arXiv subjects

Rajender Adibhatla

Publications and source records attributed to Rajender Adibhatla.

3 recordsLinked to original sources

arXiv2013

Higher congruence companion forms

For a rational prime $p \geq 3$ we consider $p$-ordinary, Hilbert modular newforms $f$ of weight $k\geq 2$ with associated $p$-adic Galois representations $ρ_f$ and $\mod{p^n}$ reductions $ρ_{f,n}$. Under suitable hypotheses on the size of the image, we use deformation theory and modularity lifting to show that if the restrictions of $ρ_{f,n} $ to decomposition groups above $p$ split then $f$ has a companion form $g$ modulo $p^n$ (in the sense that $ρ_{f,n}\sim ρ_{g,n}\otimesχ^{k-1}$).

math.NT↗

arXiv2013

Modularity of certain mod $p^n$ Galois representations

For a rational prime $p \geq 3$ and an integer $n \geq 2$, we study the modularity of continuous 2-dimensional mod $p^n$ Galois representations of $\Gal(\bar{\Q}/\Q)$ whose residual representations are odd and absolutely irreducible. Under suitable hypotheses on the local structure of these representations and the size of their images we use deformation theory to construct characteristic 0 lifts. We then invoke modularity lifting results to prove that these lifts are modular. As an application, we show that certain unramified mod $p^n$ Galois representations arise from modular forms of weight $p^{n-1}(p-1)+1$.

math.NT↗

arXiv2012

A characterization of ordinary modular eigenforms with CM

For a rational prime $p \geq 3$ we show that a $p$-ordinary modular eigenform $f$ of weight $k\geq 2$, with $p$-adic Galois representation $\rho_f$, mod ${p^m}$ reductions $\rho_{f,m}$, and with complex multiplication (CM), is characterized by the existence of $p$-ordinary CM companion forms $h_m$ modulo $p^m$ for all integers $m \geq 1$ in the sense that $\rho_{f,m}\sim \rho_{h_m,m}\otimes\chi^{k-1}$, where $\chi$ is the $p$-adic cyclotomic character.

math.NT↗