arXiv subjects

Muhammad Manji

Publications and source records attributed to Muhammad Manji.

3 recordsLinked to original sources

arXiv2026

Families of symplectic Galois representations over small parabolic eigenvarieties for Siegel cuspforms of genus $2$

We construct small parabolic eigenvarieties for holomorphic Siegel cuspforms of genus $2$ and study families of Galois representations attached to them in the spirit of Bella\"iche--Chenevier. In the course, we introduce the notion of $(\varphi, \Gamma)$-modules with $G$-structures and the notion of refined families of symplectic Galois representations by implementing the theory of symplectic Galois determinant d'apr\`es Moakher--Quast. Such families of symplectic Galois representations provide two applications: In the first application, we show that the small parabolic eigenvarieties are smooth at non-critical points by proving an infinitesimal $R=\mathbb{T}$ theorem. In the second application, we study the relationship between the geometry of the small parabolic eigenvarieties at the Saito--Kurokawa lifts for cuspidal eigenforms (both finite- and infinite-slope) and the Bloch--Kato Selmer groups of those eigenforms.

math.NT↗

arXiv2024

Iwasawa Theory for GU(2,1) at inert primes

Many problems of arithmetic nature rely on the computation or analysis of values of $L$-functions attached to objects from geometry. Whilst basic analytic properties of the $L$-functions can be difficult to understand, recent research programs have shown that automorphic $L$-values are susceptible to study via algebraic methods linking them to Selmer groups. Iwasawa theory, pioneered first by Iwasawa in the 1960s and later Mazur and Wiles provides an algebraic recipe to obtain a $p$-adic analogue of the $L$-function. In this work we aim to adapt Iwasawa theory to a new context of representations of the unitary group GU(2,1) at primes inert in the respective imaginary quadratic field. This requires a novel approach using the Schneider--Venjakob regulator map, working over locally analytic distribution algebras. Subsequently, we show vanishing of some Bloch--Kato Selmer groups when a certain $p$-adic distribution is non-vanishing. These results verify cases of the Bloch--Kato conjecture for GU(2,1) at inert primes in rank 0.

math.NT↗

arXiv2022

Euler Systems and Selmer Bounds for GU(2,1)

We investigate properties of the Euler system associated to certain automorphic representations of the unitary similitude group GU(2,1) with respect to an imaginary quadratic field $E$, constructed by Loeffler-Skinner-Zerbes. By adapting Mazur and Rubin's Euler system machinery we prove one divisibility of the ``rank 1" Iwasawa main conjecture under some mild hypotheses. When $p$ is split in $E$ we also prove a ``rank 0" statement of the main conjecture, bounding a particular Selmer group in terms of a $p$-adic distribution conjecturally interpolating complex $L$-values. We then prove descended versions of these results, at integral level, where we bound certain Bloch--Kato Selmer groups. We will also discuss the case where $p$ is inert, which is a work in progress.

math.NT↗