The Jacobian group (also known as the critical or sandpile group) is an important invariant of a finite, connected graph $X$; it finite abelian whose cardinality equal to number spanning trees $X$ (Kirchhoff's Matrix Tree Theorem). A specific type covering graph, called derived that constructed from voltage with $G$ object interest in this paper. Towers graphs are studied by using aspects class...

of the absolute Galois group of a number field F . Assume that ρ̄ is ordinary in the sense that the image of any decomposition group at a place v dividing p lies in some Borel subgroup Bv of G. Assume also that ρ̄ satisfies the conditions of [11, Section 7] which guarantee that it has a reasonable deformation theory; see Section 3.1 for details. In this paper we show that the Iwasawa invariants o...

Let $p\geq 5$ be a prime, and $\mathfrak{p}$ prime of $\bar{\mathbb{Q}}$ above $p$. $g_1$ $g_2$ $\mathfrak{p}$-ordinary, $\mathfrak{p}$-distinguished $p$-stabilized cuspidal newforms nebentype characters $\epsilon_1, \epsilon_2$ respectively, weight $k\geq 2$, whose associated have level to Assume that the residual representations at are absolutely irreducible isomorphic. Then, imprimitive $p$-...