Let p be a rational prime, and let X be a connected finite graph.In this article we study voltage covers X∞ of X attached to a voltage assignment α which takes values in some uniform p-adic Lie group G. We formulate and prove an Iwasawa main conjecture for the projective limit of the Picard groups Pic(Xn) of the intermediate voltage covers Xn, n ∈ N, and we prove one inclusion of a main conjecture for the projective limit of the Jacobians J(Xn).Moreover, we study the M H (G)-property of Zp G -modules and prove a necessary condition for this property which involves the µ-invariants of Zpsubcovers Y ⊆ X∞ of X. If the dimension of G is equal to 2, then this condition is also sufficient.
When
$p$
is an odd prime, Delbourgo observed that any Kubota–Leopoldt
$p$
-adic
$L$
-function, when multiplied by an auxiliary Euler factor, can be written as an infinite sum. We shall establish such expressions without restriction on
$p$
, and without the Euler factor when the character is non-trivial, by computing the periods of appropriate measures. As an application, we will reprove the Ferrero–Greenberg formula for the derivative
$L_p'(0,\chi )$
. We will also discuss the convergence of sum expressions in terms of elementary
$p$
-adic analysis, as well as their relation to Stickelberger elements; such discussions in turn give alternative proofs of the validity of sum expressions.
Given primes ℓ p, we record here a p-adic valued Fourier theory on a local field over Q ℓ , which is developed under the perspective of group schemes. As an application, by substituting rigid analysis for complex analysis, it leads naturally to the p-adic local functional equation at ℓ, which strongly resembles the complex one in Tate's thesis.
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