Overconvergent cohomology and quaternionic Darmon points

We develop the (co)homological tools that make effective the construction of the quaternionic Darmon points introduced by Matthew Greenberg. In addition, we use the overconvergent cohomology techniques of Pollack--Pollack to allow for the efficient calculation of such points. Finally, we provide the first numerical evidence supporting the conjectures on their rationality.Comment: Fixed some minor typos, added authors' affiliatio

Fields of definition of building blocks with quaternionic multiplication

This paper investigates the fields of definition up to isogeny of the abelian varieties called building blocks. In [Ri1] and $[\mathrm{Py}]$ a characterization of the fields of definition of these varieties together with their endomorphisms is given in terms of a Galois cohomology class canonically attached to them. However, when the building blocks have quaternionic multiplication, then the field of definition of the varieties can be strictly smaller than the field of definition of their endomorphisms. What we do is to give a characterization of the field of definition of the varieties in this case (also in terms of their associated Galois cohomology class), and we also make the computations that are needed in order to calculate in practice these fields from our characterization

Abelian varieties with many endomorphisms and their absolutely simple factors

We characterize the abelian varieties arising as absolutely simple factors of $\mathrm{GL}_2$-type varieties over a number field $k$. In order to obtain this result, we study a wider class of abelian varieties: the $k$ varieties $A / k$ satisfying that $\operatorname{End}_k^0(A)$ is a maximal subfield of $\operatorname{End}_{\bar{k}}^0(A)$. We call them Ribet-Pyle varieties over $k$. We see that every Ribet-Pyle variety over $k$ is isogenous over $\bar{k}$ to a power of an abelian $k$-variety and, conversely, that every abelian $k$-variety occurs as the absolutely simple factor of some Ribet-Pyle variety over $k$. We deduce from this correspondence a precise description of the absolutely simple factors of the varieties over $k$ of $\mathrm{GL}_2$-type

Avaluació epidemiològica dels efectes indesitjables greus dels analgèsics i antiinflamatoris no esteroïdals

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