arXiv subjects
Publications and source records attributed to James Newton.
These are notes based on four lectures given at the Heidelberg spring school on non-archimedean geometry and eigenvarieties. None of the contents are original work. Our goal is to explain the construction of eigenvarieties in various different contexts, including the prototypical example of the Coleman--Mazur eigencurve. We will also discuss some of the common geometric properties of eigenvarieties.
We give a categorical formulation of the $p$-adic local Langlands correspondence for $\mathrm{GL}_2(\mathbb{Q}_p)$,as an embedding of the derived category of locally admissible representations into the category of Ind-coherent sheaves on the moduli stack of two-dimensional representations of $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$. Moreover, we relate our version of the $p$-adic local Langlands correspondence for $\mathrm{GL}_2(\mathbb{Q}_p)$ to the cohomology of modular curves through a local-global compatibility formula.
Let $f$ be a non-CM Hecke eigenform of weight $k \geq 2$. We give a new proof of some cases of Langlands functoriality for the automorphic representation $\pi$ associated to $f$. More precisely, we prove the existence of the base change lifting, with respect to any totally real extension $F / \mathbb{Q}$, of any symmetric power lifting of $\pi$.
We prove the Ramanujan and Sato-Tate conjectures for Bianchi modular forms of weight at least 2. More generally, we prove these conjectures for all regular algebraic cuspidal automorphic representations of $\mathrm{GL}_2(\mathbf{A}_F)$ of parallel weight, where $F$ is any CM field. We deduce these theorems from a new potential automorphy theorem for the symmetric powers of 2-dimensional compatible systems of Galois representations of parallel weight.
Let $ρ$ be the $p$-adic Galois representation attached to a cuspidal, regular algebraic automorphic representation of $\mathrm{GL}_n$ of unitary type. Under very mild hypotheses on $ρ$, we prove the vanishing of the (Bloch--Kato) adjoint Selmer group of $ρ$. We obtain definitive results for the adjoint Selmer groups associated to non-CM Hilbert modular forms and elliptic curves over totally real fields.
In this paper, we establish the modularity of every elliptic curve $E/F$, where $F$ runs over infinitely many imaginary quadratic fields, including $\mathbb{Q}(\sqrt{-d})$ for $d=1,2,3,5$. More precisely, let $F$ be imaginary quadratic and assume that the modular curve $X_0(15)$, which is an elliptic curve of rank $0$ over $\mathbb{Q}$, also has rank $0$ over $F$. Then we prove that all elliptic curves over $F$ are modular. More generally, when $F/\mathbb{Q}$ is an imaginary CM field that does not contain a primitive fifth root of unity, we prove the modularity of elliptic curves $E/F$ under a technical assumption on the image of the representation of $\mathrm{Gal}(\overline{F}/F)$ on $E[3]$ or $E[5]$. The key new technical ingredient we use is a local-global compatibility theorem for the $p$-adic Galois representations associated to torsion in the cohomology of the relevant locally symmetric spaces. We establish this result in the crystalline case, under some technical assumptions, but allowing arbitrary dimension, arbitrarily large regular Hodge--Tate weights, and allowing $p$ to be small and highly ramified in the imaginary CM field $F$.
Let $F$ be a totally real field. We prove the existence of all symmetric power liftings of those cuspidal automorphic representations of $\mathrm{GL}_2(\mathbf{A}_F)$ associated to Hilbert modular forms of regular weight.
Let $F$ be a CM number field. We prove modularity lifting theorems for regular $n$-dimensional Galois representations over $F$ without any self-duality condition. We deduce that all elliptic curves $E$ over $F$ are potentially modular, and furthermore satisfy the Sato--Tate conjecture. As an application of a different sort, we also prove the Ramanujan Conjecture for weight zero cuspidal automorphic representations for $\mathrm{GL}_2(\mathbf{A}_F)$.
Let $f$ be a cuspidal Hecke eigenform of level 1. We prove the automorphy of the symmetric power lifting $\mathrm{Sym}^n f$ for every $n \geq 1$. We establish the same result for a more general class of cuspidal Hecke eigenforms, including all those associated to semistable elliptic curves over $\mathbb{Q}$.
We establish the existence of the symmetric power liftings of all holomorphic Hecke eigenforms.
We investigate local-global compatibility for cuspidal automorphic representations $π$ for GL(2) over CM fields that are regular algebraic of weight $0$. We prove that for a Dirichlet density one set of primes $l$ and any $ι: \overline{\mathbf{Q}}_l \cong \mathbf{C}$, the $l$-adic Galois representation attached to $π$ and $ι$ has nontrivial monodromy at any $v \nmid l$ in $F$ at which $π$ is special.
Recently, Andreatta, Iovita and Pilloni have constructed spaces of overconvergent modular forms in characteristic p, together with a natural extension of the Coleman-Mazur eigencurve over a compactified (adic) weight space. Similar ideas have also been used by Liu, Wan and Xiao to study the boundary of the eigencurve. This all goes back to an idea of Coleman. In this article, we construct natural extensions of eigenvarieties for arbitrary reductive groups G over a number field which are split at all places above p. If G is GL(2)/Q, then we obtain a new construction of the extended eigencurve of Andreatta-Iovita-Pilloni. If G is an inner form of GL(2) associated to a definite quaternion algebra, our work gives a new perspective on some of the results of Liu-Wan-Xiao. We build our extended eigenvarieties following Hansen's construction using overconvergent cohomology. One key ingredient is a definition of locally analytic distribution modules which permits coefficients of characteristic p (and mixed characteristic). When G is GL(n) over a totally real or CM number field, we also construct a family of Galois representations over the reduced extended eigenvariety.
We prove new automorphy lifting theorems for residually reducible Galois representations of unitary type in which the residual representation is permitted to have an arbitrary number of irreducible constituents.
Under an assumption on the existence of p-adic Galois representations, we carry out Taylor--Wiles patching (in the derived category) for the completed homology of the locally symmetric spaces associated to GL(n) over a number field. We use our construction to show that standard conjectures on completed homology imply `big R = big T' theorems. In the case that n=2 and p splits completely in the number field, we relate our construction to the p-adic local Langlands correspondence for GL(2,Q_p).
Let F be a totally real field of degree d and let p be an odd prime which is totally split in F. We define and study one-dimensional partial eigenvarieties interpolating Hilbert modular forms over F with weight varying only at a single place v above p. For these eigenvarieties, we show that methods developed by Liu, Wan and Xiao apply and deduce that, over a boundary annulus in weight space of sufficiently small radius, the partial eigenvarieties decompose as a disjoint union of components which are finite over weight space. We apply this result to prove the parity version of the Bloch--Kato conjecture for finite slope Hilbert modular forms with trivial central character (under some assumptions), by reducing to the case of parallel weight 2. As another consequence of our results on partial eigenvarieties, we show, still under the assumption that p is totally split in F, that the full (dimension 1 + d) cuspidal Hilbert modular eigenvariety has the property that many (all, if d is even) irreducible components contain a classical point with non-critical slopes and parallel weight 2 (with some character at p whose conductor can be explicitly bounded), or any other algebraic weight.
We study the basic geometry of a class of analytic adic spaces that arise in the study of the extended (or adic) eigenvarieties constructed by Andreatta--Iovita--Pilloni, Gulotta and the authors. We apply this to prove a general interpolation theorem for Langlands functoriality, which works for extended eigenvarieties and improves upon existing results in characteristic 0. As an application, we show that the characteristic p locus of the extended eigenvariety for GL(2)/F, where F is a cyclic extension of the rational numbers Q, contains non-ordinary components of dimension at least [F:Q].
We show that local-global compatibility (at split primes) away from $p$ holds at all points of the $p$-adic eigenvariety of a definite $n$-variable unitary group. The novelty is we allow non-classical points, possibly non-étale over weight space. More precisely we interpolate the local Langlands correspondence for GL(n) across the eigenvariety by considering the fibers of its defining coherent sheaf. We employ techniques of Scholze from his new approach to the local Langlands conjecture.
Let p be a rational prime and N a positive integer which is prime to p. Let W be the p-adic weight space for GL_{2,Q}. Let C_N be the p-adic Coleman-Mazur eigencurve of tame level N. In this paper, we prove that any irreducible component of C_N which is of finite degree over W is in fact finite over W. Combined with an argument of Chenevier and a conjecture of Coleman-Mazur-Buzzard-Kilford (which has been proven in special cases, and for general quaternionic eigencurves) this shows that the only finite degree components of the eigencurve are the ordinary components.