# AIS - An Introduction to p-adic Methods in Arithmetic (2023)

## Speakers and Syllabus

 Name of the Speaker with affiliation No. of Lectures Detailed Syllabus Sazzad Ali Biswas SRM University AP 05 1. Construction of p-adic numbers2. Various Properties (algebraic and Analytic) of p-adic numbers3. 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 Fields2. Global Fields3. 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 functions2. 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 Fields2. Iwasawa’s Construction of p-adic L-fucntions3. 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 Lemma2. 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 Theory2. Introduction to the work of Ribet (Converse to Herbrand)and subsequent workDetailed: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.

References:
1. A course in Arithmetic by J.-P. Serre
2. Algebraic Number Theory by J. S. Milne
3. P-adic analysis by Neal Koblitz
4. Introduction to Cyclotomic Fields by Lawrence Washington
5. On p-adic L-function, Annals of Mathematics, Kenkich Iwasawa

## Time Table

 Day Date Lecture 1 (9.30– 11.00) Tea (11.05 – 11.25) Lecture 2 (11.30– 1.00) Lunch (1.05– 2.25) Tutorial (2.30–3.30) T ea(3 .3 5- 3. 5 5) Tutorial (4.00-5.00) Snacks 5.05- 5.30 (name of the speaker) (name of the speaker) (name of the speaker + tutors) (name of the speaker + tutors) Mon June 26 CSR SAB CSR+AJ SAB+AJ Tues June 27 CSR SAB CSR+AJ SAB+AJ Wed June 28 CSR SAB CSR+AJ SAB+AJ Thu June 29 CSR SAB CSR+AJ SAB+AJ Fri June 30 MS SAB MS+AJ SAB+AJ Sat July 01 MS MKP MS+AJ PKP+RV SUNDAY: OFF Mon July 03 MS MKP MS+AJ MKP+MKP Tues July 04 MS AKM MS+AJ AKM+AJ Wed July 05 SN AKM SN+RV AKM+AJ Thu July 06 SN AKM SN+RV AKM+AJ Fri July 07 SN AKM SN+RV AKM+AJ Sat July 08 SN SN SN+RV SN+RV SUNDAY: OFF Mon July 10 SVD SN SVD+RV SN+RV Tues July 11 SVD CSR SVD+RV CSR+SAB Wed July 12 SVD CSR SVD+RV CSR+SAB Thu July 13 SVD CSR SVD+RV CSR+SAB Fri July 14 SVD CSR SVD+PV CSR+SAB Sat July 16 SVD CSR SVD+RV CSR+SAB

Tutorial Assistant

 S. Name No. Affiliation 1 Dr. Arindam Jana Postdoc, TIFR Mumbai 2 Dr. Ravitheja Vangola Postdoc, IIT Kanpur

Full forms for the abbreviations of speakers and tutors:
MS: Mihir Sheth
CSR: C S Rajan
AM: Amiya Mondal
RV: Ravitheja Vangola
AJ: Arindam Jana