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For a commutative algebraic group $G$ over a perfect field $k$, Ribet defined the set of almost rational torsion points $G_{tors,k}^{ar}$ of $G$ over $k$. For positive integers $d$, $g,$ we show there is an integer $U_{d,g}$ such that for all tori $T$ of dimension at most $d$ over number fields of degree at most $g$, $T_{tors,k}^{ar}\subseteq T\left[U_{d,g}\right]$. We show the corresponding result for abelian varieties with complex multiplication, and under an additional hypothesis, for elliptic curves without complex multiplication. Finally, we show that except for an explicit finite...

The purpose of this paper is to generalize, to certain commutative formal groups of dimension one and height greater than one defined over the ring of integers of a finite extension of ${\mathbf{Q}}_p$, some results on $p$-adic interpolation developed by Kubota, Leopoldt, Iwasawa, Mazur, Katz and others notably for the multiplicative group ${\widehat{\mathbf{G}}}_m$, and which they used to construct $p$-adic $L$-functions.

In this paper we apply the results of our previous article on the $p$-adic interpolation of logarithmic derivatives of formal groups to the construction of $p$-adic $L$-functions attached to certain elliptic curves with complex multiplication. Our results are primarily concerned with curves with supersingular reduction.