Families of bianchi modular symbols: critical base-change p-adic
L-functions and p-adic Artin formalism
L-functions and p-adic Artin formalism
File(s)
Author(s)
Salazar, Daniel Barrera
Williams, Chris
Wang-Erickson, Carl
Type
Working Paper
Abstract
Let $K$ be an imaginary quadratic field. In this article, we study the
eigenvariety for $GL(2)/K$, proving an etaleness result for the weight map at non-critical classical points and a smoothness result at base-change classical points. We give three main applications of this. (1) We construct
three-variable $p$-adic $L$-functions over the eigenvariety interpolating the
(two-variable) $p$-adic $L$-functions of classical Bianchi cusp forms in
families. (2) Let $f$ be a $p$-stabilised newform of weight $k$ at least 2
without CM by $K$. We construct a two-variable $p$-adic $L$-function attached to the base-change of $f$ to $K$ under assumptions on $f$ that we conjecture always hold, in particular making no assumption on the slope of $f$. (3) We prove that these base-change $p$-adic $L$-functions satisfy a $p$-adic Artin formalism result, that is, they factorise in the same way as the classical $L$-function under Artin formalism.
In an appendix, Carl Wang-Erickson describes a base-change deformation
functor and gives a characterisation of its Zariski tangent space.
eigenvariety for $GL(2)/K$, proving an etaleness result for the weight map at non-critical classical points and a smoothness result at base-change classical points. We give three main applications of this. (1) We construct
three-variable $p$-adic $L$-functions over the eigenvariety interpolating the
(two-variable) $p$-adic $L$-functions of classical Bianchi cusp forms in
families. (2) Let $f$ be a $p$-stabilised newform of weight $k$ at least 2
without CM by $K$. We construct a two-variable $p$-adic $L$-function attached to the base-change of $f$ to $K$ under assumptions on $f$ that we conjecture always hold, in particular making no assumption on the slope of $f$. (3) We prove that these base-change $p$-adic $L$-functions satisfy a $p$-adic Artin formalism result, that is, they factorise in the same way as the classical $L$-function under Artin formalism.
In an appendix, Carl Wang-Erickson describes a base-change deformation
functor and gives a characterisation of its Zariski tangent space.
Date Issued
2018-09-24
Citation
2018
Identifier
http://arxiv.org/abs/1808.09750v2
Subjects
math.NT
math.NT
Notes
Currently submitted to American Journal of Mathematics. Whilst this has not been accepted, I have included it for consideration as I judge it to be both considerably more original and significant than my Canad. J. Math. paper 'P-adic L-functions for GL(2)'. With over two years until the REF deadline, I hope it will be accepted by a journal by then. In this paper, we study p-adic families of Bianchi modular forms (automorphic forms for GL(2) over imaginary quadratic fields) as the weight changes. In practice, this involves studying the 'Bianchi eigenvariety'. Previous research on eigenvarieties has largely focused on the cases of those attached to reductive groups that have attached Shimura varieties, which is not the case here. This means that almost all previous methods in their study does not apply in the Bianchi case, and we were forced to develop new techniques in this paper that should apply more generally. The paper contains two main technical innovations: firstly, a non-vanishing and etaleness result for the overconvergent Bianchi cohomology in families (S4), and secondly, a smoothness result at base-change points (S5, using the deformation theory contained in the appendix). This allowed us to deduce three main arithmetic applications: 1) the construction of three-variable p-adic L-functions interpolating classical constructions in families; 2) a construction of critical slope p-adic L-functions for base-change Bianchi forms; 3) a p-adic Artin formalism result, proving a factorisation of the p-adic L-functions of a base-change form. Results (1) and (3) will hopefully have consequences for the Iwasawa theory of base-change modular forms. Indeed, such results existed in the very specialised case of trivial weight ordinary forms, and were used in Skinner-Urban's proof of the Iwasawa main conjecture for elliptic curves.
Publication Status
Submitted