Research interests

I work in an area known as:

Arithmetic geometry.

These exciting areas of modern mathematics are in the intersection of number theory, algebraic geometry and representation theory.

In Arithmetic geometry, we are interested in solutions of equations or values of interesting objects like L-functions over integers. Motives are unifying objects built out of different cohomology theories that should give us fruitful results in our endeavor. Far-reaching conjectures of Langlands and Deligne are guiding principle in our pursuit. Galois representations are explicit manifestation of these motives connected by the above-mentioned philosophy. Most recently, Scholze made

remarkable breakthrough in this fast expanding area of mathematics.

Eisenstein cycles on modular symbols

Let X be a modular curve. We call Eisenstein classes corresponding to a divisor D to be the element of the certain homology groups with boundary certain divisor and ``orthogonal to cusp forms.

In a paper with Loic Merel (that appeared in the Journal of the London Mathematical Society), we write down these Eisenstein classes for the principal congruence subgroups. This is continuation of

the work for the congruence subgroup of the form $\Gamma_0(N)$ that we computed previously and that appeared in PAMS and Pacific journal (joint with Srilakshmi Krishnamoorty).

In another work (joint with Loic Merel), we write down these classes for any subgroup of finite index. This is a generalization of the classical classical Manin-Drinfeld Theorem for subgroups of finite indices.

Self-intersection numbers of relative dualizing sheaves

In a joint work with Diganta Borah and Chitrabhanu Chowdhuri, we wrote down the Arakelov self intersection number of the relative dualizing

sheaf for the minimal regular model over integers of the modular curve $X_0(p^2)$ in terms of its genus $g_{p^2}$. In another joint paper with Chitrabhanu Chowdhuri, we found arithmetic application of our work.

Local Brauer classes and root numbers at supercuspidal primes

Let f be a primitive non-CM cusp form of weight and

let M be the Grothendieck motive associated to $f$ by Scholl.

Let X_ denote the endomorphism algebra of endomorphisms of M_f.

We can study the Brauer class of $X$ locally. In my thesis, I computed the local Brauer classes

for some `non-supercuspidal primes". With Tathagata Mandal, we managed to find the local algebra X for some supercuspidal primes. In another work joint work with Tathagata Mandal, we found the twisting properties of local root numbers at all primes (including supercuspidal primes).

Differential modular forms

Differential modular forms are developed by Buium in a series of article.

These modular forms are useful to study the lifting of modular forms in characteristic $p>0$

to characteristic zero; that in turn produces new congruences of modular forms. Abstractly, these are modular forms obtained by

applying the arithmetic $p$-jet space functor (adjoint to the $p$-typical Witt vector functor)

to the ring of $p$-adic modular forms. However, the main challenge lies in producing differential modular form of non-zero order (not classical) that should be useful to tackle Diophantine problems like Andr\'e-Oort conjecture or it's generalizations. Since these objects are obtained by taking suitable $p$-adic completion so geometrically there is a similarity between the $p$-adic (characteristic $0$) and mod $p$ objects (characteristic $p>0$).

I extended the theory of differential modular forms to the totally real fields

setting. These differential modular forms obtained by applying arithmetic $p$-jet

space functor to the ring of modular forms on Shimura curves over totally real fields.

In the above mentioned article, the examples are given of differential modular forms of non-trivial order.

However, weights of these

``new" modular forms are either zero or non-integral.

Recently, Buium discovered differential modular forms of non-zero integral weights out of mod $p$ newforms differential modular forms of small weights using companion modular forms . In a joint

work with Arnab Saha, we answered the following question:

Are there any differential modular forms of non-zero integral weights that is not of order zero over totally real fields different from $\Q$?