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Elliptic curves and the special values of L-functions (ONLINE)
Monday, 02 August 2021
Time Speaker Title Resources
09:40 to 09:50 Rajesh Gopakumar (ICTS-TIFR, India) Welcome Remarks
09:50 to 10:00 Haruzo Hida (University of California, Los Angeles, USA) Opening remarks
10:00 to 11:00 Anupam Saikia (IIT Guwahati, India) Introduction to Elliptic Curves 1

After introducing elliptic curve as a non-singular plane cubic curve over a field, the first lecture will elaborate on the group structure of points on an elliptic curve with special attention to the point at infinity. The Mordell-Weil Theorem as well as Mazur's theorem for rational torsion points will be briefly discussed too. The second lecture will introduce the notion of Neron-Tate height and the regulator of an elliptic curve. A sketch for the proof of the Mordell-Weil Theorem will be given.  The third lecture will start by focusing on elliptic curves over finite fields, and briefly discuss the zeta function. Then the Hasse-Weil L-function of an elliptic curve will be introduced. Finally, elliptic curves over the field of complex numbers and the connection with lattices will be discussed. 

References:

(i) Rational Point on Elliptic Curves, J. H. Silverman and J. Tate, Springer

(ii) The Arithmetic of Elliptic Curves, J. H. Silverman, Springer

(iii) Introduction to Elliptic Curves and Modular Forms, N. Koblitz, Springer
   
11:30 to 13:00 Shaunak Deo (IISc, India) Galois Representations 1

In his seminal paper from late 1980s, Mazur developed the theory of deformations of Galois representations which played an important role in the proof of Fermat's last theorem. Since then, the study of deformations of Galois representations has become an important theme in number theory and it has played a crucial role in proving some very important results like modularity of elliptic curves, Serre's conjecture and Sato-Tate conjecture. We will present an overview of the deformation theory of Galois representations in this mini course. We will begin with some basic notions related to Galois representations and the description of Galois representations attached to Elliptic curves and modular forms. Then we will move on to the basics of deformation theory. We will also see some R=T theorems and their significance towards the end. If time permits, we will briefly touch upon some advanced topics such as pseudo-representations and the infinite-fern of Gouvea-Mazur.

References:

1) 'Deforming Galois representations' by Mazur from 'Galois groups over Q',

2) 'An introduction to the deformation theory of Galois representations' by Mazur from 'Modular forms and Fermat's last theorem (Boston, MA, 1995)',

3) 'Deformations of Galois representations' by Gouvea from 'IAS/Park City Math. Ser., 9, Arithmetic algebraic geometry (Park City, UT, 1999)',

4) Notes of Toby Gee on 'Modularity lifting theorems' from Arizona Winter School 2013
   
14:00 to 15:00 Mahesh Kakde (IISc, India) On the Gross—Stark Conjecture 1

In the 1970s Stark made remarkable conjectures postulating existence of special elements, now known as Stark units, related to special values of Artin L-functions. We will start by recalling Stark's conjectures. In 1981 Gross gave a p-adic refinement of Stark's conjectures in the special case of totally odd characters of totally real fields. A breakthrough result of Dasgupta—Darmon—Pollack proved the Gross—Stark conjecture in the rank 1 case under certain assumptions which were soon removed by Ventullo. The general case was resolved in a joint work with Dasgupta and Ventullo. In this course we will study the proof of Gross—Stark conjecture. 
   
15:30 to 16:30 Anupam Saikia (IIT - Guwahati, India) Introduction to Elliptic Curves 2

After introducing elliptic curve as a non-singular plane cubic curve over a field, the first lecture will elaborate on the group structure of points on an elliptic curve with special attention to the point at infinity. The Mordell-Weil Theorem as well as Mazur's theorem for rational torsion points will be briefly discussed too. The second lecture will introduce the notion of Neron-Tate height and the regulator of an elliptic curve. A sketch for the proof of the Mordell-Weil Theorem will be given.  The third lecture will start by focusing on elliptic curves over finite fields, and briefly discuss the zeta function. Then the Hasse-Weil L-function of an elliptic curve will be introduced. Finally, elliptic curves over the field of complex numbers and the connection with lattices will be discussed. 

References:

(i) Rational Point on Elliptic Curves, J. H. Silverman and J. Tate, Springer

(ii) The Arithmetic of Elliptic Curves, J. H. Silverman, Springer

(iii) Introduction to Elliptic Curves and Modular Forms, N. Koblitz, Springer
   
Tuesday, 03 August 2021
Time Speaker Title Resources
10:00 to 11:00 Anupam Saikia (IIT - Guwahati, India) Introduction to Elliptic Curves 3

After introducing elliptic curve as a non-singular plane cubic curve over a field, the first lecture will elaborate on the group structure of points on an elliptic curve with special attention to the point at infinity. The Mordell-Weil Theorem as well as Mazur's theorem for rational torsion points will be briefly discussed too. The second lecture will introduce the notion of Neron-Tate height and the regulator of an elliptic curve. A sketch for the proof of the Mordell-Weil Theorem will be given.  The third lecture will start by focusing on elliptic curves over finite fields, and briefly discuss the zeta function. Then the Hasse-Weil L-function of an elliptic curve will be introduced. Finally, elliptic curves over the field of complex numbers and the connection with lattices will be discussed. 

References:

(i) Rational Point on Elliptic Curves, J. H. Silverman and J. Tate, Springer

(ii) The Arithmetic of Elliptic Curves, J. H. Silverman, Springer

(iii) Introduction to Elliptic Curves and Modular Forms, N. Koblitz, Springer
   
11:30 to 13:00 Shaunak Deo (IISc, India) Galois Representations 2

In his seminal paper from late 1980s, Mazur developed the theory of deformations of Galois representations which played an important role in the proof of Fermat's last theorem. Since then, the study of deformations of Galois representations has become an important theme in number theory and it has played a crucial role in proving some very important results like modularity of elliptic curves, Serre's conjecture and Sato-Tate conjecture. We will present an overview of the deformation theory of Galois representations in this mini course. We will begin with some basic notions related to Galois representations and the description of Galois representations attached to Elliptic curves and modular forms. Then we will move on to the basics of deformation theory. We will also see some R=T theorems and their significance towards the end. If time permits, we will briefly touch upon some advanced topics such as pseudo-representations and the infinite-fern of Gouvea-Mazur.

References:

1) 'Deforming Galois representations' by Mazur from 'Galois groups over Q',

2) 'An introduction to the deformation theory of Galois representations' by Mazur from 'Modular forms and Fermat's last theorem (Boston, MA, 1995)',

3) 'Deformations of Galois representations' by Gouvea from 'IAS/Park City Math. Ser., 9, Arithmetic algebraic geometry (Park City, UT, 1999)',

4) Notes of Toby Gee on 'Modularity lifting theorems' from Arizona Winter School 2013
   
14:00 to 15:00 Mahesh Kakde (IISc, India) On the Gross—Stark conjecture 2

In the 1970s Stark made remarkable conjectures postulating existence of special elements, now known as Stark units, related to special values of Artin L-functions. We will start by recalling Stark's conjectures. In 1981 Gross gave a p-adic refinement of Stark's conjectures in the special case of totally odd characters of totally real fields. A breakthrough result of Dasgupta—Darmon—Pollack proved the Gross—Stark conjecture in the rank 1 case under certain assumptions which were soon removed by Ventullo. The general case was resolved in a joint work with Dasgupta and Ventullo. In this course we will study the proof of Gross—Stark conjecture. 
   
15:30 to 16:30 Sudhanshu Shekhar (IIT, Kanpur, India) Introduction to Elliptic curves and Selmer groups (part 2) 1

Principal homogeneous spaces, local and global Galois cohomology groups, Selmer groups and Tate-Shafarevich groups, weak Mordell-Weil theorem, Kummer theory of elliptic curves, Selmer groups of p-adic Galois representations. 

References:

(i)  The Arithmetic of Elliptic Curves, J. H. Silverman, Springer

(ii) Elliptic Curves, J. S. Milne, World Scientific.
   
Wednesday, 04 August 2021
Time Speaker Title Resources
09:00 to 10:30 Francesc Castella (University of California, Santa Barbara, USA) Heegner Points 1

In the late 1980s, Kolyvagin obtained ground-breaking results towards the Birch and Swinnerton-Dyer conjecture for (modular) elliptic curves E/Q of analytic rank 0 or 1 by using Heegner points on E over ring class field extensions of an auxiliary imaginary quadratic field K. The goal of this mini-course is to give, as one illustration of the method of Euler systems, a more or less complete proof (assuming some background on Class Field Theory and Galois cohomology) of Koyvagin's main theorem, stating that if the Heegner point over K has infinite order, then E(K) has rank 1, and the Tate-Shafarevich group of E/K is finite.

References:

Kolyvagin, ``Finiteness of E(Q) and Sha(E,Q) for a subclass of Weil curves", 1988.

Kolyvagin, ``The Mordell-Weil and Shafarevich-Tate groups of Weil elliptic curves", 1989.

Gross, ``Kolyvagin's work on modular elliptic curves", in: L-functions and arithmetic, 1991.

Darmon, ``Rational points on modular elliptic curves" (chapter 10), 2004.
   
11:00 to 12:30 Chan-Ho Kim (Korea Institute of Advanced Studies, South Korea) Introduction to Beilinson--Kato elements and their applications 1

In his celebrated work, Kato constructed the Euler system of Beilinson--Kato elements and proved spectacular results on the Iwasawa main conjecture and the Birch and Swinnerton-Dyer conjecture by using the Euler system. We study his work in this mini-course with emphasis on the case of elliptic curves.

1. Birch and Swinnerton-Dyer conjecture and the Iwasawa main conjecture: an overview

2. Iwasawa theory for elliptic curves: a summary

3. The methods of Euler systems and Kolyvagin systems

4. The construction and properties of Kato's zeta elements: a sketch

5. Further developments (if time permits)

References:

Delbourgo, Elliptic curves and big Galois representations, 2008

Kato, p-adic Hodge theory and values of zeta functions of modular forms, Asterisque, 2004

Mazur and Rubin, Kolyvagin systems, Memoir of AMS, 2004

Rubin, Euler systems and modular elliptic curves, Galois Representations in Arithmetic Algebraic Geometry, 1998

Rubin, Euler systems, Annals of Mathematics Studies, 2000

Scholl, An introduction to Kato's Euler systems, Galois Representations in Arithmetic Algebraic Geometry, 1998
   
14:00 to 15:00 Mahesh Kakde (IISc, India) On the Gross—Stark conjecture 3

In the 1970s Stark made remarkable conjectures postulating existence of special elements, now known as Stark units, related to special values of Artin L-functions. We will start by recalling Stark's conjectures. In 1981 Gross gave a p-adic refinement of Stark's conjectures in the special case of totally odd characters of totally real fields. A breakthrough result of Dasgupta—Darmon—Pollack proved the Gross—Stark conjecture in the rank 1 case under certain assumptions which were soon removed by Ventullo. The general case was resolved in a joint work with Dasgupta and Ventullo. In this course we will study the proof of Gross—Stark conjecture. 
   
15:30 to 16:30 Sudhanshu Shekhar (IIT, Kanpur, India) Introduction to Elliptic curves and Selmer groups (part 2) 2

Principal homogeneous spaces, local and global Galois cohomology groups, Selmer groups and Tate-Shafarevich groups, weak Mordell-Weil theorem, Kummer theory of elliptic curves, Selmer groups of p-adic Galois representations. 

References:

(i)  The Arithmetic of Elliptic Curves, J. H. Silverman, Springer

(ii) Elliptic Curves, J. S. Milne, World Scientific.
   
Thursday, 05 August 2021
Time Speaker Title Resources
09:00 to 10:30 Francesc Castella (University of California, Santa Barbara, USA) Heegner Points 2

In the late 1980s, Kolyvagin obtained ground-breaking results towards the Birch and Swinnerton-Dyer conjecture for (modular) elliptic curves E/Q of analytic rank 0 or 1 by using Heegner points on E over ring class field extensions of an auxiliary imaginary quadratic field K. The goal of this mini-course is to give, as one illustration of the method of Euler systems, a more or less complete proof (assuming some background on Class Field Theory and Galois cohomology) of Koyvagin's main theorem, stating that if the Heegner point over K has infinite order, then E(K) has rank 1, and the Tate-Shafarevich group of E/K is finite.

References:

Kolyvagin, ``Finiteness of E(Q) and Sha(E,Q) for a subclass of Weil curves", 1988.

Kolyvagin, ``The Mordell-Weil and Shafarevich-Tate groups of Weil elliptic curves", 1989.

Gross, ``Kolyvagin's work on modular elliptic curves", in: L-functions and arithmetic, 1991.

Darmon, ``Rational points on modular elliptic curves" (chapter 10), 2004.
   
11:00 to 12:30 Chan-Ho Kim (Korea Institute of Advanced Studies, South Korea) Introduction to Beilinson--Kato elements and their applications 2

In his celebrated work, Kato constructed the Euler system of Beilinson--Kato elements and proved spectacular results on the Iwasawa main conjecture and the Birch and Swinnerton-Dyer conjecture by using the Euler system. We study his work in this mini-course with emphasis on the case of elliptic curves.

1. Birch and Swinnerton-Dyer conjecture and the Iwasawa main conjecture: an overview

2. Iwasawa theory for elliptic curves: a summary

3. The methods of Euler systems and Kolyvagin systems

4. The construction and properties of Kato's zeta elements: a sketch

5. Further developments (if time permits)

References:

Delbourgo, Elliptic curves and big Galois representations, 2008

Kato, p-adic Hodge theory and values of zeta functions of modular forms, Asterisque, 2004

Mazur and Rubin, Kolyvagin systems, Memoir of AMS, 2004

Rubin, Euler systems and modular elliptic curves, Galois Representations in Arithmetic Algebraic Geometry, 1998

Rubin, Euler systems, Annals of Mathematics Studies, 2000

Scholl, An introduction to Kato's Euler systems, Galois Representations in Arithmetic Algebraic Geometry, 1998
   
14:00 to 15:00 Mahesh Kakde (IISc, India) On the Gross—Stark conjecture 4

In the 1970s Stark made remarkable conjectures postulating existence of special elements, now known as Stark units, related to special values of Artin L-functions. We will start by recalling Stark's conjectures. In 1981 Gross gave a p-adic refinement of Stark's conjectures in the special case of totally odd characters of totally real fields. A breakthrough result of Dasgupta—Darmon—Pollack proved the Gross—Stark conjecture in the rank 1 case under certain assumptions which were soon removed by Ventullo. The general case was resolved in a joint work with Dasgupta and Ventullo. In this course we will study the proof of Gross—Stark conjecture. 
   
15:30 to 16:30 Shaunak Deo (IISc, India) Galois Representations 3

In his seminal paper from late 1980s, Mazur developed the theory of deformations of Galois representations which played an important role in the proof of Fermat's last theorem. Since then, the study of deformations of Galois representations has become an important theme in number theory and it has played a crucial role in proving some very important results like modularity of elliptic curves, Serre's conjecture and Sato-Tate conjecture. We will present an overview of the deformation theory of Galois representations in this mini course. We will begin with some basic notions related to Galois representations and the description of Galois representations attached to Elliptic curves and modular forms. Then we will move on to the basics of deformation theory. We will also see some R=T theorems and their significance towards the end. If time permits, we will briefly touch upon some advanced topics such as pseudo-representations and the infinite-fern of Gouvea-Mazur.

References:

1) 'Deforming Galois representations' by Mazur from 'Galois groups over Q',

2) 'An introduction to the deformation theory of Galois representations' by Mazur from 'Modular forms and Fermat's last theorem (Boston, MA, 1995)',

3) 'Deformations of Galois representations' by Gouvea from 'IAS/Park City Math. Ser., 9, Arithmetic algebraic geometry (Park City, UT, 1999)',

4) Notes of Toby Gee on 'Modularity lifting theorems' from Arizona Winter School 2013
   
Friday, 06 August 2021
Time Speaker Title Resources
09:00 to 10:30 Francesc Castella (University of California, Santa Barbara, USA) Heegner Points 3

In the late 1980s, Kolyvagin obtained ground-breaking results towards the Birch and Swinnerton-Dyer conjecture for (modular) elliptic curves E/Q of analytic rank 0 or 1 by using Heegner points on E over ring class field extensions of an auxiliary imaginary quadratic field K. The goal of this mini-course is to give, as one illustration of the method of Euler systems, a more or less complete proof (assuming some background on Class Field Theory and Galois cohomology) of Koyvagin's main theorem, stating that if the Heegner point over K has infinite order, then E(K) has rank 1, and the Tate-Shafarevich group of E/K is finite.

References:

Kolyvagin, ``Finiteness of E(Q) and Sha(E,Q) for a subclass of Weil curves", 1988.

Kolyvagin, ``The Mordell-Weil and Shafarevich-Tate groups of Weil elliptic curves", 1989.

Gross, ``Kolyvagin's work on modular elliptic curves", in: L-functions and arithmetic, 1991.

Darmon, ``Rational points on modular elliptic curves" (chapter 10), 2004.
   
11:00 to 12:30 Chan-Ho Kim (Korea Institute of Advanced Studies, South Korea) Introduction to Beilinson--Kato elements and their applications 3

In his celebrated work, Kato constructed the Euler system of Beilinson--Kato elements and proved spectacular results on the Iwasawa main conjecture and the Birch and Swinnerton-Dyer conjecture by using the Euler system. We study his work in this mini-course with emphasis on the case of elliptic curves.

1. Birch and Swinnerton-Dyer conjecture and the Iwasawa main conjecture: an overview

2. Iwasawa theory for elliptic curves: a summary

3. The methods of Euler systems and Kolyvagin systems

4. The construction and properties of Kato's zeta elements: a sketch

5. Further developments (if time permits)

References:

Delbourgo, Elliptic curves and big Galois representations, 2008

Kato, p-adic Hodge theory and values of zeta functions of modular forms, Asterisque, 2004

Mazur and Rubin, Kolyvagin systems, Memoir of AMS, 2004

Rubin, Euler systems and modular elliptic curves, Galois Representations in Arithmetic Algebraic Geometry, 1998

Rubin, Euler systems, Annals of Mathematics Studies, 2000

Scholl, An introduction to Kato's Euler systems, Galois Representations in Arithmetic Algebraic Geometry, 1998
   
14:00 to 15:00 Sudhanshu Shekhar (IIT, Kanpur, India) Introduction to Elliptic curves and Selmer groups (part 2) 3

Principal homogeneous spaces, local and global Galois cohomology groups, Selmer groups and Tate-Shafarevich groups, weak Mordell-Weil theorem, Kummer theory of elliptic curves, Selmer groups of p-adic Galois representations. 

References:

(i)  The Arithmetic of Elliptic Curves, J. H. Silverman, Springer

(ii) Elliptic Curves, J. S. Milne, World Scientific.
   
15:30 to 16:30 Shaunak Deo (IISc, India) Galois Representations 4

In his seminal paper from late 1980s, Mazur developed the theory of deformations of Galois representations which played an important role in the proof of Fermat's last theorem. Since then, the study of deformations of Galois representations has become an important theme in number theory and it has played a crucial role in proving some very important results like modularity of elliptic curves, Serre's conjecture and Sato-Tate conjecture. We will present an overview of the deformation theory of Galois representations in this mini course. We will begin with some basic notions related to Galois representations and the description of Galois representations attached to Elliptic curves and modular forms. Then we will move on to the basics of deformation theory. We will also see some R=T theorems and their significance towards the end. If time permits, we will briefly touch upon some advanced topics such as pseudo-representations and the infinite-fern of Gouvea-Mazur.

References:

1) 'Deforming Galois representations' by Mazur from 'Galois groups over Q',

2) 'An introduction to the deformation theory of Galois representations' by Mazur from 'Modular forms and Fermat's last theorem (Boston, MA, 1995)',

3) 'Deformations of Galois representations' by Gouvea from 'IAS/Park City Math. Ser., 9, Arithmetic algebraic geometry (Park City, UT, 1999)',

4) Notes of Toby Gee on 'Modularity lifting theorems' from Arizona Winter School 2013
   
Saturday, 07 August 2021
Time Speaker Title Resources
09:00 to 10:30 Francesc Castella (University of California, Santa Barbara, USA) Heegner Points 4

In the late 1980s, Kolyvagin obtained ground-breaking results towards the Birch and Swinnerton-Dyer conjecture for (modular) elliptic curves E/Q of analytic rank 0 or 1 by using Heegner points on E over ring class field extensions of an auxiliary imaginary quadratic field K. The goal of this mini-course is to give, as one illustration of the method of Euler systems, a more or less complete proof (assuming some background on Class Field Theory and Galois cohomology) of Koyvagin's main theorem, stating that if the Heegner point over K has infinite order, then E(K) has rank 1, and the Tate-Shafarevich group of E/K is finite.

References:

Kolyvagin, ``Finiteness of E(Q) and Sha(E,Q) for a subclass of Weil curves", 1988.

Kolyvagin, ``The Mordell-Weil and Shafarevich-Tate groups of Weil elliptic curves", 1989.

Gross, ``Kolyvagin's work on modular elliptic curves", in: L-functions and arithmetic, 1991.

Darmon, ``Rational points on modular elliptic curves" (chapter 10), 2004.
   
11:00 to 12:30 Chan-Ho Kim (Korea Institute of Advanced Studies, South Korea) Introduction to Beilinson--Kato elements and their applications 4

In his celebrated work, Kato constructed the Euler system of Beilinson--Kato elements and proved spectacular results on the Iwasawa main conjecture and the Birch and Swinnerton-Dyer conjecture by using the Euler system. We study his work in this mini-course with emphasis on the case of elliptic curves.

1. Birch and Swinnerton-Dyer conjecture and the Iwasawa main conjecture: an overview

2. Iwasawa theory for elliptic curves: a summary

3. The methods of Euler systems and Kolyvagin systems

4. The construction and properties of Kato's zeta elements: a sketch

5. Further developments (if time permits)

References:

Delbourgo, Elliptic curves and big Galois representations, 2008

Kato, p-adic Hodge theory and values of zeta functions of modular forms, Asterisque, 2004

Mazur and Rubin, Kolyvagin systems, Memoir of AMS, 2004

Rubin, Euler systems and modular elliptic curves, Galois Representations in Arithmetic Algebraic Geometry, 1998

Rubin, Euler systems, Annals of Mathematics Studies, 2000

Scholl, An introduction to Kato's Euler systems, Galois Representations in Arithmetic Algebraic Geometry, 1998
   
14:00 to 15:00 Shaunak Deo (IISc, India) Galois Representations 5

In his seminal paper from late 1980s, Mazur developed the theory of deformations of Galois representations which played an important role in the proof of Fermat's last theorem. Since then, the study of deformations of Galois representations has become an important theme in number theory and it has played a crucial role in proving some very important results like modularity of elliptic curves, Serre's conjecture and Sato-Tate conjecture. We will present an overview of the deformation theory of Galois representations in this mini course. We will begin with some basic notions related to Galois representations and the description of Galois representations attached to Elliptic curves and modular forms. Then we will move on to the basics of deformation theory. We will also see some R=T theorems and their significance towards the end. If time permits, we will briefly touch upon some advanced topics such as pseudo-representations and the infinite-fern of Gouvea-Mazur.

References:

1) 'Deforming Galois representations' by Mazur from 'Galois groups over Q',

2) 'An introduction to the deformation theory of Galois representations' by Mazur from 'Modular forms and Fermat's last theorem (Boston, MA, 1995)',

3) 'Deformations of Galois representations' by Gouvea from 'IAS/Park City Math. Ser., 9, Arithmetic algebraic geometry (Park City, UT, 1999)',

4) Notes of Toby Gee on 'Modularity lifting theorems' from Arizona Winter School 2013
   
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