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See detailModular forms and the inverse Galois problem for PSL_2( Z /p^n Z )
Adibhatla, Rajender UL

Presentation (2013, August 28)

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See detailA characterization of ordinary modular eigenforms with CM
Adibhatla, Rajender UL; Tsaknias, Panagiotis UL

in Arithmetic and Geometry (2013, July)

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See detailHigher companion forms via Galois deformation theory
Adibhatla, Rajender UL

Presentation (2013, May 13)

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See detailModularity of certain 2-dimensional mod p^n representations of Gal(Qbar/Q
Adibhatla, Rajender UL

Presentation (2013, March 07)

For an odd rational prime p and integer n>1, we consider certain continuous representations rho_n of G_Q into GL_2(Z/p^nZ) with fixed determinant, whose local restrictions "look" like they arise from ... [more ▼]

For an odd rational prime p and integer n>1, we consider certain continuous representations rho_n of G_Q into GL_2(Z/p^nZ) with fixed determinant, whose local restrictions "look" like they arise from modular Galois representations, and whose mod p reductions are odd and irreducible. Under suitable hypotheses on the size of their images, we use deformation theory to lift rho_n to rho in characteristic 0. We then invoke a modularity lifting theorem of Skinner-Wiles to show that rho is modular. [less ▲]

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See detailHigher congruence companion forms
Adibhatla, Rajender UL

in Acta Arithmetica (2012), 156(2), 17

For a rational prime p≥3 we consider p-ordinary, Hilbert modular newforms f of weight k≥2 with associated p-adic Galois representations \rho_f and mod p^n reductions \rho_{f,n}. Under suitable hypotheses ... [more ▼]

For a rational prime p≥3 we consider p-ordinary, Hilbert modular newforms f of weight k≥2 with associated p-adic Galois representations \rho_f and mod p^n reductions \rho_{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 \rho_{f,n} to decomposition groups above p split then f has a companion form g modulo pn (in the sense that \rho_{f,n} \sim \rho_{g,n}\otimes \chi^{k−1}). [less ▲]

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See detailModularity of certain mod p^n Galois representations
Adibhatla, Rajender UL

E-print/Working paper (2012)

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 $\Gal(\overline{\Q}/\Q)$ whose residual representations ... [more ▼]

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 $\Gal(\overline{\Q}/\Q)$ 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 $p^{n-1}(p-1)+1$. [less ▲]

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