NCMW - Elliptic Curves (2023)

Speakers and Syllabus

 Syllabus:

Name of the Speaker with affiliation Number of Lectures Syllabus/ Topics
Dr. Sarbeswar Pal IISER-TVM Lectures 4 hrs 30 mins
Tutorial - 1 hr		 Affine and Projective varieties : Rational Maps between varieties Algebraic Curves : Curves, Maps between curves, Divisors, Di↵erentials and Riemann Roch Theorem
R. Thangadurai Professor HRI, Prayagraj Lectures - 7 hrs
Tutorial - 1 hr		 The Geometry of Elliptic Curves : Elliptic Curves : Weierstrass Equations, The Group Law, Isogenies, Discriminant and multiplicative reduction, The Tate Module, The Weil Pairing The Endomorphism Ring The Automorphism Group, Formal Groups, Frey Curves
Neil Dummigan Professor Sheffield University, U.K. Lectures -6 hrs
Tutorial - 2 hrs		 Elliptic Curves over finite fields : Rational Points, The Weil Conjectures, The Endomorphism Ring, Hasse Invariant calculations.
Dr. Jayanta ManoharmayumUniversity, U.K. Sheffield Lectures - 6 hrs
Tutorial - 1 hr 2		 Elliptic Curves over C : The isomorphism E/C ⇠ = C⇤ /q Z, Tate Curve, Elliptic Integrals, Construction of Elliptic Functions Analytic and Algebraic Maps, Uniformization, The Lefschetz Principle Elliptic Curves over
Ram Murty Professor Queen’s University, Canada. Lectures - 6 hrs
Tutorial - 1 hr		 Global Fields: The Weak Mordell-Weil Theorem, The Kummer Pairing via Cohomology, The Descent Procedure, The Mordell-Weil Theorem over Q, Heights, Torsion Points, The Minimal Discriminant, The Rank of an Elliptic Curve, Szpiro Conjecture and ABC.
Dr. Narasimha Kumar, IIT Hyderabad. Lectures - 5 hrs
Tutorial - 1 hr		 Integral Points on Elliptic Curves - I: Diophantine Approximation, Distance Functions, Siegel Theorem,
Dr. Srilakshmi Krishnamoorthy IISER-TVM. Lectures - 1 hr 30 mins 
Tutorial - 1 hr		 Integral Points on Elliptic Curves - II: Shafarevich Theorem, Roth Theorem.
Dr. Shaunak Deo IISc, Bangalore. Lectures - 6 hrs Tutorial - 1 hr Computing the Mordell-Weil Group: Examples, Twisting-General Theory Homogeneous Spaces, The Selmer and Shafarevich-Tate Groups Twisting-Elliptic Curves.
Kalyan Chakraborty Professor KSOM, Calicut. Lectures - 6 hrs Tutorial - 1 hr 3 Algorithmic Aspects of Elliptic Curves: Elliptic Curve Factorization Algorithm (by Lenstra), Elliptic Curve Cryptography, Pairing-Based Cryptography, Applications to Cryptography, Double-and-Add Algorithms, Counting the Number of Points in E(Fq ), Solving the ECDLP: The General Case and special case, Computing the Weil Pairing, The Tate-Lichtenbaum Pairing.
Lawrence Washington Professor University of Maryland, USA. Lectures - 6 hrs Introduction to Iwasawa Theory: Basic facts, The structure of ⇤-modules, Iwasawa’s theorem, consequences, Logarithmic derivatives, Local units modulo cyclotomic units.
Sujatha Ramdorai Professor University of British Columbia, Canada Lectures - 6 hrs Tutorial - 1 hr Iwasawa modules for elliptic curves: Bloch–Kato Selmer groups, Selmer structures,The extension F1 /F , Selmer groups over F1 , Control theorems, Construction of the p-adic L-functions, Some p-adic Measure Theory, Formulation of the Main Conjecture.
Dr. Bharathwaj Palvannan IISc, Bangalore. Lectures - 6 hrs  Tutorial - 1 hr p-adic L-functions of elliptic curves : Periods, modular symbols, measures, algebraicity and integrality of L-values.

Time Table

 Week 1

Time/ Day 03/04 04/04 05/04 06/04 07/04 08/04
09:30- 11:00 SP SP RT RT LW LW
11:00- 11:30 Tea Tea Tea Tea Tea Tea
11:30- 13:00 RT RT JM JM JM JM
13:00- 14:30 Lunch Lunch Lunch Lunch Lunch Lunch
14:30- 16:00 SP ND ND ND ND NK
16:00- 16:30 Tea Tea Tea Tea Tea Tea
16:40- 17:40 RT+SKP +SB SP+SKP +SB ND+SKP +SB JM+SKP +SB ND+SKP +SB NK+SKP +SB
18:00- 19:00 LW LW LW RT NK NK

 Week 2

Time/ Day 10/04 11/04 12/04 13/04 14/04 15/04
09:00- 10:30 SR SR SR SK RM RM
10:30- 11:00 Tea Tea Tea Tea Tea Tea
11:00- 12:30 SR SD SD RM SD SD
12:30- 14:00 Lunch Lunch Lunch Lunch Lunch Lunch
14:00- 15:30 NK BP BP BP BP RM
15:30- 16:00 Tea Tea Tea Tea Tea Tea
16:00- 17:00 KC KC KC KC KC KC
17:10- 18:10 KC+TD +RV BP+TD +RV SD+TD +RV RM+TD +RV SK+TD +RV SR+TD +RV

 

Full forms for the abbreviations of Speakers and Tutors

speakers:

  • SP: Dr. Sarbeswar Pal
  • RT: Prof. Ravindranathan Thangadurai
  • ND: Prof. Neil Dummigan
  • JM: Dr. Jayanta Manoharmayum
  • RM: Prof. Ram Murty
  • SK: Dr. Srilakshmi Krishnamoorthy
  • SD: Dr. Shaunak Deo
  • KC: Prof. Kalyan Chakraborty
  • NK : Dr. Narasimha Kumar
  • LW: Prof. Lawrence Washington
  • SR: Prof. Sujatha Ramdorai
  • BP: Dr. Bharathwaj Palvannan

Tutors :

  • SKP: Mr. Sunil Pasupulati (Final year PhD student at IISER-TVM)
  • SZ: Dr. Ravitheja Vangala (Postdoctoral fellow, IISc, Bangalore)
  • SB: Dr. Subham Bhakta (Visitor, IISER-TVM)
  • TD: Dr. Tarun Dalal (Postdoctoral Fellow, HRI, Allahabad).

References

  • Joseph H. Silverman, The arithmetic of elliptic curves, second ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009.
  • J. Coates and R. Sujatha, Cyclotomic fields and zeta values, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2006.
  • Lawrence C. Washington, Introduction to cyclotomic fields, second ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997.
  • Lawrence C. Washington, Elliptic curves, second ed., Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2008, Number theory and cryptography.
  • Ralph Greenberg, Introduction to Iwasawa theory for elliptic curves, Arithmetic algebraic geometry (Park City, UT, 1999), IAS/Park City Math. Ser., vol. 9, Amer. Math. Soc.,Providence, RI, 2001, pp. 407–464.
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