Journal of the Ramanujan Mathematical Society

Chandan Singh Dalawat ¹, Jung-Jo Lee ²

Affiliations
1 Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, IN
2 Department of Mathematics, Seoul National University, Shillim-dong, Gwanak-gu, Seoul 151-742, KP

Abstract

We give an intrinsic parametrisation of the set of tamely ramified extensions of a local field with finite residue field and bring to the fore the role played by group cohomology. We show that two natural definitions of the cohomology class of a tamely ramified finite galoisian extension coincide, and can be recovered from the parameter. We also give an elementary proof of Serre’s mass formula in the tame case and in the simplest wild case, and we classify tame galoisian extensions of degree the cube of a prime.

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Chandan Singh Dalawat ¹

Affiliations
1 Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad-211019, IN

Abstract

Using Kummer theory for a finite extension K of Q_p(ζ) (where p is a prime number and ζ a primitive p-th ischolar_main of 1), we compute the ramification filtration and the discriminant of an arbitrary elementary abelian p-extension of K. We also develop the analogous Artin-Schreier theory for finite extensions of F_p((π)), where π is transcendental, and derive similar results for their elementary abelian p-extensions.

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Chandan Singh Dalawat ¹

Affiliations
1 Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad-211019, IN

Abstract

We show how the ramification filtration on the maximal elementary abelian p-extension (p prime) on a local number field of residual characteristic p can be derived using only Kummer theory and a certain orthogonality relation for the Kummer pairing, even in the absence of a primitive p-th ischolar_main of 1; the case of other local fields was treated earlier. In all cases, we compute the contribution of cyclic extensions to Serre’s degree-p mass formula.

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Chandan Singh Dalawat ¹

Affiliations
1 Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad-211019, IN

Abstract

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 Q_p 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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Chandan Singh Dalawat ¹

Affiliations
1 Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad-211019, IN

Abstract

We review connections between the group K₂ of a field and universal central extensions, quadratic forms, central simple algebras, differential forms, abelian extensions, abelian coverings, explicit reciprocity laws, special values of zeta functions, and special values of L-functions. No proofs, minimal bibliography.

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