#### Date of Submission

9-22-2020

#### Date of Award

9-22-2021

#### Institute Name (Publisher)

Indian Statistical Institute

#### Document Type

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

#### Subject Name

Computer Science

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Bangalore)

#### Supervisor

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 âŠ‚ X be the smooth projective completion. An interesting and challenging problem is to understand the Ã©tale fundamental group Ï€1(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 âˆ’ 1 elements where g is the genus of X, and r is the number of points in X âˆ’ U. But when k has prime characteristic p > 0, these statements are no longer true. The full structure of Ï€1(U) is not known in this case. Now onward, assume that k has characteristic p > 0. By the definition of Ï€1(U), the set Ï€A(U) of isomorphic classes of finite (continuous) group quotients of Ï€1(U) is in bijective correspondence with the finite Galois Ã©tale 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 Ï€A(U).

#### Control Number

ISILib-TH479

#### Creative Commons License

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

#### DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

#### Recommended Citation

Das, Soumyadip Dr., "On the Inertia Conjecture and Its Generalizations." (2021). *Doctoral Theses*. 455.

https://digitalcommons.isical.ac.in/doctoral-theses/455

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843875