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V. Suresh

Publications and source records attributed to V. Suresh.

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↗
arXiv2024

Local-global principle for groups of type An over semi global fields

Let F be the function field of a curve over a complete discretely valued field K. Let G be a semisimple simply connected linear algebraic group over F of type An. We give a description of the obstruction to local global principle for principal homogeneous spaces under G over F with respect to discrete valuations of F in terms of R-equivalence classes of G over some suitable over fields. Using this description we prove that this obstruction vanishes under some conditions on the residue field K.

math.AG↗
arXiv2023

Failure of the local-global principle for isotropy of quadratic forms over function fields

We prove the failure of the local-global principle, with respect to discrete valuations, for isotropy of quadratic forms over function fields of transcendence degree at least 2 over algebraically closed fields. Our construction involves generalized Kummer varieties as well as a new nontriviality result for the unramified cohomology of products of elliptic curves over discretely valued fields, which can be viewed as an arithmetic version of a theorem of Gabber.

math.AG↗
arXiv2023

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.

math.AG↗
arXiv2023

Optimized Machine Learning for CHD Detection using 3D CNN-based Segmentation, Transfer Learning and Adagrad Optimization

Globally, Coronary Heart Disease (CHD) is one of the main causes of death. Early detection of CHD can improve patient outcomes and reduce mortality rates. We propose a novel framework for predicting the presence of CHD using a combination of machine learning and image processing techniques. The framework comprises various phases, including analyzing the data, feature selection using ReliefF, 3D CNN-based segmentation, feature extraction by means of transfer learning, feature fusion as well as classification, and Adagrad optimization. The first step of the proposed framework involves analyzing the data to identify patterns and correlations that may be indicative of CHD. Next, ReliefF feature selection is applied to decide on the most relevant features from the sample images. The 3D CNN-based segmentation technique is then used to segment the optic disc and macula, which are important regions for CHD diagnosis. Feature extraction using transfer learning is performed to extract features from the segmented regions of interest. The extracted features are then fused using a feature fusion technique, and a classifier is trained to predict the presence of CHD. Finally, Adagrad optimization is used to optimize the performance of the classifier. Our framework is evaluated on a dataset of sample images collected from patients with and without CHD. The results show that the anticipated framework accomplishes elevated accuracy in predicting the presence of CHD. either a particular user with a reasonable degree of accuracy compared to the previously employed classifiers like SVM, etc.

cs.CV↗
arXiv2021

Local triviality for G-torsors

Let C $\rightarrow$ Spec(R) be a relative proper flat curve over an henselian base. Let G be a reductive C-group scheme. Under mild technical assumptions, we show that a G-torsor over C which is trivial on the closed fiber of C is locally trivial for the Zariski topology.

math.AG↗
arXiv2020

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.

math.AG↗
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.

math.NT↗
arXiv2018

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↗
arXiv2018

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.

math.NT↗
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.

math.NT↗
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.

math.RA↗
arXiv2013

Colliot-Thélène's conjecture and finiteness of u-invariants

We show that Colliot-Thélène'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.

math.AG↗
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.

math.AG↗
arXiv2012

Admissibility of groups over function fields of p-adic curves

Let K be a field and G a finite group. The question of 'admissibility' of G over K was originally posed by Schacher, who gave partial results in the case K = Q. In this paper, we give necessary conditions for admissibility of a finite group G over function fields of curves over complete discretely valued fields. Using this criterion, we give an example of a finite group which is not admissible over Qp(t). We also prove a certain Hasse principle for division algebras over such fields.

math.RA↗