorphic to the $G$-module $( \mathbb{Z} / p \mathbb{Z} )^2$ and there exists $n \in \mathbb{N}$ such that the local-global divisibility by $p^n$ does not hold over ${\mathcal{E}} ( k )$.
Let $k$ be a number field and let ${\mathcal{A}}$ be a ${\rm GL}_2$-type variety defined over $k$ of dimension $d$. We show that for every prime number $p$ satisfying certain conditions (see Theorem 2), if the local-global divisibility principle by a power of $p$ does not hold for ${\mathcal{A}}$ over $k$, then there exists a cyclic extension $\widetilde{k}$ of $k$ of degree bounded by a constant depending on $d$ such that ${\mathcal{A}}$ is $\widetilde{k}$-isogenous to a ${\rm GL}_2$-type variety defined over $\widetilde{k}$ that admits a $\widetilde{k}$-rational point of order $p$. Moreover, we explain how our result is related to a question of Cassels on the divisibility of the Tate-Shafarevich group, studied by Ciperiani and Stix and Creutz.
Let $p \geq 2$ be a prime number and let $k$ be a number field. Let $\mathcal{A}$ be an abelian variety defined over $k$. We prove that if ${\rm Gal} ( k ( {\mathcal{A}}[p] ) / k )$ contains an element $g$ of order dividing $p-1$ not fixing any non-trivial element of ${\mathcal{A}}[p]$ and $H^1 ( {\rm Gal} ( k ( {\mathcal{A}}[p] ) / k ), {\mathcal{A}}[p] )$ is trivial, then the local-global divisibility by $p^n$ holds for ${\mathcal{A}} ( k )$ for every $n \in \mathbb{N}$. Moreover, we prove a similar result without the hypothesis on the triviality of $H^1 ( {\rm Gal} ( k ( {\mathcal{A}}[p] ) / k ) , {\mathcal{A}}[p] )$, in the particular case where ${\mathcal{A}}$ is a principally polarized abelian variety. Then, we get a more precise result in the case when ${\mathcal{A}}$ has dimension $2$. Finally we show with a counterexample that the hypothesis over the order of $g$ is necessary. In the Appendix, we explain how our results are related to a question of Cassels on the divisibility of the Tate-Shafarevich group, studied by Ciperani and Stix.
Let $p$ be a prime number and let $k$ be a number field. Let $E$ be an elliptic curve defined over $k$. We prove that if $p$ is odd, then the local-global divisibility by any power of $p$ holds for the torsion points of $E$. We also show with an example that the hypothesis over $p$ is necessary. We get a weak generalization of the result on elliptic curves to the larger family of ${\rm GL}_2$-type varieties over $k$. In the special case of the abelian surfaces $A/k$ with quaternionic multiplication over $k$ we obtain that for all prime $p$, except a finite number depending on $A$, the local-global divisibility by any power of $p$ holds for the torsion points of $A$
Let $p$ be a prime number and let $ k $ be a number field, which does not contain the field $\mathbb{Q} (ζ_p + \bar{ζ_p})$. Let $\mathcal{E}$ be an elliptic curve defined over $k$. We prove that if there are no $k$-rational torsion points of exact order $p$ on $\E$, then the local-global principle holds for divisibility by $p^n$, with $n$ a natural number. As a consequence of the deep theorem of Merel, for $p$ larger than a constant depending only on the degree of $k$, there are no counterexamples to the local-global divisibility principle. Nice and deep works give explicit small constants for elliptic curves defined over a number field of degree at most 5 over $\mathbb{Q}.
We prove that only for powers of 2 and 3 could occur counterexamples to the local-global divisibility principle for elliptic curves defined over the rationals. For we refine our previous criterion for the validity of the principle. We also give an example that shows that the assumptions of our criterion are necessary.
Let $ p $ be a prime lager than 3. Let $k$ be a number field, which does not contain the subfield of $\mathbb{Q} (ζ_{p^2})$ of degree $p$ over $\mathbb{Q}$. Suppose that $\mathcal{E}$ is an elliptic curve defined over $k$. We prove that the existence of a counterexample to the local-global divisibility by $p^2$ in $\mathcal{E}$, assures the existence of a $k$-rational point of exact order $p$ in $\mathcal{E}$. Using the Merel Theorem, we then shrunk the known set of primes for which there could be a counterexample to the local-global divisibility by $p^2$.
Inspired by Warren Sinnott 's method we prove a linear independence result modulo p for the Iwasawa power series associated to Kubota-Leopoldt p-adic L-functions.