A first course in Local arithmetic
Notes for a course at the H.-C. R. I., Allahabad, 15 August 2008 -- 26 January 2009
math.HO↗
arXiv subjects
Chandan Singh Dalawat
Publications and source records attributed to Chandan Singh Dalawat.
Wilson's theorem
We show that there are four possibilities for the product of all elements in the multiplicative group of a quotient of the ring of integers in a number field, and give precise conditions for each of the possibilities to occur. This generalisation of Wilson's theorem turns out to have been first discovered by M. Laššák (2000), but our proof is simpler and more direct.
math.NT↗
Congruent numbers, elliptic curves, and the passage from the local to the global
The ancient unsolved problem of congruent numbers has been reduced to one of the major questions of contemporary arithmetic: the finiteness of the number of curves over $\bf Q$ which become isomorphic at every place to a given curve. We give an elementary introduction to congruent numbers and their conjectural characterisation, discuss local-to-global issues leading to the finiteness problem, and list a few results and conjectures in the arithmetic theory of elliptic curves.
math.HO↗
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.
math.HO↗
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.
math.HO↗
The Tao of Mathematics, and Think Locally
An informal discussion of Serre's conjecture on the modularity of odd irreducible representations of Gal(\bar Q|Q) into GL_2(\bar F_p), using Ramanujan's tau-function as an illustrative example. Also, a word about the importance of thinking locally.
math.HO↗
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.
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Restriction, corestriction and the characteristic homomorphism for 0-cycles of degree 0
We show that the characteristic homomorphism of Colliot-Th{é}l{è}ne and Sansuc from the Chow group of 0-cycles of degree~0 is compatible with restriction and corestriction maps.
math.AG↗
The Chow group of a Ch{â}telet surface over a number field
We compute the Chow group of a Ch{â}telet surface over a dyadic field. Combined with the previous work of Bloch, Colliot-Th{é}l{è}ne, Coray, Ischebeck, Sansuc, Swinnerton-Dyer, and the author, this allows one to compute the Chow group of any Ch{â}telet surface over any number field.
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Groupe de Chow d'une surface rationnelle sur un corps local
We determine the Chow group of zero-cycles on a rational surface X defined over a finite extension K of the field of p-adic numbers (p a prime) when X is split by an unramified extension of K.
math.AG↗
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.
math.AG↗
Le groupe de Chow d'une surface de Ch{â}telet
Colliot-Th{é}l{è}ne has determined the Chow group of zero-cycles on a Ch{â}telet surface X defined over a finite extension K of the field of p-adic numbers (p an odd prime) when X is split by an unramified extension of K. Using similar techniques, we extend this calculation to the ramified case, simplifying his result in the process. We also determine how this group changes when we pass from K to a finite extension of K.
math.AG↗
Arithmetic on curves
A telegraphic survey of some of the standard results and conjectures about the set $C({\bf Q})$ of rational points on a smooth projective absolutely connected curve $C$ over ${\bf Q}$.
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