Iwasawa Theory of Symmetric Rankin-Selberg Products of Modular Forms and Applications

Artin formalism for motives dictates that if a motive M splits into two submotives as M = M1 ⊕ M2, then the corresponding L-functions factorize as L(M, s) = L(M1, s)L(M2, s). If a p-adic V representation of Gal(Q/Q) arises as a p-adic realization of such motive, then V can be written as V = V1 ⊕ V2 for some p-adic representation V1 and V2, thus we can rephrase this property in terms of p-adic Galois representations. Conjecturally, to a pure geometric p-adic representation V , one can attach a p-adic L-function, which interpolates L(V, s) for critical values s of V in the sense of Deligne. Then it is natural to ask if the factorization above could be generalized to the p-adic L-functions of these representations. In the case where V does not have any critical values, it is a non-trivial result. One could approach the problem by defining the p-adic L-function Lp(V ) as the product Lp(V1)Lp(V2), however, one then needs to make sense of why Lp(V ) is the “correct” p-adic L-function. For a newform f , one can attach a 2-dimensional Galois representation Vf , and a rank-4 representation Vf ⊗ Vf , which splits into a direct sum of a 3-dimensional representation Sym2(Vf ) and a 1-dimensional representation ∧2Vf . In this case, there exists no critical values for Vf ⊗ Vf , hence the factorization problem above becomes non-trivial. In [23], Dasgupta showed the expected factorization for the p-adic L-functions is true when f is p-ordinary. He used Hida’s 3-variable p-adic L-function attached to the Rankin-Selberg product of Hida families Lp(F, F, σ), which interpolates L(Vf ⊗ Vg, s) at certain weights, to make sense of Lp(Vf ⊗ Vf ), and utilized the Beilinson-Flach elements to show that the factorization holds. In [1], Arlandini and Loeffler generalized Dasgupta’s results to the p-non-ordinary setting, using Loeffler and Zerbes’ “geometric p-adic L-function” Lgeom p (F, F, σ), where F is a Coleman family passing through f . Iwasawa-Greenberg Main Conjecture for a p-adic representation V relates the p-adic L-function of V to a cohomological object attached to V , called Selmer group. The characteristic ideals of these Selmer groups are called algebraic p-adic L-functions. In [38], Palvannan showed that the factorization formula holds for the algebraic counterparts of the p-adic L-functions occurring in Dasgupta’s factorization. In this thesis, we investigate the factorization problem for the algebraic p-adic L-functions for certain twists of rank-4 adjoint representation of a Coleman family F over an affinoid algebra A passing through a p-non-ordinary newform f . To achieve this, we utilize Pottharst’s analytic Selmer complexes. We also prove a perfectness result for these Selmer complexes, which generalizes [19, Corollary 4.49] to our setting. At the end, we also discuss under which conditions the Selmer complex attached to certain twists of VF b⊗E V ∗ F specializes to the Selmer complex attached to Ad(VF ).