Name of the Speaker with affiliation

No. of Lectures

Detailed Syllabus

Sazzad Ali Biswas

SRM University AP

05

1. Construction of p-adic numbers
2. Various Properties (algebraic and Analytic) of p-adic numbers
3. Arithmetic operations of p-adic numbers

Detailed:
 Absolute Value on field, Ostrowski’s Theorem, The field of p-adic numbers, The ring of p-adic integers Z_p and its properties (e.g. The fraction field of Z_p),

Mihir Sheth

IISc Bangalore

04

1. Local Fields
2. Global Fields
3. p-adic algebraic numbers

Detailed:
Extension of non-Archimedian Absolute Values, Locally Compact Fields, Local Fields, Number Fields, Class groups, p-adic Algebraic number Theory, Algebraic closure of Q_p, C_p.

Amiya Kumar Mondal

IISER Berhampur

04

Modular Forms

Detailed:
Modular forms and their properties, Modular Elliptic Curves

Santosh Nadimpally

IIT Kanpur

06

1. Construction of p-adic zeta functions
2. Various Properties of p-adic Zeta function

Detailed:
 p-adic interpolation of the function f(x)=a^x, p-adic distributions, Bernoulli distribuitions, Measures and Integration, p-adic zeta functions as a Mellin-Mazur Transformation,  Functional equation of p-adic zeta functions and root numbers, p-adic analytic expression for zeta function.

Manish Kumar Pandey

SRM University AP

02

Special values of the Riemann zeta functions

Detailed:
Riemann zeta functions and its analytic continuation, Bernoulli numbers, special values of Riemann zeta functions

Shaunak Deo

IISC Bangalore

06

1. Cyclotomic Fields
2. Iwasawa’s Construction of p-adic L-fucntions
3. p-adic Family of Modular Forms

Detailed:
Various properties of Cyclotomic fields, The Knonecker-Weber Theorem, Mod-p congruence between modular forms, and Iwasawa construction of p-adic L-functions, p-adic Family of Modular Forms

C S Rajan

Ashoka University

(4+5)=09

(4 talks, in the first week of the program)
1. Hensel’s Lemma
2. Local- Global Principle

Detailed: Polynomial over p-adic integers and their Solution, Quadratic Forms, Local-
Global  Principle, Hasse-Minkowski’s theorem

(5 talks, in the last week of the program)
1. Main Conjecture of Iwasawa Theory
2. Introduction to the work of Ribet (Converse to Herbrand)
and subsequent work
Detailed:
Introduction to the work of Iwasawa on the class groups of cyclotomic number fields; the Iwasawa algebra; modules over the Iwasawa algebra; construction of the arithmetic $p$-adic $L$-function and some consequences; statement of the Iwasawa Main conjecture.

Stickelberger elements and annihilators of class groups.

Ribet's converse to Herbrand and an indication of further developments towards the Main conjecture of Iwasawa theory.

References: the Seminaire Bourbaki article of Serre on the work of Iwasawa; the two volumes of Washington; Ribet's article in Inv. on converse to Herbrand.