arXiv subjects
Publications and source records attributed to Chandan Singh Dalawat.
This update to my article on Congruent numbers, elliptic curves, and the passage from the local to the global, which appeared in Resonance, December 2009, pp. 1183--1205 (https://www.ias.ac.in/describe/article/reso/014/12/1183-1205) and was posted here as arXiv:0704.3783, covers a few recent advances in the arithmetic of elliptic curves with special reference to the congruent number problem.
A finite separable extension of a field is called primitive if there are no intermediate extensions. The most interesting primitive extensions of a local field with finite residue field are the wildly ramified ones, and our aim here is to parametrise them in a canonical manner.
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.
Let $p$ be a prime, $k$ a finite extension of $\mathbf{F}_p$ of cardinal $q$, $l$ a finite extension of $k$ of group $Σ=\mathrm{Gal}(l|k)$, and $T$ a subgroup of $l^\times$. Using the method of "little groups", we classify irreducible $\mathbf{F}_p$-representations of the group $G=T\times_qΣ$, the twisted product of $Σ$ with the $Σ$-module $T$. We then use these results to classify irreducible continuous $\mathbf{F}_p$-representations of the profinite group of automorphisms $\mathrm{Gal}(\tilde K|K)$ of the maximal galoisian extension $\tilde K$ of a local field $K$ with residue field $k$.
Let $p$ be a prime number, let $K$ be a $p$-field (a local field with finite residue field of characteristic $p$), let $L$ be a finite galoisian tamely ramified extension of $K$, and let $G=\mathrm{Gal}(L|K)$. Suppose that $L$ is split over $K$ in the sense that the short exact sequence $1\to T\to G\to G/T\to1$ has a section, where $T$ is the inertia subgroup of $G$. We determine the structure of the $\mathbf{F}_p[G]$-module $L^\times\!/L^{\times p}$ in characteristic $0$ when the $p$-torsion subgroup ${}_pL^\times$ of $L^\times$ has order $p$, and of the $\mathbf{F}_p[G]$-modules $L^\times\!/L^{\times p}$ and $L^+\!/\wp(L^+)$ in characteristic $p$, where $\wp(x)=x^p-x$. Let $\tilde K$ be a maximal galoisian extension of $K$, let $V$ be the maximal tamely ramified extension of $K$ in $\tilde K$, let $Γ=\mathrm{Gal}(V|K)$, and let $B$ be the maximal abelian extension of exponent $p$ of $V$ in $\tilde K$. We determine the structure of the $\mathbf{F}_p[[Γ]]$-module $\mathrm{Gal}(B|V)$, and show how this leads in characteristic $0$ to a simple proof of the fact that the profinite group $\mathrm{Gal}(\tilde K|K)$ is generated by $[K:\mathbf{Q}_p]+3$ elements.
A finite separable extension $E$ of a field $F$ is called primitive if there are no intermediate extensions. It is called solvable if the group $\mathrm{Gal}(\hat E|F)$ of automorphisms of its galoisian closure $\hat E$ over $F$ is solvable, and a $p$-extension ($p$ prime) if the degree $[E:F]$ is a power of $p$. We show that a solvable primitive $p$-extension $E$ of $F$ is uniquely determined (up to $F$-isomorphism) by $\hat E$ and characterise the extensions $D$ of $F$ such that $D=\hat E$ for some solvable primitive $p$-extension $E$ of $F$.
Let $p$ be a prime and let $K$ be a finite extension of the field ${\bf Q}_p$ of $p$-adic numbers such that the group ${}_pK^\times$ has order $p$. The ${\bf F}_p$-space $K^\times\!/K^{\times p}$ carries a natural filtration coming from the valuation on $K$, and a natural bilinear pairing coming from the reciprocity isomorphism for the exponent~$p$. We determine the orthogonal filtration for this pairing. We also prove the analogous result for $p$-fields of characteristic $p$.
We show that the recent result of Castañeda 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.
Rousseau's simple proof of the quadratic reciprocity law, followed by the proof of its equivalence with Hilbert's product formula. The Hilbert symbol is explained in terms of the reciprocity isomorphism, and the places of Q are determined.
Using the quadratic reciprocity law as the motivating example, we convey an understanding of classical reciprocity laws.
For a local field with finite residue field of characteristique p, we give some refinements of Serre's mass formula in degree p which allow us to compute for example the contribution of cyclic extensions, or of those whose galoisian closure has a given group as group of automorphisms, or has ramification properties given in advance.
For a local field F with finite residue field of characteristic p, we describe completely the structure of the filtered F_p[G]-module K^*/K^*p in characteristic 0 and $K^+/\wp(K^+) in characteristic p, where K=F(\root{p-1}\of F^*) and G=\Gal(K|F). As an application, we give an elementary proof of Serre's mass formula in degree p. We also determine the compositum C of all degree p separable extensions with solvable galoisian closure over an arbitrary base field, and show that C is K(\root p\of K^*) or K(\wp^{-1}(K)) respectively, in the case of the local field F. Our method allows us to compute the contribution of each character G\to\F_p^* to the degree p mass formula, and, for any given group Γ, the contribution of those degree p separable extensions of F whose galoisian closure has group Γ.
We give an elementary introduction, through illustrative examples but without proofs, to one of the basic consequences of the Langlands programme, namely the law governing the primes modulo which a given irreducible integral polynomial splits completely. Some recent results, such as the modularity of elliptic curves over the rationals, or the proof of Serre's conjecture by Khare and Wintenberger, are also illustrated through examples.
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.
An expository account of Ribet's modular construction
Using Kummer theory for a finite extension K of \Qp(ζ)(where p is a prime number and ζa 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((π)) and derive similar results for their elementary abelian p-extensions.
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.
Some thoughts on the congruence D=0,1(mod 4) for the absolute discriminant D of a number field