arXiv subjects
Publications and source records attributed to Aftab Pande.
We compute the semisimplifications of the mod-$p$ reductions of $2$-dimensional crystalline representations of the absolute Galois group of the p-adic numbers of slope $(2,3)$ and arbitrary weight, building on work of Bhattacharya-Ghate
Given a reducible Galois representation $\overlineρ: G_{\mathbb{Q}} \rightarrow GL_2( \mathbb{F}_q)$ we show there exists an irreducible deformation $ρ: G_{\mathbb{Q}} \rightarrow GL_2 (\mathbb{W} [[T_1, T_2,.., T_r,....,]])$ of $\overlineρ$ ramified at infinitely many primes, where $\mathbb{W}$ denotes the ring of Witt vectors of $\mathbb{F}_q$. This is a modification of Ramakrishna's result for the irreducible case.
We study the rank of the $p$-Selmer group $Sel_p(A/k)$ of an abelian variety $A/k$, where $k$ is a function field. If $K/k$ is a quadratic extension and $F/k$ is a dihedral extension and the $\mathbb{Z}_p$-corank of $Sel_p (A/K)$ is odd, we show that the $\mathbb{Z}_p$-corank of $Sel_p(A/F) \geq [F:K]$. The result uses the theory of local constants developed by Mazur-Rubin for elliptic curves over number fields.
We extend previous work of the author using an idea of Buzzard and give an elementary construction of non-ordinary $p$-adic families of Hilbert Modular Eigenforms.
We construct infinitely ramified Galois representations $ρ$ such that the $a_l (ρ)$'s have distributions in contrast to the statements of Sato-Tate, Lang-Trotter and others. Using similar methods we deform a residual Galois representation for number fields and obtain an infinitely ramified representation with very large image, generalising a result of Ramakrishna.
We use a method of Buzzard to study p-adic families of different types of modular forms - classical, over imaginary quadratic fields and totally real fields. In the case of totally real fields of even degree, we get local constancy of dimensions of spaces of fixed slope and varying weight. For imaginary quadratic fields we obtain bounds independent of the weight on the dimensions of such spaces.