A modular proof of the properness of the Coleman-Mazur eigencurve

The Coleman-Mazur eigencurve $\mathscr{E}$ is a rigid analytic space parametrizing overconvergent $p$-adic modular Hecke eigenforms with nonzero $U_{p}$-eigenvalues. It has a projection map $w:\mathscr{E}\to\mathscr{W}$, where $\mathscr{W}$ is the weight space parametrizing continuous characters of $\mathbb{Z}_{p}^{\times}$. It was constructed by Coleman and Mazur in [9] for $p>2$ and tame level $1$, and generalized to all primes and tame levels by Buzzard in [4].

In their original 1998 paper, Coleman and Mazur asked whether there exist $p$-adic analytic families of overconvergent eigenforms of finite slope parametrized by a punctured disc, converging at the puncture to an overconvergent eigenform of infinite slope. Buzzard and Calegari suggested in [6] that this can be reframed as the question of whether $w:\mathscr{E}\to\mathscr{W}$ satisfies the valuative criterion for properness, and proved by explicit means that the eigencurve is indeed “proper” in this sense for $p=2$ and tame level $1$. (Note that $w:\mathscr{E}\to\mathscr{W}$ has infinite degree, so is not proper in the “usual” sense.) Calegari in [7] generalized the Buzzard-Calegari argument to show that the eigencurve is proper at algebraic weights for all primes and tame levels.

The question was eventually resolved by Diao and Liu, who proved in 2014 ([12]) that the eigencurve is indeed proper. Their proof is completely different from the method of Buzzard-Calegari and Calegari. It proceeds by analyzing families of Galois representations over the eigencurve, relying on deep, powerful Galois- and Hodge- theoretic machinery developed by Berger-Colmez, Kedlaya-Liu, Bellovin, Kedlaya-Pottharst-Xiao, Liu, and others.

In this paper, we give a new proof of the theorem of Diao-Liu based on the original method of Buzzard and Calegari. Our proof is short and explicit, and it uses no Galois theory, instead relying only on elementary properties of overconvergent modular forms. A precise statement of the theorem follows.

We now describe our method. It suffices to prove the statement over the center disc of $\mathscr{W}$, consisting of analytic weights (i.e. characters of $\mathbb{Z}_{p}^{\times}$ that are analytic on $l+p\mathbb{Z}_{p}$ for any integer $l$). This is because the geometry of $\mathscr{E}$ over the remainder (“boundary annulus”) of $\mathscr{W}$ is relatively simple. In particular, by Theorem 1.2.1 of [20], in that region, $\mathscr{E}$ decomposes into a countable disjoint union of pieces that are finite over $\mathscr{W}$. (For philosophical completeness, we remark that Ren and Zhao’s proof of this theorem is also based on concrete computation with no use of Galois theory.) On the other hand, the geometry of $\mathscr{E}$ over the center disc of $\mathscr{W}$ is expected to be very complicated.

Now the argument of Buzzard-Calegari and Calegari for algebraic weights can be summarized as follows. First, it is standard that a finite-slope eigenform of any overconvergence radius analytically continues to a finite-slope eigenform of “large” overconvergence radius. Next, it is not too hard to check that the family $h:D^{\times}\to\mathscr{E}$ of finite-slope overconvergent eigenforms extends to an overconvergent eigenform $\tilde{h}(0)$ at the puncture, also of “large” overconvergence radius; the question is whether $\tilde{h}(0)$ is also finite-slope. Finally, one shows that $U_{p}$ is in fact injective on the space of forms of “large” overconvergence radius, so indeed $U_{p}\tilde{h}(0)\neq 0$.

The limitation of this argument is its reliance on Katz’s geometric definition of an overconvergent modular form of algebraic weight $k$ as a section of the line bundle $\omega^{k}$ on a certain rigid subset of the modular curve $X:=X_{1}(N)$, along with Coleman’s ad hoc definition of an overconvergent modular form of general weight as a Katz overconvergent form of weight $0$ multiplied by the $p$-deprived Eisenstein series of the desired weight. In order to generalize it, we need to be able to write overconvergent modular forms of arbitrary weight as sections of line bundles on modular curves.

Fortunately, Pilloni provides such a definition in [19], constructing a weight-$w$ line bundle $\omega^{w}$ on a suitable rigid subset of $X_{1}(N)$ for every weight $w$. (Andreatta, Iovita, and Stevens independently constructed similar sheaves using a different method in [1].) However, the subset on which Pilloni defines $\omega^{w}$ is not quite large enough: for $w$ analytic, $\omega^{w}$ is defined on the locus where the Hasse invariant has valuation less than $\frac{p-1}{p}$, whereas Buzzard and Calegari require the forms they work with to be well-defined even on elliptic curves that are “too supersingular” (meaning they have Hasse invariant of valuation at least $\frac{p}{p+1}$).

In light of this, we begin by constructing extensions of Pilloni’s line bundles on higher-level modular curves of the form $X_{0}(p^{m}):=X(\Gamma_{0}(p^{m})\cap\Gamma_{1}(N))$ (which parametrize tuples $(E,\psi_{N},C^{m})$ where $E$ is an elliptic curve, $\psi_{N}$ is a tame level structure, and $C^{m}$ is a cyclic subgroup of $E$ of order $p^{m}$). These extensions are well-defined in particular over the closure of the locus where $C^{m}[p]$ is canonical, which is sufficient for the application to properness, but in fact go even farther. For $v\in[0,1]$, one can define rigid subsets $X_{0}(p^{m})[0,v]$ of $X_{0}(p^{m})$ such that $X_{0}(p^{m})[0,v]\subset X_{0}(p^{m})[0,v^{\prime}]$ if $v<v^{\prime}$, $X_{0}(p^{m^{\prime}})[0,v]$ is the preimage of $X_{0}(p^{m})[0,v]$ under the forgetful projection if $m^{\prime}>m$, and $\bigcup_{v<1}X_{0}(p^{m})[0,v]$ contains, among other things, the entire locus where $E$ is supersingular. Then we have the following.

While we do not need such aggressive extensions to prove properness, we hope that they may be of independent interest for other applications. For algebraic weights, Buzzard has used analytic continuations of overconvergent forms over the supersingular locus to establish modularity of Galois representations ([3]).

Once we can work on the closure of the canonical locus of $X_{0}(p^{m})$, we can use Buzzard and Calegari’s method; in fact, in Pilloni’s setup, the argument becomes even cleaner than theirs was originally. In particular, we have the following result about the injectivity of $U_{p}$.

Understanding these linear-algebraic properties of $U_{p}$ on spaces of forms of large overconvergence radius is useful for many reasons. The Newton polygon upper bounds in Liu-Wan-Xiao’s proof of the spectral halo conjecture for definite quaternion algebras over $\mathbb{Q}$ ([17]) and the author’s analysis of slopes in eigenvarieties for definite unitary groups of arbitrary rank ([21]) rely on checking that $U_{p}$ is nonzero on certain classical subspaces. Calegari has conjectured (see [8]) the much stronger statement that in some cases $U_{p}$ admits a convergent spectral expansion (this has only been proven for $p=2$, $N=1$, and $w=0$, by Loeffler in [18]).

We hope that our new proof of the properness of the eigencurve along with its various intermediate results may expand the range of available techniques for analyzing the geometry of eigenvarieties. Note that Hattori [14] has successfully applied the Buzzard-Calegari method to some Hilbert modular eigenvarieties over algebraic weights. The author hopes to extend these methods to the question of properness for these or other higher-dimensional eigenvarieties in the future.

Verifying properness in other situations would likely have interesting additional implications for the structure of $p$-adic eigenforms. One has to be cautious here; it is not true, for example, that limits of sequences of finite-slope modular forms must be finite-slope—counterexamples are constructed by Coleman and Stein in [10]. However, properness can be used to show, as Hattori and Newton do in [15], that an irreducible component of the Coleman-Mazur eigencurve of finite degree over weight space must actually be finite.

In Section 2, we construct our extended sheaves of overconvergent modular forms. These sheaves are given by functions on spaces of images of Hodge-Tate maps on Tate modules of elliptic curves. In Section 2.1, we compute the images of these Hodge-Tate maps. In Section 2.2, we construct the rigid spaces corresponding to these images. In Section 2.3, we define and describe the admissible open subsets over which our invertible sheaves will be defined. In Section 2.4, we construct our sheaves and show that they are invertible. In Section 2.5, we construct the $U_{p}$ operator on our extended overconvergent forms.

In Section 3, we prove Theorem 1.1.1. In Section 3.1, we prove that infinite-slope forms must have relatively small overconvergence radius, whereas finite-slope forms must have relatively large overconvergence radius. In Section 3.2, we complete the proof by showing that families of finite-slope forms of large overconvergence radius over a punctured disc extend to a form of large overconvergence radius over the puncture.

I am very grateful to Richard Taylor for regular discussions and guidance on this topic, and to Vincent Pilloni for suggesting the method of extending sheaves of overconvergent modular forms by renormalizing the Hodge-Tate map. I would also like to thank Rebecca Bellovin, George Boxer, Kevin Buzzard, Frank Calegari, Mark Kisin, Joe Kramer-Miller, James Newton, James Upton, and Daqing Wan for valuable conversations about the Coleman-Mazur eigencurve.

This work was done during the support of the National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship.

Let $E$ be an elliptic curve over $\mathcal{O}_{\mathbb{C}_{p}}$, $e:\operatorname{spec}\mathcal{O}_{\mathbb{C}_{p}}\to E$ its identity, $\omega_{E}=e^{*}\Omega_{E/\mathcal{O}_{\mathbb{C}_{p}}}^{1}$ its space of invariant differentials, and $T_{p}E=\varprojlim_{n}E[p^{n}](\mathcal{O}_{\mathbb{C}_{p}})$ its Tate module (note that $E[p^{n}](\mathcal{O}_{\mathbb{C}_{p}})=E[p^{n}](\mathbb{C}_{p})$ since $E[p^{n}]$ is finite flat over $\operatorname{spec}\mathcal{O}_{\mathbb{C}_{p}}$). Let $HT:T_{p}E\to\omega_{E}$ be the Hodge-Tate map, defined as follows: if $x=(\dotsc,x_{n},x_{n-1},\dotsc,x_{1})\in T_{p}E$ (where $x_{n}\in E[p^{n}](\mathcal{O}_{\mathbb{C}_{p}})$ and $x_{n-1}=px_{n}$), let $x^{\vee}:E[p^{\infty}]\to\mu_{p^{\infty}}$ be the map taking $P\in E[p^{n}]$ to the Weil pairing of $x_{n}$ and $P$ (also defined over $\mathcal{O}_{\mathbb{C}_{p}}$); then

(The reader is warned that we have inserted a dualization into the usual definition of this map, because for our purposes it looks cleaner this way.)

Let $v$ be the standard valuation on $\mathbb{C}_{p}$ (so $v(p)=1$). We will compute $v(HT(x))$ for any $x\in T_{p}E\setminus pT_{p}E$. Let $h(E)$ be the valuation of the Hasse invariant of $E$ if that valuation lies in $[0,1]$, and $1$ otherwise. If $h(E)>0$, let $n(E)\in\mathbb{Z}_{\geq 0}$ be such that $\frac{1}{p^{n(E)-1}(p+1)}\leq h(E)<\frac{1}{p^{n(E)-2}(p+1)}$; if $h(E)=0$, let $n(E)=\infty$.

Let $n=n(E)$; recall that if $n\geq 1$, then $E$ has canonical subgroups $H_{can}^{1}\subset\dotsc\subset H_{can}^{n}$, where $H_{can}^{i}$ has order $p^{i}$, defined as follows. $H_{can}^{1}$ is the subgroup of order $p$ defined by Lubin and Katz in Chapter 3 of [16], and once $H_{can}^{i-1}$ has been defined, $H_{can}^{i}$ is the preimage of $H_{can}^{i-1}$ under the isogeny $E\mapsto E/H_{can}^{1}$. When $E$ is ordinary, $H_{can}^{1}$ is the unique lift of the kernel of Frobenius on the reduction of $E$, and $H_{can}^{n}$ exists for all $n$.

In order to make this computation, we first need to understand the $p$-torsion points of the completion $\widehat{E}$ of $E$ at the identity. Recall that $\widehat{E}$ is associated to a formal group over $\mathcal{O}_{\mathbb{C}_{p}}$ of height $2$ if $E$ is supersingular and $1$ if $E$ is ordinary, and that we have an embedding $\widehat{E}[p^{\infty}](\mathcal{O}_{\mathbb{C}_{p}})\hookrightarrow E[p^{\infty}](\mathcal{O}_{\mathbb{C}_{p}})$ which is an isomorphism if $E$ is supersingular and an isomorphism onto the canonical subgroup if $E$ is ordinary.

We will compute the valuation $v(P)$ of a point $P\in\widehat{E}[p^{\infty}](\mathcal{O}_{\mathbb{C}_{p}})$, as defined in Section 1.2.1 of [13]: one chooses an isomorphism $\widehat{E}\xrightarrow{\sim}\operatorname{Spf}(\mathcal{O}_{\mathbb{C}_{p}}\llbracket T\rrbracket)$ taking the identity section of $\widehat{E}$ to $T=0$, and defines $v(P)$ to be the valuation of the $T$-coordinate $T(P)$ of the image of $P$ under this isomorphism. The choice of isomorphism does not matter because a ring automorphism of $\mathcal{O}_{\mathbb{C}_{p}}\llbracket T\rrbracket$ fixing the ideal $(T)$ must fix valuations of $T$-coordinates.

When convenient, we will also use $P$ to refer to the image of $P$ in $E[p^{\infty}](\mathcal{O}_{\mathbb{C}_{p}})$.

With this calculation in hand, we may apply formulas from [13] to compute the values of $v(HT)$ asserted in Proposition 2.1.1.

Let $\widehat{\mathcal{E}}$ be a formal model for $E$ over $\mathcal{O}_{\mathbb{C}_{p}}$. Let $\omega_{E}^{int}$ be the pullback of the sheaf of integral differential forms on $\widehat{\mathcal{E}}$, so that $\omega_{E}^{int}\cong\mathcal{O}_{\mathbb{C}_{p}}$. Let $\omega_{E}^{+}$ be the $\mathcal{O}_{\mathbb{C}_{p}}$-submodule of $\omega_{E}^{int}$ generated by the image of $HT:T_{p}E\to\omega_{E}^{int}$. It will be useful to work with $\omega_{E}^{+}$ while constructing the fiber bundle of unions of balls on which our forms will be defined.