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R. Parimala

Publications and source records attributed to R. Parimala.

At least 19 recordsLinked to original sources
arXiv2024

Rost injectivity for classical groups over function fields of curves over local fields

Let F be a complete discretely valued field with residue field a global field or a local field with no real orderings. Let G be an absolutely simple simply connected group of outer type A_n. If 2 and the index of the underlying algebra of G are coprime to the characteristic of the residue field of F, then we prove that the Rost invariant map from the first Galois cohomology set of G to the degree three Galois cohomology group is injective. Let L be the function field of a curve over a local field K and G an absolutely simple simply connected linear algebraic group over L of classical type. Suppose that the characteristic of the residue field of K is a good prime for G. As a consequence of our result and some known results we conclude that the Rost invariant of G is injective.

math.NT↗
arXiv2022

Period -Index problem for hyperelliptic curves

Let $C$ be a smooth projective curve of genus 2 over a number field $k$ with a rational point. We prove that the index and exponent coincide for elements in the 2-torsion of $\Sha(Br(C))$. In the appendix, an isomorphism of the moduli space of rank 2 stable vector bundles with odd determinant on a smooth projective hyperelliptic curve $C$ of genus $g$ with a rational point over any field of characteristic not two with the Grassmannian of $(g-1)$-dimensional linear subspaces in the base locus of a certain pencil of quadrics is established, making a result of (\cite{De-Ra}) rational. We establish a twisted version of this isomorphism and we derive as a consequence a weak Hasse principle for the smooth intersection $X$ of two quadrics in ${\mathbb P}^5$ over a number field: if $X$ contains a line locally, then $X$ has a $k$-rational point.

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arXiv2021

Local-global principles for constant reductive groups over semi-global fields

We study local-global principles for torsors under reductive linear algebraic groups over semi-global fields; i.e., over one variable function fields over complete discretely valued fields. We provide conditions on the group and the semiglobal field under which the local-global principle holds, and we compute the obstruction to the local-global principle in certain classes of examples. Using our description of the obstruction, we give the first example of a semisimple simply connected group over a semi-global field where the local-global principle fails. Our methods include patching and R-equivalence.

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arXiv2020

Local-Global Principle for Unitary Groups Over Function Fields of p-adic Curves

Let K be a p-adic field and F the function field of a curve over K. Let G be a connected linear algebraic group over F of classical type. Suppose the prime p is a good prime for G. Then we prove that projective homogeneous spaces under G over F satisfy a local global principle for rational points with respect to discrete valuations of F . If G is a semisimple simply connected group over F , then we also prove that principal homogeneous spaces under G over F satisfy a local global principle for rational points with respect to discrete valuations of F.

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arXiv2019

Local-global principles for tori over arithmetic curves

In this paper we study local-global principles for tori over semi-global fields, which are one variable function fields over complete discretely valued fields. In particular, we show that for principal homogeneous spaces for tori over the underlying discrete valuation ring, the obstruction to a local-global principle with respect to discrete valuations can be computed using methods coming from patching. We give a sufficient condition for the vanishing of the obstruction, as well as examples were the obstruction is nontrivial or even infinite. A major tool is the notion of a flasque resolution of a torus.

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arXiv2017

Local-global Galois theory of arithmetic function fields

We study the relationship between the local and global Galois theory of function fields over a complete discretely valued field. We give necessary and sufficient conditions for local separable extensions to descend to global extensions, and for the local absolute Galois group to inject into the global absolute Galois group. As an application we obtain a local-global principle for the index of a variety over such a function field. In this context we also study algebraic versions of van Kampen's theorem, describing the global absolute Galois group as a pushout of local absolute Galois groups.

math.RA↗
arXiv2017

Local-Global Principles for Zero-Cycles on Homogeneous Spaces over Arithmetic Function Fields

We study the existence of zero-cycles of degree one on varieties that are defined over a function field of a curve over a complete discretely valued field. In particular, we show that local-global principles hold for such zero-cycles provided that local-global principles hold for the existence of rational points over extensions of the function field. This assertion is analogous to a known result concerning varieties over number fields. We also show that our results hold more generally in the henselian case.

math.AG↗
arXiv2016

Local-global principle for reduced norms over function fields of p-adic curves

Let F be a function field in one variable over a p-adic field and D a central division algebra over F of degree n coprime to p. We prove that Suslin invariant detects whether an element in F is a reduced norm. This leads to a local-global principle for reduced norms with respect to all discrete valuations of F.

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arXiv2014

Degree three cohomology of function fields of surfaces

Let F be a finite field and l a prime not equal to the characteristic of F. Let K be the function field of a surface over F. Assume that K contains a primitive lth root of unity. In the paper we prove a certain local-global principle for elements of H^3(K, μ_l) in terms of symbols in H^2(K, μ_l) with respect to discrete valuations of K. We also show that this local global principle is equivalent to the vanishing of certain unramified cohomology groups of 3-folds over finite fields. Using this local-global principle we show that every element in H^3(F, μ_l) is a symbol. The vanishing of the unramified cohomology groups has consequences in the study of integral Tate conjecture and Brauer-Manin obstruction for existence of zero-cycles.

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arXiv2014

On the u-Invariant of Function Fields of Curves Over Complete Discretely Valued Fields

Let K be a complete discretely valued field with residue field k. If char(K) = 0, char(k) = 2 and the 2-rank of k is d, we prove that there exists an integer N depending on d such that the u-invariant of any function field in one variable over K is bounded by N. The method of proof is via introducing the notion of uniform boundedness for the p-torsion of the Brauer group of a field and relating the uniform boundedness of the 2-torsion of the Brauer group to finiteness of the u-invariant. We prove that the 2-torsion of the Brauer group of function fields in one variable over K are uniformly bounded.

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arXiv2012

Hasse principle for G-trace forms

Let k be a global field of characteristic not 2. We prove a local-global principle for the existence of self-dual normal bases, and more generally for the isomorphism of G-trace forms, of G-Galois algebras over k.

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arXiv2012

Colliot-Th\'el\`ene's conjecture and finiteness of u-invariants

We show that Colliot-Th\'el\`ene's conjecture on 0-cycles of degree 1 implies finiteness for the u-invariant of the function field of a curve over a totally imaginary number field and period-index bounds for the Brauer groups of arbitrary fields of transcendence degree 1 over the rational numbers.

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arXiv2012

Quadric surface bundles over surfaces

Let T -> S be a finite flat morphism of degree two between regular integral schemes of dimension at most two (and with 2 invertible), having regular branch divisor D. We establish a bijection between Azumaya quaternion algebras on T and quadric surface bundles over S with simple degeneration along D. This is a manifestation of the exceptional isomorphism A_1^2 = D_2 degenerating to the exceptional isomorphism A_1 = B_1. In one direction, the even Clifford algebra yields the map. In the other direction, we show that the classical algebra norm functor can be uniquely extended over the discriminant divisor. Along the way, we study the orthogonal group schemes, which are smooth yet nonreductive, of quadratic forms with simple degeneration. Finally, we provide two surprising applications: constructing counter-examples to the local-global principle for isotropy, with respect to discrete valuations, of quadratic forms over surfaces; and a new proof of the global Torelli theorem for very general cubic fourfolds containing a plane.

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arXiv2010

The Kneser-Tits conjecture for groups with Tits-index E_{8,2}^{66} over an arbitrary field

We prove: (1) The group of multipliers of similitudes of a 12-dimensional anisotropic quadratic form over a field K with trivial discriminant and split Clifford invariant is generated by norms from quadratic extensions E/K such that q_E is hyperbolic. (2) If G is the group of K-rational points of an absolutely simple algebraic group whose Tits index is E_{8,2}^{66}, then G is generated by its root groups, as predicted by the Kneser-Tits conjecture.

math.GR↗
arXiv2008

On the Pfister Number of Quadratic Forms

The generic quadratic form of even dimension n with trivial discriminant over an arbitrary field of characteristic different from 2 containing a square root of -1 can be written in the Witt ring as a sum of 2-fold Pfister forms using n-2 terms and not less. The number of 2-fold Pfister forms needed to express a quadratic form of dimension 6 with trivial discriminant is determined in various cases.

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