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| Paper IPM / M / 17606 | ||||||||||||||||
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For K/F a finite Galois extension of number fields, the relative P\'olya group \Po(K/F) is the subgroup of the ideal class group of K generated by all the strongly ambiguous ideal classes in K/F. Recently, the notion of Ostrowski quotient \Ost(K/F), as the cokernel of the capitulation map into \Po(K/F), has been introduced.
In this paper, using some results of Gonz\'alez-Avil\'es, we find a new approach to define \Po(K/F) and \Ost(K/F) which is the main motivation for us to investigate analogous notions in the elliptic curve setting. For E an elliptic curve defined over F, we define the Ostrowski quotient \Ost(E,K/F) and the coarse Ostrowski quotient \Ostc(E,K/F) of E relative to K/F, for which in the latter group we do not take into account primes of bad reduction. Our main result is a non-trivial structure theorem for the group \Ostc(E,K/F) and we analyze this theorem, in some details, for the class of curves E over quadratic extensions K/F.
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