AIS - Modular Forms (2019)

Speakers and Syllabus


The goal of this instructional school is to provide an exposure to the PhD students working in number theory to the fundamental aspects of modular forms and then highlight some of the important results.

 

Module-1: Modular Groups               (six lectures in the first week)
    1. The full modular group
    2. Congruence subgroups
    3. Action on the upper half plane, elliptic points, cusps
    4. Fundamental domain, compactification, volume  
Speaker: Dr.  Anupam Saikia, IIT Guwahati

 

Module-2: Modular forms of level one     (six lectures in the first week and two in the second week)
    1. Fourier expansion
    2. Eisenstein series
    3. Dimension formulae, zeros and poles using contour integrals, valence formula
    4. Hecke operators
    5. Ramanujan’s tau function, j-function and its relevance
Speaker: Dr. Rupam Barma, IIT Guwahati

Module-3: Modular forms of higher level(six lectures in the second week)
    1. Modular forms for congruence subgroups
    2. Eisenstein series of higher level for $Gamma_0(N),\Gamma_1(N)$
    3. Peterson inner product
Speaker: Dr. Soumya Das, IISc Bangalore

Module-4: Hecke operators of higher level
(four lectures in the second week and two in the third)
    1. Double coset operators
    2. Oldforms and newforms, Atkin-Lehner Theory
    3. Hecke eigneform
Speaker: Dr. Debargha Banerjee, IISER Pune

Module-5: Modular forms of half-integral weight(six lectures in the third week)
    1. Definition and examples
    2. Non-vanishing of Fourier coefficients, number theoretic applications
    3. Shimura Correspondence
    4. Hecke Theory for half-integral weight forms 
    5. Fourier coefficients and it’s relation with L-values of the associated Shimura lift
Speaker: Dr. Narasimha Kumar, IIT Hyderabad


Module-6
:  L-function attached to modular forms (four lectures in the third week)
    1. Euler product
    2. Functional equation, relation between elliptic curves and modular forms
    3. Converse Theorems of Hecke

 

Speakers (tentative):
    1. B. Ramakrishnan, HRI, Allahabad
    2. Debargha Banerjee, IISER Pune
    3. Soumya Das, IISc Bangalore
    4. Narasimha Kumar, IIT Hyderabad
    5. Anupam Saikia, IIT Guwahati
    6. Rupam Barman, IIT Guwahati
Tutors
    1. Nilanjan Bag (IITG),
    2. Pramath Anamby (IISc Bangalore)
    3. Manish Kumar Pandey (HRI Allahabad)

 

 

 


Time Table

 

Day

Date

Lecture 1

(09.30-11.00)

Tea

 

Lecture 2

(11.30-01.00)

Lunch

 

Tutorial

(02.30-3.30)

Tea

Tutorial

(4-5)

Snacks

MON

13.05.2019

AS-1

 

RB-1

L

(AS, NB)

 

(AS, NB)

S

TUE

14.05.2019

AS-2

T

RB-2

U

(RB, NB)

T

(RB, NB)

N

WED

15.05.2019

RB-3

E

AS-3

N

(AS, NB)

E

(AS, NB)

A

THU

16.05.2019

RB-4

A

AS-4

C

(RB, NB)

A

(RB, NB)

C

FRI

17.05.2019

AS-5

 

RB-5

H

(AS, NB)

 

(AS, NB)

K

SAT

18.05.2019

AS-6

 

RB-6

 

(RB, NB)

 

(RB, NB)

S

 

MON

20.05.2019

SD-1

 

RB-7

L

(SD, PA)

 

(SD, PA)

S

TUE

21.05.2019

SD-2

T

RB-8

U

(RB, NB)

T

(RB, NB)

N

WED

22.05.2019

DB-1

E

SD-3

N

(SD, PA)

E

(SD, PA)

A

THU

23.05.2019

SD-4

A

DB-2

C

(DB, PA)

A

(DB, PA)

C

FRI

24.05.2019

DB-3

 

SD-5

H

(SD, PA)

 

(SD, PA)

K

SAT

25.05.2019

SD-6

 

DB-4

 

(DB, PA)

 

(DB, PA)

S

 

MON

27.05.2019

NK-1

 

DB-5

L

(DB, MKP)

 

(DB, MKP)

S

TUE

28.05.2019

NK-2

T

DB-6

U

(NK, MKP)

T

(NK, MKP)

N

WED

29.05.2019

BR-1

E

NK-3

N

(BR, MKP)

E

(BR, MKP)

A

THU

30.05.2019

BR-2

A

NK-4

C

(NK, MKP)

A

(NK, MKP)

C

FRI

31.05.2019

BR-3

 

NK-5

H

(BR, MKP)

 

(BR, MKP)

K

SAT

1.06.2019

BR-4

 

NK-6

 

(NK, MKP)

 

(NK, MKP)

S

 

AS: Anupam Saikia (IITG),
RB: Rupam Barman (IITG),
SD: Soumya Das (IISc Bangalore),
DB: Debargha Banerjee (IISER Pune),
NK: Narasimha Kumar (IIT Hyderabad)
BR: B. Ramakrishnan (HRI Allahabad)
NB: Nilanjan Bag (IITG), PA: Pramath Anamby (IISc Bangalore)
MKP: Manish Kumar Pandey (HRI Allahabad)

 

 

 

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