The Ostrowski quotient of an elliptic curve

Abstract

For [Formula: see text] a finite Galois extension of number fields, the relative Pólya group [Formula: see text] is the subgroup of the ideal class group of [Formula: see text] generated by all the strongly ambiguous ideal classes in [Formula: see text]. The notion of Ostrowski quotient [Formula: see text], as the cokernel of the capitulation map into [Formula: see text], has been recently introduced in [E. Shahoseini, A. Rajaei and A. Maarefparvar, Ostrowski quotients for finite extensions of number fields, Pacific J. Math. 321 (2022) 415–429, doi: 10.2140/pjm.2022.321.415]. In this paper, using some results of [C. D. González-Avilés, Capitulation, ambiguous classes and the cohomology of the units, J. Reine Angew. Math. 613 (2007) 75–97], we find a new approach to define [Formula: see text] and [Formula: see text] which is the main motivation for us to investigate analogous notions in the elliptic curve setting. For [Formula: see text] an elliptic curve defined over [Formula: see text], we define the Ostrowski quotient [Formula: see text] and the coarse Ostrowski quotient [Formula: see text] of [Formula: see text] relative to [Formula: see text], for which in the latter group we do not take into account primes of bad reduction. Our main result is a nontrivial structure theorem for the group [Formula: see text] and we analyze this theorem, in some details, for the class of curves [Formula: see text] over quadratic extensions [Formula: see text].