Topics in Local Langlands Correspondence for GL2(Qp) : Galois Orbits of Newforms and Extensions of p -adic Representations

Abstract

This thesis investigates questions arising from the arithmetic of modular forms, Galois representations, and the local Langlands correspondence. The first part studies the global arithmetic of modular forms through the Galois action on Hecke eigenforms, while the second part explores the local representation-theoretic aspects of the $p$-adic Langlands correspondence. In the first part, we establish a lower bound for the number of non-CM Galois orbits of newforms with non-trivial quadratic nebentypus for sufficiently large weights. Extending the work of Dieulefait, Pacetti, and Tsaknias in the trivial nebentypus setting, we analyze the restrictions imposed by the quadratic character on local inertial types and determine the number of admissible Galois orbits of such types. We further prove that Atkin-Li pseudo-eigenvalues are Galois equivariant and hence, up to a natural equivalence relation, define a global Galois invariant. Together with existence results for newforms having prescribed local behavior, these invariants yield a lower bound for the number of non-CM Galois orbits by counting compatible pairs of local-global invariants. Finally, computations in small weights show that this lower bound is not always attained, indicating that certain local equivalences are not realized globally by Galois conjugation over the coefficient field of the modular form. The second part of the thesis concerns the $p$-adic local Langlands correspondence. We compute extension groups in the category of duals of $p$-adic Banach space representations of $\mathrm{GL}_2(\mathbb{Q}_p)$. Focusing on representations arising from the $p$-adic local Langlands correspondence for generic Galois representations, we classify these extensions completely. These results are then applied to prove the vanishing of extensions between the dual $p$-adic Banach space representations attached to reducible Galois representations and supercuspidal Galois isotypic components of the $p$-adic etale cohomology of the finite-level Drinfeld spaces.