Pseudo-modularity and Iwasawa theory

We prove, assuming Greenberg's conjecture, that the ordinary eigencurve is Gorenstein at an intersection point between the Eisenstein family and the cuspidal locus. As a corollary, we obtain new results on Sharifi's conjecture. This result is achieved by constructing a universal ordinary pseudodeformation ring and proving an $R = \mathbb T$ result.Comment: Changes to section 5.9; typos corrected. To appear in Amer. J. Math. 54 page

Class groups and local indecomposability for non-CM forms

In the late 1990's, R. Coleman and R. Greenberg (independently) asked for a global property characterizing those $p$-ordinary cuspidal eigenforms whose associated Galois representation becomes decomposable upon restriction to a decomposition group at $p$. It is expected that such $p$-ordinary eigenforms are precisely those with complex multiplication. In this paper, we study Coleman-Greenberg's question using Galois deformation theory. In particular, for $p$-ordinary eigenforms which are congruent to one with complex multiplication, we prove that the conjectured answer follows from the $p$-indivisibility of a certain class group.Comment: 40 pages, with a 11-page appendix by Haruzo Hida. v3: improvements to exposition, minor correction

Families of Bianchi modular symbols: critical base-change p-adic L-functions and p-adic Artin formalism

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; let $f$ be a regular $p$-stabilised newform of weight $k$ at least 2 without CM by $K$. (1) 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 $f$. (2) We construct three-variable $p$-adic $L$-functions over the eigenvariety interpolating the $p$-adic $L$-functions of classical base-change Bianchi cusp forms in families. (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.Comment: 26 pages, with a 3 page appendix by Carl Wang-Erickson. Comments welcome! Changes for v5: added contents, minor changes to exposition. v4: corrected funding acknowledgements. v3: This version has a new introduction, has been reorganised and greatly shortened, and incorporates minor corrections. v2: minor correction

Moduli of Galois Representations

The theme of this thesis is the study of moduli stacks of representations of an associative algebra, with an eye toward continuous representations of profinite groups such as Galois groups. The central object of study is the geometry of the map $\bar{\psi}$ from the moduli stack of representations to the moduli scheme of pseudorepresentations. The first chapter culminates in showing that $\bar{\psi}$ is very close to an adequate moduli space of Alper. In particular, $\bar{\psi}$ is universally closed. The second chapter refines the results of the first chapter. In particular, certain projective subschemes of the fibers of $\bar{\psi}$ are identified, generalizing a suggestion of Kisin. The third chapter applies the results of the first two chapters to moduli groupoids of continuous representations and pseudorepresentations of profinite algebras. In this context, the moduli formal scheme of pseudorepresentations is semi-local, with each component Spf B_\bar{D} being the moduli of deformations of a given finite field-valued pseudorepresentation $\bar{D}$. Under a finiteness condition, it is shown that $\bar{\psi}$ is not only formally finite type over Spf B_\bar{D}, but arises as the completion of a finite type algebraic stack over Spec B_\bar{D}. Finally, the fourth chapter extends Kisin's construction of loci of coefficient spaces for p-adic local Galois representations cut out by conditions from p-adic Hodge theory. The result is extended from the case that the coefficient ring is a complete Noetherian local ring to the more general case that the coefficient space is a Noetherian formal scheme.Mathematic