B. Ramakrishnan


Research Summary:

1. (with M. Manickam) Let $F$ be a Siegel cusp form of integral weight $k$ on the Siegel modular group $\Gamma_g:= Sp_{g}({\mathbb Z})$ of degree $g$ and let $a(T)$ be its Fourier coefficient, where $T$ is a positive definite, half-integral, symmetric matrix of size $g$. For $g=2$, W. Kohnen proved that

\begin{displaymath} a(T) \ll_{\epsilon,F} ({\rm min}T)^{5/18+\epsilon} (\det T)^{(k-1)/2 +\epsilon} \quad (\epsilon >0), \end{displaymath} (1)

where ${\rm min} T$ is the least positive integer represented by $T$. By reduction theory, the above estimate becomes
\begin{displaymath} a(T) \ll_{\epsilon,F} (\det T)^{k/2 - 13/36 +\epsilon} \quad (\epsilon >0). \end{displaymath} (2)

The method of Kohnen is based on estimates for the Fourier coefficients of Jacobi Poincaré series as well as for the Petersson norms of the Fourier-Jacobi coefficients of $F$.

Later, S. Böcherer and W. Kohnen generalized this method to obtain similar estimates for higher genus.

In this work, we generalize Kohnen's estimate for half-integral weight Siegel cusp forms of genus $2$ and arbitrary level. More precisely, we prove the following theorem.

Theorem 0.1
\begin{thm} Let $N, k$\ be natural numbers, $k>2$. Let $F(Z)$\ be a Siegel cusp ... ...psilon} (\det T)^{k/2-1/4+\epsilon} \quad (\epsilon >0). \end{equation}\end{thm}


Since $({\rm min}T) \ll (\det T)^{1/2}$ by reduction theory, we obtain the following corollary.
Corollary 0.1
\begin{cor} Under the notations of \thmref{thm:1}, we have \begin{equation} a(T)... ...on, F} (\det T)^{k/2-1/18+\epsilon} \quad (\epsilon >0). \end{equation}\end{cor}

2. (with R. Thangadurai)  In this work, we prove certain divisibility properties of the Fourier coefficients of a class of normalized Eisenstein series modulo certain prime powers.


Publications:

  1. Saito-Kurokawa correspondence of degree two for arbitrary level, (with M. Manickam), J. Ramanujan Math. Soc. 17, No. 3 (2002) 149-160.
  2. A Note on Certain Divisibility Properties of the Fourier Coefficients of Normalised Eisenstein Series, (with R. Thangadurai), Expo. Math. 21, No. 1 (2003) 75-82.

Preprints:

  1. A Note on the Estimates for Fourier Coefficients of Siegel Cusp Forms of Half-integral Weight, (with M. Manickam).
  2. An Introduction to Modular Forms and Hecke Operators, (with M. Manickam).


Visits to other Institutes:

1. Visited Department of Mathematics, University of Paris VI during June 2002.


Other Activities:

1. Taught a one semester course on Elementary Number Theory for the first year students.

2. Coordinating the JEST for selecting students for our Ph.D programme.

 



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