Slope classicality in higher Coleman theory via highest weight vectors in completed cohomology

We give a proof of the slope classicality theorem in classical and higher Coleman theory for modular curves of arbitrary level using the completed cohomology classes attached to overconvergent modular forms. The latter give an embedding of the quotient of overconvergent modular forms by classical modular forms, which is the obstruction space for classicality in either cohomological degree, into a unitary representation of GL2(ℚp). The Up operator becomes a double coset, and unitarity yields slope vanishing.

Fix a sufficiently small compact open subgroup K p ≤ GL 2 (A (p) f ) and let C p be the completion of an algebraic closure of Q p . Let X 1 (p n )/C p be the smooth compactification of the modular curve parameterizing elliptic curves with a point of exact order p n and level K p structure. Everywhere below, we view X 1 (p n ) as an adic space over C p . The closed canonical ordinary locus X 1 (p n ) e is the topological closure of the locus of rank one points parameterizing elliptic curves of ordinary reduction equipped with a point generating the canonical subgroup of level p n . We write X 1 (p n ) w = X 1 (p n )\X 1 (p n ) e for its open complement (the subscripts e and w refer to the trivial and nontrivial elements of the Weyl group for GL 2 ).
Writing ω for the modular sheaf, the space H 0 (X 1 (p n ) e , ω k ) is naturally identified with the direct sum of spaces of overconvergent modular forms of weights κ such that κ = z k χ where χ is a character of (Z/p n Z) × . From the perspective of the higher Coleman theory of Boxer and Pilloni (1,2), it is natural to also consider the compactly supported cohomology H 1 c (X 1 (p n ) w , ω k ). These groups are related by the exact sequence of compactly supported topological sheaf cohomology arising from the following exact sequence of sheaves on the topological space X 1 (p n ) obtained via push-pull of ω k along the inclusions j : X 1 (p n ) w → X 1 (p n ) and i : X 1 (p n ) e → X 1 (p n ): As in refs. 1 and 2 (see also section 2.2 below), there is an operator U p on each of these spaces induced by a cohomological correspondence and extending a classical double-coset Hecke operator U p on H • (X 1 (p n ), ω k ) (up to matching choices of the normalization). For any s ∈ R and C p vector space V equipped with an action of a linear operator U p , we can pass to the part V <s of slope less than s, defined to be the span of all generalized eigenspaces of U p for eigenvalues λ with |λ| > p −s .
In cohomological degree zero, this is a result of Coleman (3,4). In degree one, this is a result of Boxer and Pilloni (1, 2) (who also reprove Coleman's result). * We give a short proof using the connection between overconvergent modular forms and the completed cohomology of modular curves established in refs. 5 and 6. This provides a perspective on a fundamental result in the p-adic theory of automorphic forms: We recall that Coleman's proof (in the degree zero case) is based on an analysis of the de Rham cohomology of modular curves and a clever dimension counting, while the proof of Boxer and Pilloni is based on slope estimates established via an analysis of cohomological correspondences and integral structures on coherent cohomology. Our proof, by contrast, proceeds by embedding the defect to classicality in completed cohomology so that the necessary slope estimates are a trivial consequence of unitarity, itself a trivial consequence of the construction of completed cohomology from integral singular cohomology (see Remark 1 for the origins of this approach in Emerton's classicality for Jacquet modules). This depends on strong nondegeneracy results of refs. 5 or 6, but the actual construction of the cohomology classes is completely explicit, so that our proof of Theorem 1 reduces to elementary matrix computations.

Proof of Theorem 1
Theorem 1 is an immediate consequence of Lemma 1 below, itself an immediate consequence of the results of refs. 5 or 6.
Let X be the infinite-level (compactified) modular curve of prime-to-p-level K p . It admits an action of GL 2 (Q p ) and, by Scholze's primitive comparison (see ref. 6, corollary 4.4.3), H 1 (X , O X ) is identified with the C p -completed cohomology of the tower of modular curves of prime-to-p-level K p . We need only that it is a Banach space with a unitary action of GL 2 (Q p ), which follows because the unit ball, that is, the image of

Proof of Theorem 1, assuming Lemma 1:
Consider . Because the action of GL 2 (Q p ) is unitary, it has operator norm ≤ 1/p |t| , so the slope < |t| part vanishes in H 1 (X , O X ) N (Zp ) . If we write then, combining the above with Lemma 1, we find that, for t = 0, Q <|t| t = 0. To obtain Theorem 1, we split Eq. 1 for k = 1 + t into two short exact sequences, Taking the slope < |t| part yields the isomorphisms-for the first sequence this is immediate, since this functor is always left exact, and, for the second sequence, right exactness follows from compactness of the U p operator on overconvergent forms. It thus remains only to prove Lemma 1. This is essentially immediate from the results of refs. 5 or 6, once the GL 2 (Q p ) actions are matched up. This matching is actually a bit subtle, as there are multiple possible conventions for the Hodge-Tate period map and the equivariant structure on the modular sheaf. Any set of choices gives the same GL 2 (Q p )-action modulo inverse transpose and some determinants, so, often, the precise choices are irrelevant. Here, however, we must follow a power of p coming from the action of diag(p, 1), so it is crucial to screw our heads on exactly right on this point. In the next section, we fix normalizations, then prove Lemma 1.

Choices.
We fix the action of GL 2 (Q p ) on X so that, over the noncompactified infinite-level curve Y , GL 2 (Q p ) = Aut(Q 2 p ) acts by composition with the trivialization of the Tate module of the universal elliptic curve; that is, we use the action on the homological normalization of the moduli problem. This differs by an inverse transpose from the cohomological normalization, where the action is on the trivialization of the firstétale cohomology of the universal elliptic curve.
We take the Hodge-Tate period map π HT : X → P 1 so that π HT | Y is the classifying map for the Hodge-Tate line inside of the firstétale cohomology of the universal elliptic curve. Thus, over Y , we have a GL 2 (Q p )-equivariant commuting diagram, Here e 1 and e 2 are the universal basis for the Tate module V p (E ) = T p (E )[1/p], x and y are the standard basis for H 0 (P 1 , O P 1 (1)) so that homogeneous coordinates are [x : y], and E ∨ denotes the dual of the universal elliptic curve. Of course, there is a canonical isomorphism E ∨ ∼ = E inducing ω E ∨ ∼ = ω E ; however, this isomorphism does not respect the natural GL 2 (Q p )-equivariant structures! Equivariantly, where, here, | det | comes from the action of isogenies on H 1 (E , Ω E ). Note that this twist is actually on the entire , so that the distinction between these equivariant structures also determines the normalization of the prime-to-p Hecke operators. Below we will continue, as in the introduction, to write simply ω for the modular sheaf, with the understanding that we have adopted the equivariant structure described above.
Under the natural map to X → X 1 (p n ), X 1 (p n ) e is the image of π −1 HT ([0 : 1]), and the action of GL 2 (Q p ) on x , y = H 0 (P 1 , O P 1 (1)) is by the standard representation a b c d · x = ax + cy and a b c d · y = bx + dy.

The U naive p
Operator. The operator U naive p at level X 1 (p n ) of refs. 1 and 2 is defined using the correspondence C parameterizing degree p isogenies ψ : (E 1 , P 1 ) → (E 2 , P 2 ) (here we suppress prime-to-p-level structure from the notation). Writing the two obvious projections as p 1 , p 2 : C → X 1 (p n ), U naive p is defined on ω k as tr • p 1! • ψ * • p * 2 . Given a geometric point (E , P ) that is not a cusp and a nonzero differential η on E , we can compute (U naive p f )(E , P , η) as follows: First, choose a basis (e 1 , e 2 ) of T p (E ) such that e 1 reduces to P mod p n . Then, for 0 ≤ i ≤ p − 1, write where e i denotes the image of We will now realize this same U naive p as a double-coset operator: Let B denote the upper triangular Borel in GL 2 . The space of overconvergent modular forms of weight k at any finite level Γ 1 (p n ) is naturally embedded as the B 1 (p n ) = Γ 1 (p n ) ∩ B (Q p ) invariants in the B (Q p ) representation M † k := H 0 ([0 : 1], (π HT * π * HT O(k )) sm ), where, here, the superscript sm denotes the subsheaf of π HT * π * HT O(k ) = π HT * ω k whose sections over a quasi-compact open V are those with locally constant orbit map for the action of the stabilizer of V in GL 2 (Q p ) (i.e., sections fixed by some compact open subgroup of GL 2 (Q p ), i.e., sections coming from finite level). For more on this construction, see ref. 5, section 3.1.