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Local Discriminants, Kummerian Extensions, and Elliptic Curves
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Starting from Stickelberger’s congruence for the absolute discriminant of a number field, we ask a series of natural questions which ultimately lead to an orthogonality relation for the ramification filtration on K(p√ K×), where K is any finite extension of Qp containing a primitive p-th ischolar_main of 1. An extensive historical survey of discriminants and primary numbers is included. Among other things, we give a direct proof of Serre’s mass formula in the case of quadratic extensions. Incidentally, it is shown that every unit in a local field is the discriminant of some elliptic curve.
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