Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name

Computer Science


Theoretical Statistics and Mathematics Unit (TSMU-Bangalore)


Kumar, Manish (TSMU-Bangalore; ISI)

Abstract (Summary of the Work)

This thesis concerns problems related to the ramification behaviour of the branched Galois covers of smooth projective connected curves defined over an algebraically closed field of positive characteristic. Our first main problem is the Inertia Conjecture proposed by Abhyankar in 2001. We will show several new evidence for this conjecture. We also formulate a certain generalization of it which is our second problem, and we provide evidence for it. We give a brief overview of these problems in this introduction and reserve the details for Chapter 4.Let k be an algebraically closed field, and U be a smooth connected affine k-curve. Let U ⊂ X be the smooth projective completion. An interesting and challenging problem is to understand the étale fundamental group π1(U). We only consider this as a profinite group up to isomorphism, and so the base point is ignored. When k has characteristic 0, it is well known that this group is the profinite completion of the topological fundamental group. In particular, it is a free profinite group, topologically generated by 2g + r − 1 elements where g is the genus of X, and r is the number of points in X − U. But when k has prime characteristic p > 0, these statements are no longer true. The full structure of π1(U) is not known in this case. Now onward, assume that k has characteristic p > 0. By the definition of π1(U), the set πA(U) of isomorphic classes of finite (continuous) group quotients of π1(U) is in bijective correspondence with the finite Galois étale covers of U. For a finite group G, let p(G) denote the subgroup of G generated by all its Sylow p-subgroups. In 1957 Abhyankar conjectured on what groups can occur in the set πA(U).


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Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.


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