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I am currently a Research Fellow at Mathematics Institute,
University of Warwick. I completed my PhD under the supervision of Professor Samir Siksek.
My research interests broadly lie in number theory and I have been working on modular forms of half-integral weight. Here is a link to my thesis.
Previously, I did M.Math. degree at Indian Statistical Institute, Bangalore
and worked under the supervision of Professor B.Sury.
Research Abstract
Let k be an odd integer and N be a positive integer divisibe by 4. Let χ be a Dirichlet character modulo N. Shimura decomposes the space of half-integral cusp forms of weight k/2, level N and character χ as a direct sum of subspace generated by single-variable theta-series and direct sum of subspaces of forms that are ''Shimura-equivalent'' to φ; here φ runs through the newforms of weight k-1, level dividing N/2 and character, square of χ. In my thesis I gave an explicit algorithm for computing this decomposition.
Once we have the decomposition, we can explore Waldspurger's theorem expressing the critical values of the L-functions of twists of an elliptic curve in terms of the coefficients of modular forms of half-integral weight. Following Tunnell, this often allows us to give a criterion for the n-th twist of an elliptic curve to have positive rank in terms of the number of representations of certain integers by certain ternary quadratic forms.
The Waldspurger's theorem comes with a hypothesis that the newforms of integral weight must have either level divisible by 16 or their corresponding local automorphic representation at the prime 2 must not be supercuspidal. I am currently interested in extending Waldspurger kind of result in general case by allowing supercuspidal representations at the prime 2 with small conductors. The other things I am interested in are questions regarding algebraicity of Fourier coefficients of half integral weight modular forms and distribution of the Fourier coefficients of half integral weight Hecke eigenforms.
Publications
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(Joint with N. Kumar) A note on the Fourier coefficients of half-integral weight modular forms. To appear in Archiv der Mathematik . on arXiv
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Hecke operators in half-integral weight. To appear in Journal de Théorie des Nombres de Bordeaux. on arXiv
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Explicit application of Waldspurger's theorem, LMS Journal of Computation and Mathematics, Vol. 16 (2013), 216-245.
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On Shimura's decomposition, International Journal of Number Theory, Vol. 9, no. 06, (2013), 1431-1445.
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(Joint with B. Sury) Some vanishing sums involving binomial coeffcients in the denominator, Albanian Journal of Mathematics,Vol. 2 (2008), 27-32.
Some MAGMA Codes
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For computing Shimura's decomposition (shimuraspacewtk.m)
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For computing the Shimura equivalent space of an elliptic curve that comes from theta series of ternary quadratic forms; it includes implementation of Dickson's algorithm (quadratic.m)
Teaching
I was a lecturer for MA3H1 Topics in Number Theory. Here is my lecture note.
Previously, I was teaching assistant for Topics in Number Theory during Autumn 2011 and for
Analysis II during Spring 2011.
I have been supervisor of undergraduate first and second year Mathematics students since October 2008 to April 2012.
Recent Visits, Conferences and Invited Talks
October 14-18, 2013: Women in Numbers workshop, CIRM Luminy.
September 16-20, 2013: Minicourses on Algebraic and Explicit Methods in Arithmetic, Laboratoire de Mathematiques de Besancon.
March 20-22, 2013: Attending Explicit Methods for Modular Forms, University of Warwick.
March 17-19, 2013: Attending Automorphic Forms and L-Functions, University of Darmstadt.
March 7, 2013: Invited speaker at OberSeminar, MPIM Bonn.
February 25-March 02, 2013: Visiting University of Luxembourg.
December 12, 2012: Invited speaker at Algebra and Number Theory Seminar, University of Luxembourg.
January 22, 2013: Invited speaker at Number Theory Seminar, University of Cologne.
January 01-March 31,2013: Visiting MPIM, Bonn, Germany.
December 12, 2012: Invited speaker at Number Theory Seminar, Tel Aviv University.
December 08-22, 2012: Visiting Technion - Israel Institute of Technology, Israel.
November 15 2012: Invited speaker at Oxford Number Theory Seminar.
October 01-03,2012: Attending Arithmetic of L-functions, University of Bristol.
September 24-28,2012: Attending Selmer Groups, Descent and the Distribution of Ranks, University of Warwick.
September 12, 2012: Speaking at Number Theory Seminar in Kyushu University, Japan.
August 06-10, 2012: Attending Building Bridges: 1st EU-US conference on Automorphic Forms and related topics in RWTH, Aachen.
June 17-22, 2012: Attending CNTA XII in Lethbridge.
June 10-17, 2012: Attending BIRS summer school on Contemporary methods for solving Diophantine equations.
April 2, 2012: Invited speaker at ISI Colloquium in Indian Statistical Institute, Bangalore.
February 22, 2012: Invited speaker at Bristol Heilbronn Seminar.
November 18, 2011: Giving a talk at London Mathematical Society GSM 2011.
August 29-September 8, 2011: Attending Summer School and Conference on "Computations with Modular Forms 2011" in Heidelberg.
July 12-14, 2011: Invited speaker at Foundation of Computational Mathematics 2011 in Budapest.
May 4-6, 2011: Attending the BSD conference in Cambridge.
April 18-21, 2011: Attending British Mathematical Colloquium 2011 in University of Leicester.
April 14-16, 2011: Giving a talk at Young Researchers in Mathematics 2011 in Warwick.
Photos
I love photography. Here is a small selection.
Curriculum Vitae
Here is my CV.
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