The Chow group of a del Pezzo surface over a local field

Detailed illustration of the method for calculating the Chow group of a rational surface over a local field [math.AG/0302157 (th.~4)], applied to a certain del Pezzo surface of degree~4. Involves the construction of a regular integral model and the determination of the specialisation map.Comment: 8 page

Some aspects of the functor K_2 of fields

A review of the connections between K_2 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.Comment: Final version. To appear in the Journal of the Ramanujan Mathematical Societ

Ribet's modular construction of unramified p-extensions of Q(\mu_p)

An expository account of Ribet's modular constructionComment: Notes for a series of lectures delivered at Gauhati, 22--30 September 200

Final remarks on local discriminants

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 root of 1; the case of other local fields with finite residue fields was treated earlier. In all cases, we compute the contribution of cyclic extensions to Serre's degree-$p$ mass formula.Comment: 14 pages; new section 6 on the contribution of cyclic extensions to Serre's degree-p mass formul

The ramification filtration in certain pp-extensions

We show that the recent result of Casta\~neda and Wu about the ramification filtration in certain $p$-extensions of function fields of prime characteristic $p$ is equally valid over local fields of mixed characteristic $(0,p)$. Apart from being applicable to both equicharacteristic and mixed characteristic cases, our method has the advantage of being purely local, purely conceptual, more natural, and much shorter.Comment: 5 pages, reorganised, filled in the details, and wrote a summar

Further remarks on local discriminants

Using Kummer theory for a finite extension K of \Qp(\zeta)(where p is a prime number and \zeta a primitive p-th root 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 \Fp((\pi)) and derive similar results for their elementary abelian p-extensions.Comment: 26 page

Primary units in cyclotomic fields

We investigate the interrelationships of three notions of primary units in the local cyclotomic field of $p$-th roots of~1($p$ being an odd prime number), especially with reference to global units.Comment: 7 pages, suppresed cor. 4 in v2, to appear in the Annales des sciences math\'ematiques du Qu\'ebe

Good reduction, bad reduction

We give some general properties of good and bad reduction, and some recent examples (worked out with Dipendra Prasad) of varieties having bad reduction not accounted for by their cohomology. We include some consequences of our remarks for varieties over number fields having good reduction everywhere.Comment: Text of a talk at Madras, August 200

The Chow group of a Ch{\^a}telet surface over a number field

We compute the Chow group of a Ch{\^a}telet surface over a dyadic field. Combined with the previous work of Bloch, Colliot-Th{\'e}l{\`e}ne, Coray, Ischebeck, Sansuc, Swinnerton-Dyer, and the author, this allows one to compute the Chow group of any Ch{\^a}telet surface over any number field.Comment: 8 page

Numbers and periods

A somewhat pretentious presentation of number systems (N, Z, Q, R, C, Q_p, >...). The problem of a p-adic characterisation of good-reduction p-adic curves is posed.Comment: Notes from a talk at Allahabad, February 200