A unipotent circle action on $p$-adic modular forms

Following a suggestion of Peter Scholze, we construct an action of $\widehat{\mathbb{G}_m}$ on the Katz moduli problem, a profinite-\'{e}tale cover of the ordinary locus of the $p$-adic modular curve whose ring of functions is Serre's space of $p$-adic modular functions. This action is a local, $p$-adic analog of a global, archimedean action of the circle group $S^1$ on the lattice-unstable locus of the modular curve over $\mathbb{C}$. To construct the $\widehat{\mathbb{G}_m}$-action, we descend a moduli-theoretic action of a larger group on the (big) ordinary Igusa variety of Caraiani-Scholze. We compute the action explicitly on local expansions and find it is given by a simple multiplication of the cuspidal and Serre-Tate coordinates $q$; along the way we also prove a natural generalization of Dwork's equation $\tau=\log q$ for extensions of $\mathbb{Q}_p/\mathbb{Z}_p$ by $\mu_{p^\infty}$ valid over a non-Artinian base. Finally, we give a direct argument (without appealing to local expansions) to show that the action of $\widehat{\mathbb{G}_m}$ integrates the differential operator $\theta$ coming from the Gauss-Manin connection and unit root splitting, and explain an application to Eisenstein measures and $p$-adic $L$-functions.


Introduction and analogy
In this work, following a suggestion of Peter Scholze, we descend the unipotent quasi-isogeny action on a component Caraiani-Scholze's [1, Section 4] ordinary (big) Igusa formal scheme for GL 2 to construct an action of the formal p-adic torus G m on the Katz moduli problem over the ordinary locus. Suitably interpreted, this action is a local, p-adic analog of the global, archimedean phenomena whereby the horizontal translation action of R on the complex upper half plane H descends to an action of S 1 on the image of {Imτ > 1} ⊂ H in the complex modular curve.
The space of functions on the Katz moduli problem that are holomorphic at the cusps is equal the completion of classical modular forms for the q-expansion topology (Serre's space of p-adic modular functions 1 ). Thus we may think of our G m -action as a unipotent circle action on p-adic modular functions. The analogy with the archimedean circle action is stronger than one might first guess, and leads, e.g., to interesting representation-theoretic consequences.
After constructing the G m -action, we study its properties and interaction with other classical notions in the p-adic theory of modular curves and modular forms such as the unit root splitting, Dwork's equation τ = log q, the differential operator θ, Gouvea's twisting measure, and Katz's Eisenstein measures.
We highlight one application to explain the significance of this construction: Via p-adic Fourier theory, the G m -action is equivalent to the p-adic interpolation of powers of polynomials in the differential operator θ. This allows us to introduce a twisting direction into any p-adic family of modular forms. In particular, when applied to Eisenstein series, it allows the construction of certain two-variable p-adic L-functions studied by Katz [9] starting from single-variable Eisenstein families. These p-adic L-functions are not new, but the method of construction by first interpolating the differential operators into a group action is original to this work.
A key advantage of this method is that we construct the G m -action and then relate it to differential operators obtained from the Gauss-Manin connection without ever using a cuspidal or Serre-Tate ordinary local expansion. In particular, we expect that our method will generalize to the p-adic interpolation of certain differential operators constructed by Eischen and Mantovan [3] on the µ-ordinary locus of more general PEL Shimura Varieties (where local expansions are unavailable or difficult to work with) into actions of Lubin-Tate formal groups.
In the present work we have focused on exploring the ramifications of the existence of a large quasi-isogeny action on the Caraiani-Scholze Igusa formal scheme for the classical space of Katz/Serre p-adic modular functions. In a sequel [5], we study the action of the quasi-isogeny group on the space of functions on the big Igusa formal scheme itself as a natural space of p-adic automorphic forms in the context of the p-adic Langlands program. Ordinary p-adic modular forms (in the sense of Hida) play an important role in this study, and in [5] we also explain how Hida's finiteness and classicality theorems for ordinary p-adic modular forms can be understood from this perspective. clarify the p-adic constructions that follow. Consider the complex manifold Two important observations about Y ∞−ord follow immediately from (1.1.0.1): (1) Modular forms of level Γ 1 (N ) (for any N ) restrict to 1 Z 0 1 -invariant functions {Imτ > 1}, and thus induce holomorphic functions on Y ∞−ord . (2) The action of R by horizontal translation on H descends to a (real analytic) action of the circle group S 1 on Y ∞−ord . This action integrates the vector field d dτ . We can decompose any holomorphic function f on Y ∞−ord according to this S 1 action uniquely as a Fourier series f (q) = n∈Z a n q n , q = e 2πiz .
In other words, the space of functions on Y ∞−ord is a Frechet completion of the direct sum of the character spaces for the S 1 -action, with each character appearing exactly once.
1.1.1. Fourier coefficients and representation theory. The Fourier coefficients a n of classical modular forms play an important role in the global automorphic representation theory for GL 2 . In particular, for a Hecke eigenform, the constant coefficient a 0 is non-vanishing if and only if the corresponding global automorphic representation is globally induced (i.e. the modular form is Eisenstein). Suitably interpreted, the constant term a 0 is a functional that realizes the induction. The non-constant coefficients, on the other hand, are Whittaker functionals.
1.1.2. The slope formalism on metrized tori. While the construction of Y ∞−ord above may at first seem ad hoc, it has a natural moduli interpretation, which we explain now. The key point is to use the slope formalism for metrized tori, or, equivalently, lattices, as explained, e.g., in Casselman's survey [2].
A metrized torus is a finite dimensional torus (compact real abelian Lie group) T together with a translation invariant metric, or, equivalently, a positive definite inner product on LieT ∼ = H 1 (T, R). There is a natural slope formalism on metrized tori: the rank function is dimension, and the degree function is given by deg T := logVol(T ).
If a two-dimensional metrized torus T is unstable (i.e., not semi-stable), then it contains a unique circle of shortest length.
If E/C is an elliptic curve, the underlying real manifold of E(C) is a twodimensional metrized torus when equipped with the metric coming from the canonical principal polarization. We compute the values of τ ∈ D for which E τ is semistable: The metric induced by the principal polarization is identified with 1/Imτ times the metric induced by the identity R1 + Rτ = C and the standard metric on C. Semistability is preserved by scaling the metric, so we may eliminate the scaling and consider just the metric induced by the standard metric on C. The length of a shortest circle in E τ (C) is equal to the length of a shortest vector in H 1 (E τ , Z), which is 1. The area of the entire torus E τ (C), on the other hand, is Imτ . Thus, the slope of the full torus is 1 2 log Imτ , while the smallest slope of a circle inside is 0. We conclude that for τ ∈ D, E τ is semi-stable when Imτ ≤ 1, and otherwise is unstable with shortest circle given by 1.1.4. Moduli of unstable elliptic curves. Using the slope formalism, we may consider the moduli space of unstable elliptic curves E/C equipped with a trivialization of the shortest circle, S 1 ֒→ E(C). From Example 1.1.3, we find that this space is naturally identified with Y ∞−ord . In this moduli interpretation, the space Imτ > 1 is the cover where the trivialization of the shortest circle is extended to an oriented trivialization . From the moduli perspective, the fact that we can evaluate modular forms to obtain functions on Y ∞−ord comes from two facts: (1) Given a point of Y ∞−ord , there is a unique holomorphic differential ω can whose pullback to S 1 along the trivialization of the shortest circle integrates to 1. Thus, the modular sheaf ω is canonically trivialized over Y ∞−ord , and modular forms can be evaluated along this trivialization,. (2) Using the polarization, the trivialization of the shortest circle also gives rise to a trivialization of the quotient torus E(C)/S 1 , so that E(C) is equipped with the structure of an extension of real tori The basis 1/N for the torsion on S 1 = R/Z then gives rise to a canonical Γ 1 (N )-level structure on E for any level N .
which is flat, we find that ∇(ω can ) is in the span of u can . Thus, ∇ωcan ucan is a holomorphic differential form on Y ∞−ord . For the elliptic curve E τ as in Example 1.1.3, if we denote by e 1 and e τ the natural basis elements for H 1 (E(C), Z) and by e * 1 and e * τ the dual basis, we find that ω can = e * 1 + τ e * τ , and u can = e * τ , so that ∇ω can u can = dτ = d log q.
In particular, the S 1 action integrates the vector field d dτ dual to ∇ωcan ucan .
where E/SpecS is an elliptic curve up to prime-to-p isogeny, ϕ is a trivialization of the formal group of E, f is a trivialization of the adelic prime-to-p Tate module.
By work of Katz [8], the moduli problem is represented by a p-adic formal scheme where V Katz is a p-adically complete flat (torsion-free) Z p -algebra. For a R a padically complete Z p -algebra, we write There is a natural moduli action of is a natural space of p-adic modular forms of weight κ; in particular, classical modular forms of integral weight and prime-to-p level are embedded GL 2 (A (p) f )-equivariantly (up to a twist) in this space for the character z → z k . The embedding is given by evaluation on the trivialization of the modular sheaf ω given by the canonical differential for the universal elliptic curve up-to-prime-to-p-isogeny. We have the relative de Rham cohomology and Gauss-Manin connection ∇. Note that the moduli problem classified by I Katz is equivalent to the moduli problem classifying triples (E, ϕ, α) where E and ϕ are before, and α is a trivialization of the prime-to-p Tate module all considered up to isomorphism of E. Using this equivalence, we obtain a welldefined Weil pairing on E[p ∞ ], and combining this with the trivialization ϕ, we obtain the structure of an extension This is analogous to the archimedean extension (1.1.4.1); in particular, pulling back via the map E[p ∞ ] → Q p /Z p and using the the crystalline-de Rham comparison, we obtain a canonical flat section u can in H 1 dR (E univ ) (spanning the unit-root filtration). Together ω can and u can are a basis for H 1 dR (E univ ), and in this basis ∇ is lower nilpotent and thus is determined by a single differential form By the theory of Kodaira-Spencer, the differential form dτ is non-vanishing, and thus admits a dual vector field d dτ such that dτ, d dτ = 1.
Theorem A. There is an action of G m on I Katz whose derivative is the vector field d dτ defined above. Moreover, this action combines with the action of f )) where the semi-direct product is formed with the respect to the conjugation action (z, g) · ζ · (z, g) −1 = ζ z 2 detur(g) . Remark 1.2.5. The G m -action of Theorem A is uniquely determined by the condition that it integrate d dτ . We note that d dτ acts as the derivation −θ = −q d dq on cuspidal q-expansions; however, in our proof we construct the action and prove it integrates d dτ without using cuspidal (or Serre-Tate) q-expansions, which is an important point for future generalizations.
The key observation in the construction of this G m -action and subsequent computations is that we may work on a very ramified cover, a component of the (big) Igusa variety of Caraiani-Scholze [1,Section 4], where the extension structure (1.2.3.1) extends to a trivialization of the p-divisible group At the price of the ramification, life is simplified on this cover: for example, computations with the crystalline connection are reduced to computing the crystalline realization of maps Q p /Z p → G m . Most importantly, the obvious action of automorphisms of G m × Q p /Z p on this cover extends to an action of a much larger group of quasi-isogenies of G m × Q p /Z p . This quasi-isogeny group contains a very large unipotent subgroup, the quasiisogenies from Q p /Z p to G m , or, the universal cover G m ∼ in the language of Scholze-Weinstein [15]. The action of this large group of quasi-isogenies is the ultimate source of the G m -action on I Katz . Indeed: I Katz is the quotient by the subgroup of isomorphisms, i.e. the Tate module T p G m , and thus picks up a residual action of Remark 1.2.6. The action of a larger group of quasi-isogenies on this cover is a natural characteristic p analog of the prime-to-characteristic phenomenon where, when full level is added at l = p, there is an isogeny moduli interpretation that gives an action of GL 2 (Q l ) extending the action of GL 2 (Z l ) in the isomorphism moduli interpretation. Rigidifying in characteristic p using isomorphisms to an ordinary p-divisible group provides both more and less structure than when l = p: on the one hand, the isogeny group is solvable, and thus appears more like the subgroup of upper triangular matrices, but on the other hand the unipotent subgroup has a much richer structure than any groups that appear when l = p. If we instead rigidified using a height two formal group, we would obtain a super-singular Igusa variety, which has more in common with the l = p case (the isogeny action is by the invertible elements of the non-split quaternion algebra over Q p ); in [6] we use this structure to compare p-adic modular forms and continuous p-adic automorphic forms on the quaternion algebra ramified at p and ∞. We will not use this connection to perfectoid modular curves in the present work, however, it will play an important role in [5].
1.2.8. Local expansions. An important aspect of our proof of Theorem A is that we make no appeal to local expansions at cusps or ordinary points, so that our approach is well-suited for generalization to other PEL Igusa varieties. After proving Theorem A, however, we also give a direct computation of the action on local expansions: we find that at ordinary points the action is given by multiplication of a Serre-Tate coordinate, and at the cusps it is given by multiplication of the inverse of the standard cuspidal coordinate q.
1.2.9. Dwork's equation. While developing some of the machinery used to compute the local expansions of the G m -action, and using the same philosophy of base change to a very ramified cover, we also give a new proof of Dwork's equation τ = log q on the formal deformation space of G m × Q p /Z p over F p which is valid for a larger family of Kummer p-divisible groups (which include not only the deformations of G m × Q p /Z p over Artinian F p -algebras, but also, e.g., the p-divisible group of the Tate curve, and other interesting groups when the base is not Artinian). These results can be found in Section 3.
1.2.10. Other constructions. After preparing an earlier version of this article, we learned that Gouvea [4, III.6.2] had already some time ago constructed a twisting measure equivalent to our G m -action (interpreted as an algebra action via p-adic Fourier theory as described in 1.2.11 below). In 7.2 we recall Gouvea's construction and explain how it can be rephrased as an alternate construction of the G m -action via the exotic isomorphisms of Katz [9, 5.6]. Gouvea's construction has the advantage of using only classical ideas, but is conceptually more opaque. In particular, we note that the interaction of the G maction with the prime-to-p group action (equivalently, Hecke operators away from p) is considerably clarified by our construction.
There is a third, even simpler (and even more opaque) approach in which one builds the G m -action algebraically starting with the differential operator θ and the q-expansion principle; we explain this in Remark 1.2.12 below.
1.2.11. The algebra action. Via p-adic Fourier theory, the action of G m described in Theorem A is equivalent to an action of Cont(Z p , Z p ) on V Katz . This action admits a particularly simple description on cuspidal q-expansions: f ∈ Cont(Z p , Z p ) acts as multiplication by f (n) on the coefficient of q n (cf. Theorem 7.1.1). As remarked above, the existence of this algebra action was first established by Gouvea [4, Corollary III.6.8], who interpreted it as a twisting measure.
From this perspective, the action of the monomial function z k is by the derivation θ k (recall θ = q d dq ), and thus we may view our G m -action as interpolating the differential operators θ k into an algebra action. In Section 8 we adopt this perspective to reinterpret some of the results of Katz [9] on two-variable Eisenstein measures.
Remark 1.2.12. In fact, we can construct the G m -action by applying the qexpansion principle [9, 5.2] to complete the action of polynomials in θ on V Katz to an action of Cont(Z p , Z p ). Note that polynomials are not dense Cont(Z p , Z p ), so the q-expansion principle needed here says not just that the q-expansion map is injective, but also that the cokernel is flat over Z p .
In order to use this method, one must first show that the operator θ on qexpansions preserves the space of p-adic modular forms (instead of deducing this by differentiating the G m action). One way this can be done is by showing it is the effect on q-expansions of the differential operator dual to the image of ω 2 can under the Kodaira-Spencer isomorphism, which can be verified by a computation over C, as explained by Katz [9,5.8].
1.2.13. Ordinary p-adic modular forms. The action of Cont(Z p , Z p ) interacts naturally with the Z × p action on V Katz , and thus we may view V Katz as a Z × p -equivariant quasi-coherent sheaf on the profinite set Z p (viewed as a formal scheme whose ring of functions is Cont(Z p , Z p )). As Z p is the space of characters of G m , this viewpoint is analogous to thinking of functions on Y ∞−ord in the global, archimedean setting as Fourier series.
A straightforward computation with q-expansions implies that restriction induces an isomorphism between the fiber at 0 ∈ Z p of the subsheaf V Katz,hol of p-adic modular function with q-expansion holomorphic at all cusps and the space of ordinary p-adic modular formsà la Hida. Note that the fiber at zero is the maximal trivial quotient for the G m -action, and ordinary modular forms are those such that the corresponding p-adic Banach representation of GL 2 (Q p ) admits a map to a unitary principal series. Thus, our statement is a local, p-adic analog of the global, archimedean statement that the global automorphic representation attached to a classical modular form is globally induced if and only if its Fourier expansion has a non-zero constant term.
We do not discuss this phenomenon further in the present work, but this characterization of ordinary p-adic modular forms will play an important role in our study of functions on I CS as a natural space of p-adic automorphic forms in [5]. Moreover, this perspective also leads to representation-theoretic proofs of Hida's finiteness and classicality results for ordinary p-adic modular forms, as will be explained in [5].
1.3. A remark on notation. Over a ring in which p is topologically nilpotent, the formal group G m is equivalent to the p-divisible group µ p ∞ . In the introduction so far we have only used the notation G m , because we wanted to emphasize in our discussion of the action that this is not a torsion group (e.g., the Z p -points are 1 + pZ p ). In the remainder of the article, however, it will be convenient to prefer the notation µ p ∞ when we are speaking about p-divisible groups appearing, e.g., in a moduli problem, and to generally reserve the notation G m for when we are discussing the action on I Katz . This is especially convenient to avoid the oversized notation G m ∼ when discussing universal covers! 1.4. Outline. In Section 2 we collect some results on p-divisible groups that will be needed in the rest of the paper. In Section 3 we study extensions of Q p /Z p by µ p ∞ ; in particular, we introduce Kummer p-divisible groups (following a construction of Katz-Mazur [11, 8.7]) and prove our generalization of Dwork's formula τ = log q.
In Section 4 we recall the Katz and Caraiani-Scholze moduli problems over the ordinary locus, and explain the relation between them.
In Section 5 we construct the action of G m and prove Theorem A. In Section 6 we compute the action on local expansions, and show that there is no global Serre-Tate coordinate on I Katz (dispelling some myths in the literature).
In Section 7 we explain how to obtain the algebra action of Cont(Z p , Z p ) using padic Fourier theory, and compare our construction to Gouvea's original construction of this algebra action. Finally, in Section 8 we explain an application to Eisenstein measures and p-adic L-functions.

Preliminaries on p-divisible groups
In this section we collect some results on p-divisible groups that will be useful in our construction. Our principal references are [14] and [15]; we also provide some complements.
For the proof Theorem A, the most important result in this section is Lemma 2.5.1. It computes, for I a nilpotent divided powers ideal in a ring R where p is nilpotent, the action of (1) We write µ p ∞ for the inductive system (µ p i ) i≥0 , where µ p n is the kernel of multiplication by p n on G m , and the inclusion maps are the obvious ones; it is a p-divisible group of height 1. When p is topologically nilpotent on R we also write µ p ∞ = G m , notation that will be explained below. (2) We write Q p /Z p for the inductive system (1/p n Z p /Z p ) with the obvious inclusions; it is a p-divisible group of height 1.
Given a p-divisble group, each of the G i defines a presheaf in abelian groups on Alg op R , and we will also denote by G the presheaf colimG i so that for S an R-algebra. With this notation, we have a canonical identification Remark 2.1.2. Note that the maps are injective as maps of presheaves, so that in any faithful topology where the objects of Alg op R are all quasi-compact (e.g. fppf), (2.1.1.1) is also the colimit as sheaves by [16,Lemma 7.17.5]. In particular, one could instead define a p-divisible group as, e.g., an fppf sheaf satisfying certain properties, as is often done in the literature. We prefer the given definition because we will have occasion later on to consider finer topologies.
Remark 2.1.3. We will usually consider p-divisible groups over a ring R where p is nilpotent, or over an affine formal scheme SpfR where p is topologically nilpotent in R. In the latter case, there are two natural ways one might try to define G(S) for S a topological R-algebra: one could first algebraize to obtain a p-divisible group over SpecR, then apply the definition above, or one could take the limit of G(S/I) where I runs over the ideals defining the topology on R. The latter is the correct definition for our purpose. For example, if R = O Cp with the p-adic topology and G = G m (= µ p ∞ ), then, the second, correct, definition gives G m (R) = 1 + m where m is the maximal ideal in O Cp while the first, incorrect, definition gives only the p-power roots of unity.

Formal neighborhoods and Lie algebras.
For G a presheaf in abelian groups on Alg op R , we define the formal neighborhood of the identity G by and the Lie algebra LieG by Note that, by definition LieG(S) = Lie G(S). We have the following important structural result:

Universal covers.
For any presheaf in abelian groups G, we define and its sub-functor For A ∈ Nilp op R we will write an element of G ∼ (A) as a sequence (g 0 , g 1 , . . .) such that p(g i+1 ) = g i for all i ≥ 0; the elements of T p G are those such that g 0 = 1. In particular, we have an exact sequence of presheaves When G is a p-divisible group, we call G ∼ the universal cover, following [15]. In this case, we have is an exact sequence of sheaves in the fpqc topology.
Proof. We must verify that G ∼ → G is surjective as a map of fpqc sheaves. Note Remark 2.3.2. Exactness at the right in 2.3.1.1 typically fails in the fppf topology. For example, if G = µ p ∞ and R is finitely generated of characteristic p, then µ p ∞ ∼ (R) = 1. Any fppf cover of such an R is by finitely generated rings of characteristic p, thus µ p ∞ ∼ is the trivial sheaf on the small fppf site of SpecR. On the other hand, if R contains any nilpotents (e.g. R = k[ǫ]/ǫ 2 ), then µ p ∞ (R) = 1, and thus the map µ p ∞ ∼ → µ p ∞ is not surjective in the fppf topology.

2.3.3.
Crystalline nature of the universal cover. Suppose G 0 is a p-divisible group over a ring R in which p is nilpotent, R ′ → R is a nilpotent thickening, and G is a lift of G 0 to R. Then, the reduction map is an isomorphism: the inverse sends (g 0 , g 1 , . . .) to (g ′ 0 , g ′ 1 , . . .) where g ′ i is defined to be p n (g i+n ) for n sufficiently large and any liftg i+n ∈ G(S) of g i+n . Note that these lifts exist by the formal smoothness of Theorem 2.2.1 and the p n th multiple is independent of lift for n sufficiently large by a lemma of Drinfeld [7, Lemma 1.1.2].
2.4. The universal vector extension. For R in which p is nilpotent, and G/R a p-divisible group, we denote by EG the universal vector extension of G, which is an extension There is a natural map s G : G ∼ → EG sending (g 0 , g 1 , . . .) ∈ G ∼ (S) to p n g ′ n for n sufficiently large and g ′ n any lift of g n to EG(S); this is well-defined since ω G ∨ is annihilated by the same power of p that annihilates R. Remark 2.4.1. From the construction of the universal vector extension in [14], we find that EG is the push-out of the extension 2.3.1.1 by the natural map T p G → ω G ∨ sending x to x * dt t where we think of x as a map from G ∨ to G m . Note that the map T p G → ω G ∨ factors through G[p n ] for n sufficiently large (such that p n annihilates R and thus ω G ∨ ), so that EG can be constructed as an fppf pushout (avoiding issues with fpqc sheafification in showing the pushout exists). These considerations lead to the following question: is there a natural topology suitable for constructions such as in the previous remark involving T p G and G ∼ , but avoiding the set theoretic issues of the fpqc topology? 2.4.2. Crystalline nature. If R ′ → R is a nilpotent divided powers thickening, G 0 and H 0 are p-divisible groups over R, and G and H are lifts of G 0 and H 0 , respectively, to R ′ , and ϕ : G 0 → H 0 is a morphism, then we obtain a morphism Eϕ(R) : EG 0 → EH 0 by the universality of EG 0 (using that ϕ * EH 0 is a vector extension of H 0 ). Messing [14,Theorem IV.2.2] shows that there is a functorial lift By [15, Lemma 3.2.2], the following diagram commutes: then, passing to Lie algebras, we obtain a (nilpotent) crystal in locally free O crys -modules DG 0 whose value on a nilpotent divided powers thickening R ′ → R is LieEG ∨ where G is any lift of G 0 to R ′ . Given such an R ′ and G, we obtain a filtered vector bundle on SpecR ′ The assignment G 0 → DG 0 is a contravariant functor: given ϕ : G 0 → H 0 we obtain a map DH 0 → DG 0 from the construction Eϕ ∨ .
2.5. An important example. We now explain how to compute the maps in dia- with µ p ∞ and ω G ∨ with G a using the basis dt t , and Z p is included anti-diagonally, i.e. by z → (z, −z). Here s G is the map a → (a, 0).
For H = µ p ∞ , EH = H, and s H is the map ). Here we have writtenφ for the composition of the arrows at the top of the diagram (2.4.2.1) and the subscript 0 to denote its zeroth component. The exponential and logarithm make sense because g ′ 0 is congruent to 1 mod the kernel I of R ′ → R, which is a nilpotent divided powers ideal. Because exp(z log(g ′ 0 )) = (g ′ 0 ) z for z ∈ Z p , we find that the map is zero on the anti-diagonally embedded Z p . In particular, we deduce the following lemma, which we will use in our verification of Theorem A.
is multiplication by log g 0 .

2.6.
Comparing the Gauss-Manin and crystalline connections. Let S be a scheme where p is locally nilpotent, let π : A → S be an an abelian scheme, and write A ∨ for the dual abelian scheme. We have the relative de Rham cohomology We also have the universal extension of EA[ Given a vector bundle with connection (V, ∇) over S, and a vector field t, viewed as a map t : D × S → S, we obtain an isomorphism of vector bundles on D × S where 0 is the zero vector field. It will be useful to make this isomorphism explicit when S = SpecR and M is the R-module of sections of V over SpecR. Then the map t is given by and the zero section is given by The isomorphism ∇ t is then given in coordinates by where by abuse of notation we have also written ∇ t for the derivation M → M associated to t by ∇.

2.7.
Serre-Tate lifting theory. For R a ring in which p is nilpotent, and R 0 = R/I for I a nilpotent ideal, let Def(R, R 0 ) be the category of triples (E 0 , G, ǫ) where E 0 /R 0 is an elliptic curve, G is a p-divisible group, and ǫ : G| R0 We denote by Ell(R) the category of elliptic curves over R. There is a natural functor from Ell(R) to Def(R, R 0 ) where ǫ E is the canonical isomorphism The following result is due to Serre- Tate In this section we study extension of Q p /Z p by µ p ∞ . In particular, we recall a construction from [11, 8.7] of certain extensions which we call Kummer p-divisible groups, and prove our generalization of Dwork's equation τ = log q (Theorem 3.3.1 below).
3.1. The canonical trivialization. Suppose given an extension of p-divisible groups E : µ p ∞ → G → Q p /Z p over a scheme S where p is locally nilpotent. The inclusion µ p ∞ → G induces an isomorphism ω G = ω µ p ∞ , and we denote by ω can the image of dt t in ω G . The map G → Q p /Z p induces an injection The image is the unit root filtration, which splits the Hodge filtration; we write u can for the image of t∂ t ∈ Lieµ p ∞ in LieEG ∨ .
We thus obtain a trivialization where the first term spans the Hodge filtration and the second the unit root filtration. The elements t∂ t and dt t are flat for the connections on LieE(Q p /Z p ) ∨ and LieEµ ∨ p ∞ , respectively, and thus we find that in the basis (3.1.0.1), ∇ crys is lower nilpotent, i.e.
In particular, the extension determines a differential form The notation is a slight abuse, as in general there is no function τ E in O(S) whose differential is equal to dτ E ; nevertheless, as we will see below, it is natural to think of this as the differential of Dwork's divided powers coordinate τ .

3.2.
Kummer p-divisible groups. For R a ring and q ∈ R × , we will construct an extension of p-divisible groups over SpecR, We call the extensions E q arising from this construction Kummer p-divisible groups (for reasons explained below in Remark 3.2.5). This construction is due to Katz-Mazur [11, 8.7] (who work in the univeral case over Z[q, q −1 ]), but because it will be useful later we give the details and some complements below. We first consider the fppf sheaf in groups consisting of pairs (x, m) such that for k sufficiently large, x p k = q p k m . Projection to the second component gives a natural map Roots q → Z[1/p]. The kernel is identified with µ p ∞ , and the projection admits a canonical section over Z by 1 → (q, 1). We consider the quotient by the image of this section G q := Roots q /Z.
Proof. If we let Roots ′ q be the subsheaf of Roots q of elements (x, m) with m ∈ Z[1/p], 0 ≤ m < 1, then the group law induces an isomorphism Roots ′ q × Z → Roots q . Thus, Roots ′ q as a sheaf of sets is isomorphic to Roots q /Z, and for A an R-algebra with SpecA connected, G q (A) = Roots q (A)/(q, 1) Z and any element of G q (A) has a unique representative of the form (x, m) ∈ Roots q (A) with 0 ≤ m < 1. Such an element is p k -torsion if and only if m ∈ 1/p k Z and x p k = q p k m . In particular, we find that G q = colimG q [p k ]. Moreover, multiplication by p is an epimorphism because taking a pth root of x gives an fppf cover. Thus, to see that G q is a p-divisible group, it remains only to see that G q [p] is a finite flat group scheme. In fact, for any k, our description of elements shows that G q [p k ] is represented by with multiplication given by "carrying," i.e. for x 1 a root of q a1 and x 2 a root of q a2 , in the group structure This is a finite flat group scheme.
Finally, the extension structure is clear from definition.
Remark 3.2.2. Let Roots q,k ⊂ Roots q be the elements (x, m) such that p k m ∈ Z and x p k = q p k m , so that Roots q,k /Z = G q [p k ]. We have a natural pairing Roots q,k × Roots q −1 ,k → µ p k given by (g, a), (h, b) = g p k b h p k a , which induces a perfect pairing It realizes an isomorphism of extensions Note that at the level of groups G q ∼ = G q −1 ; the extension structures E q and E q −1 differ by composition with an inverse on either Q p /Z p or µ p ∞ . (2) For A an Artin local ring with perfect residue field k of characteristic p, any lift of the trivial extension µ p ∞ × Q p /Z p over k to A is uniquely isomorphic to E q for a unique q ∈ G m (A), and q −1 is the Serre-Tate coordinate of the lift (cf. Remark 3.2.6 below). (3) The formation of E q commutes with base change. In particular, there is a universal Kummer p-divisible group, E quniv /G m,Z = SpecZ[q ±1 univ ], so that for any q ∈ R × , E q /SpecR is given via pullback of E quniv through the map SpecR → G m given by q ∈ R × = G m (R).

Remark 3.2.4.
Over a general R, not every extension of Q p /Z p by µ p ∞ is a Kummer p-divisible group, and for those which are, there may not be a canonical choice of q as in the Artin local case. In particular, the extension given by the p-divisible group of the universal trivialized elliptic curve over V Katz,Fp is not a Kummer p-divisible group, as we explain in 6.4 below.
Remark 3.2.5. For any k ≥ 0, consider the Kummer sequence We may take the pull-back by Z → G m , 1 → q to obtain an extension µ p k → p k − Roots q → Z. Equivalently, this extension is the image of q under the coboundary map Indeed, an element of p k − Roots q is a pair (x, a) ∈ G m × Z such that x p k = q a , and this is mapped to the pair which lies in Roots q . This is an isomorphism of p k − Roots q onto its image, which consists of all (x, m) such that m ∈ 1 p k Z and x p k = q p k m -this is what we denoted by Roots q,k in Remark 3.2.2. In particular, the map Roots q → G q induces an isomorphism It is for this reason that we refer to E q as a Kummer p-divisible group.
Note that there are also natural maps between the Kummer sequences as k varies inducing the obvious inclusions as sub-functors of Roots q , and we find Roots q = colim k p k − Roots q .
To construct G q we can also take the colimit already at the level of the Kummer sequences. If we do so, we obtain the (exact) exponential sequence

There is a map
α : Z → G m sending 1 to q which extends uniquely to a map
Remark 3.2.6. In this remark we explain a third construction of G q and the connection to Serre-Tate coordinates: Consider the extension (3.2.6.1) We obtain an extension of Q p /Z p by G m , A q , as the push-out of (3.2.6.1) by We claim there is a natural isomorphism G q ∼ = A q [p ∞ ] respecting the extension structure. To see this, note that the push-out A q is constructed as the quotient of G m × Z[1/p] by the subgroup generated by (q, 1). Then, the p ∞ -torsion is just the image of Roots q in A q , as desired. We note that if q ∈ G m (R), then taking the push-out and passing to p ∞ torsion is equivalent to just taking the pushout under 3.2.6.2 viewed as a map to G m . Thus, when restricted to q ∈ G m (R) for Artin local R with perfect residue field, our construction gives the extension of Q p /Z p by µ p ∞ with Serre-Tate coordinate q −1 (cf. [14, Appendix 2.4-2.5]).
We will need the following structural result on maps between Kummer p-divisible groups: Proof. Let t = q ′ /q. Suppose given a compatible system of roots t 1/p n of t. We obtain an isomorphism between G q [p n ] and G ′ q [p n ] respecting the extension structure by sending an element (a, k/p n ) to (at k/p n , k/p n ), and these are compatible for varying n.
Conversely, given an isomorphism ψ : G q [p ∞ ] → G q ′ [p ∞ ] compatible with the extension structures, if we restrict to ψ n : G q [p n ] → G q ′ [p n ], then for any (a, 1/p n ) ∈ G q [p n ], ψ(a, 1/p n ) = (a ′ , 1/p n ) for a ′ such that a ′p n = q ′ , and a ′ /a is p n th root of t that is independent of a because two choices of a differ by an element of µ p n ; it thus comes from an element of R × , and the isomorphism at level p n is as above; the roots of t chosen by varying the level then must also be compatible, giving an element ofG m mapping to t.

Dwork's equation
for a unique q ≡ 1 mod (t, p), and q −1 is the Serre-Tate coordinate (cf. Example 3.2.3-(2) and Remark 3.2.6). The W (F p ) point x can with q = 1 parameterizes the unique split lift to W (F p ), the canonical lifting, and we can extend the canonical basis ω can | xcan , u can | xcan of EG| xcan at this point to a flat basis over the divided powers envelope of x can (the extension of u can | xcan is just u can itself, but ω can is not flat so the flat extension of ω can | xcan is not equal to ω can ). The position of the Hodge filtration with respect to this basis then defines a divided powers function τ , and a conjecture of Dwork proven by Katz [7] states 2 τ = log q −1 .
As observed by Katz [8], this is equivalent to computing, in the language of 3.1, We now give a simple proof of this result by using a very ramified base-change to split E q . The result is valid for any Kummer p-divisible group: Proof. By reduction to the universal case, it suffices to prove this for E q over In this case, Ω S is free with basis d log q = dq q , thus it suffices to show that ∇ crys,q∂q (dτ Eq ) = −1.
The vector field q∂q, thought of as a map t : D × S → S is given by the map of rings R → R[ǫ]/ǫ 2 , q → (1 + ǫ)q, and we can compute the isomorphism induced by ∇ crys as follows: First, we observe that t * E q = E (1+ǫ)q and 0 * E q = E q , where q is thought of an element of R[ǫ], and under these identifications the isomorphism Thus, using the description of 2.6.2, it suffices to show that the induced map is given in the canonical bases by It suffices to verify this after flat base change, so we may adjoin roots q 1/p ∞ and (1 + ǫ) 1/p ∞ to obtain a ring R ∞ /(R[ǫ]/ǫ 2 ).

Moduli problems for ordinary elliptic curves
In this section, we discuss various moduli problems for ordinary elliptic curves over a base S where p is locally nilpotent.

Level structures.
4.1.1. Prime-to-p level structure. For T a topological space, we write T for the functor on Sch sending S to Cont(|S|, T ).
Given an elliptic curve E/S over a scheme S, we define the prime-to-p Tate module as a functor on Sch/S, where the transition map from E[n ′ ] to E[n] for n|n ′ is multiplication by n ′ /n. The transition maps are affine, so the prime-to-p Tate module is representable. We define the adelic prime-to-p Tate module as the sheaf on S Zar The prime-to-p Tate module is functorial for quasi-p-isogenies, and the prime-to-p adelic Tate module is functorial for quasi-isogenies. An integral prime-to-p infinite level structure on E is a trivialization An rational prime-to-p infinite level structure on E is a trivialization The degree of a rational prime-to-p infinite level structure is the index

Structures at p.
If R is a ring in which p is nilpotent, and E/SpecR is an elliptic curve, we will consider the following presheaves on Nilp op R : (as defined already in 2.3.) The formal group and p-divisible group of E are functorial with respect to quasiprime-to-p-isogenies of E, and the universal cover of E[p ∞ ] is functorial with respect to quasi-isogenies of E.

Infinite level structure at p. An integral ordinary infinite level structure on E[p ∞ ] is a trivialization
A rational ordinary infinite level structure on E[p ∞ ] is a trivialization The degree of a rational ordinary infinite level structure is the degree of the corresponding quasi-isogeny E[p ∞ ] → µ p ∞ × Q p /Z p .

4.2.
Polarization and the Weil pairing. Our moduli problems will need to take into account a polarization, so we first recall some notation. For R a ring in which p is nilpotent and E/SpecR an elliptic curve, the p n -Weil pairing is a perfect antisymmetric pairing e p n ,E : E[p n ] × E[p n ] → µ p n . It induces an anti-symmetric Q p -bilinear pairing for i + j = s + t + k and t large enough that a i , b j ∈ E[p t ] so that the right-hand side is defined. 1. If f : E → E ′ is an isogeny or quasi-prime-to-p isogeny, then Proof. The first equation for isogenies is a well-known property of the Weil pairing, and the second equation for isogenies is then immediate from the definition of e ∼ . Once the isogeny statements are established, the quasi-isogeny statements follow as raising to a prime-to-p integer power is invertible on µ p n and raising to any integer power is invertible on µ p ∞ ∼ .
In particular, we note that the p n Weil pairings e p n are functorial in degree one quasi-prime-to-p-isogenies of E, and the universal cover Weil pairing e ∼ is functorial in degree one quasi-isogenies of E.
Below we will also consider the standard pairing  [1], this moduli problem is represented by an affine p-adic formal scheme over Z p , The ring V CS is flat over Z p ; indeed, it is the Witt vectors of a perfect ring over F p . For the finite level variant this follows from [1, p.718] (who work over W (F p ) instead of Z p , but this is not necessary here). Their argument applies equally well to infinite level at p by taking the Witt lift of the perfect ring representing the corresponding mod p moduli problem, which is just the colimit of the perfect rings representing the finite level moduli problems (equivalently, one takes the colimit of the Witt lifts of these and then p-adically completes).

Remark 4.3.2.
We will explain the construction of V CS in more detail from a classical perspective below.

A polarized variant.
For our purposes, we will also need the polarized variant of this moduli problem. To state it, we first observe that any triple (E, ϕ p , ϕ A (p) f ) as above, we can choose a representative for the isogeny class of E such that ϕ p and ϕ A (p) f are both degree one, and that such a representative is determined up to degree one quasi-isogeny. Because the Weil pairing of an elliptic curve is preserved under degree one quasi-isogeny, we obtain a well-defined Weil pairing e The polarized moduli problem then parameterizes triples as above where we additionally require that where , std is the pairing defined in (4.2.1.1). The polarized moduli problem is represented by a closed formal subscheme I 1 CS = SpfV 1 CS ⊂ I CS .

4.3.4.
A p-integral version. The corresponding p-integral moduli problem where E is an elliptic curve up to quasi-prime-to-p-isogeny and ϕ p is integral ordinary infinite level structure is equivalent to the rational moduli problem via the natural inclusion, and thus is also represented by I CS (cf., e.g., [ where E/R is an elliptic curve up to quasi-prime-to-p-isogeny, ϕ p is Katz level structure on E[p ∞ ], and ϕ A (p) f is a rational infinite 4 prime-to-p level structure.

4.4.1.
Representatiblity. By work of Katz [8] this moduli problem is represented by an affine p-adic formal scheme over Z p Remark 4.4.3. We note that to give the data ϕ • and ϕé t is equivalent to equipping E[p ∞ ] with the structure of an extension

Group actions.
4.5.1. Automorphism groups at p. We consider the twisted Borel B p , the presheaf on Nilp op Zp defined by Because there are no non-zero maps from µ p ∞ ∼ to Q p and Hom(Q p , µ p ∞ ∼ ) = µ p ∞ ∼ , we can write this as the matrix group We write M p for the diagonal subgroup, and U p = µ p ∞ ∼ for the unipotent subgroup. We write det : B p → Q × p for the product of the diagonal entries. We also consider the integral variants

4.5.2.
Moduli action on I CS and I 1 CS . We write G (p) = GL 2 (A (p) f ). Composition with the level structures gives an action of B p × G (p) on I CS .
We would like to understand the subgroup preserving I 1 CS : First, it is a straightforward computation to check that the Weil pairing constructed above transforms via the character On the other hand, the standard pairing , std on µ p ∞ ∼ × Q p transforms via Thus, writing det ∼ = det ur · det p (whose image is contained in Z × p ), we find that the group B p × G (p) 1 := ker det ∼ is the stabilizer of I 1 CS . 4.5.3. p-integal moduli action of the unipotent subgroup. It will be useful for computations with the crystalline connection for us to have a more explicit description of the action of U p on the p-integral moduli problem represented by I CS . We give such a description now; the key point is that any unipotent automorphism of the universal cover lifts a unipotent automorphism of the p-divisible group µ p ∞ ×Q p /Z p modulo a nilpotent ideal. Let u ∈ U p (R) = µ p ∞ ∼ (R), and x ∈ I CS (R). Write u = (ζ k ) ∈ µ p ∞ ∼ (R) and let I be any nilpotent ideal of R containing ζ 0 − 1. Then, ) is a triple corresponding to x under the p-integral moduli interpretation then the corresponding covers are finiteétale, we find that the Katz (resp. Caraiani-Scholze, resp. polarized Caraiani-Scholze) moduli problem with finite prime-to-p level K p is represented by the ring Example 4.5.6. If Γ 1 (N ) denotes the subgroup of GL 2 ( Z (p) ) congruent to  Katz represents the moduli problem classifying over R the triples (E, ϕ, P ) where E is an elliptic curve over R, ϕ : E ∼ − → G m , and P ∈ E[N ](R) is a point of (fiberwise) exact order N . This is the moduli problem most commonly considered in the literature on p-adic modular forms. 4.6. First presentation of I Katz as a quotient. Recall from above that I CS has a p-integral moduli interpretation as parameterizing triples where E is an elliptic curve up to quasi-prime-to-p-isogeny, ϕ p is integral ordinary infinite level structure on E[p ∞ ], and ϕ A (p) f is rational infinite prime-to-p level structure. Then there is a natural projection map to I Katz given by where ϕ p is the isomorphism induced by ϕ p after restriction: It can be verified that this map is an fpqc torsor for the action of  on I CS (cf Lemma 5.1.1 below). Thus, I Katz is an fpqc quotient of I CS for this group action, and we obtain a residual action of the quotient of the normalizer of this group in B p × G p by the group itself. This induces the action of Z × p × G (p) given by the first moduli interpretation of I Katz , but nothing further.
The surprising observation that allows us to construct our G m -action on I Katz is that, if we restrict this projection map to I 1 CS , then the normalizer grows, and we obtain a non-trivial residual action from the unipotent part U ∼ p . We describe this in the next section.

5.
The G m -action 5.1. Second presentation of I Katz as a quotient. If we restrict the projection to I 1 CS , then in terms of the second moduli interpretation of I Katz , it is given by where ϕ p is as described above in 4.6 and ϕé t p is induced by subgroup Z × p × G (p) agrees with the standard moduli action on I Katz , and thus we immediately obtain the compatibility between the G m -action and the standard moduli action described in Theorem A.
Remark 5.2.1. The subgroup p Z acts as powers of the classical diamond operator at p. Similarly, we obtain the Hecke operator U p and the canonical frobenius lift from the Hecke action of diag(p, 1) and diag(1, p).

Differentiating the action.
We have now constructed the G m -action and proved the desired compatibility with the standard moduli action on I Katz . To complete the proof of Theorem A, we must prove a claim about the derivative of the G m -action. We recall the setup now: We consider the action map To differentiate it, we compose with the tangent vector t∂ t at the identity in G m . The latter is given by a map D → G m which in coordinates is Thus, the composition of the action map with t∂ t gives a vector field on I Katz described as a map t u : On the other hand, we have the universal extension E Euniv[p ∞ ], ϕuniv,ϕé t univ : G m → E univ [p ∞ ] → Q p /Z p (cf. (4.4.3.1)), and, as explained in 3.1, this extension gives rise to a differential dτ ∈ Ω IKatz .
To complete our proof of Theorem A, we show Theorem 5.3.1. Notation as above, Proof. It suffices to work over Z/p n for arbitrary n. We abbreviate S = I Katz | Z/p n and R = V Katz /p n so that S = SpecR. We write π : E → S for the universal elliptic curve up to prime-to-p-isogeny over S and ϕ, ϕé t for the universal trivializations of E and E[p ∞ ]/ E. We recall the definition of dτ (mod p n ): we have the canonical extension  (1 + ǫ) · (E 0 , ϕ 0 , ϕé t 0 , α 0 ) = (E u , ϕ u , ϕé t u , α u ) In particular, ∇ crys,tu is identified with the Messing isomorphism induced by the isomorphism E u mod ǫ = E 0 mod ǫ given by 1 + ǫ = 1 mod ǫ and (5.3.1.2).
To compute this, we pass to the flat cover and thus over S ∞ , we have a canonical splitting of E E0[p ∞ ], ϕ0,ϕé t which gives an isomorphism If we let our description of the unipotent action in 4.5.3 then shows that over S ∞ , E u is the Serre-Tate lift to S ∞ corresponding to the isomorphism Thus, the Messing isomorphism in the canonical basis is identified over S ∞ with the map If we write this in the canonical basis we get a map and, by Lemma 2.5.1, it is given by  By construction, these bases are identified with the bases ω can , u can , and thus we obtain equation (5.3.1.1), concluding the proof.

Local expansions
In this section we compute the G m -action on Serre-Tate ordinary and cuspidal q-expansions (i.e. on the formal neighborhood of an F p -point of I Katz and on the punctured formal neighborhood of a cusp). In both cases, the action is given by a simple multiplication of the canonical coordinate (cf. Corollaries 6.5.1 and 6.3.1 below for precise statements).
In fact, both computations are special cases of a more general statement, Theorem 6.2.1 below, which computes the action on a point whenever the associated p-divisible group is a Kummer p-divisible group (as defined in 3.2).
The statement of Theorem 6.2.1 begs the question: is the p-divisible group of the universal elliptic curve over I Katz (with its extension structure) a Kummer p-divisible group? Indeed, one can find claims in the literature that the local Serre-Tate coordinates extend to a function q on I Katz , which would imply this group was Kummer. However, these claims are flawed, and indeed the universal extension is not Kummer; we take a brief detour in 6.4 to dispel these myths.
6.1. Another moduli interpretation for I Katz . To state our general computation cleanly, we introduce a third moduli interpretation for I Katz that puts the emphasis on p-divisible group and its extension structure (as in Remark 4.4.3).
Let R be a p-adically complete ring and let π be topologically nilpotent for the p-adic topology on R. Combining the second moduli interpretation of I Katz (cf. 4.5.4) and Serre-Tate lifting theory (cf. 2.7), we obtain an identification where E 0 is an elliptic curve up to quasi-prime-to-p-isogeny over R/π, E is an extension of p-divisible groups over R f is infinite prime-to-p level structure on E 0 , all subject to a compatibility with the Weil pairing as in the second moduli interpretation of I Katz .
6.2. Computing the action on Kummer extensions. Recall from 3.2 that there is a Kummer construction which, given q ∈ G m (R) produces an extension of p-divisible groups over R Using the explicit description of the G m -action in 4.5.3, we find Theorem 6.2.1. Suppose ζ ∈ G m (R) and π ∈ R is such that ζ ≡ 1 mod π, and x ∈ I Katz (R) is represented by the quadruple (E 0 , E q , ψ, ϕ A (p) f ) (in the sense of 6.1) for some q ∈ G m (R). Then where ψ ′ is the composition of ψ with the canonical identification E q | R/π = E ζ −1 q | R/π coming from q ≡ ζ −1 q mod π.
Proof. If we write y for the point represented by (E 0 , E ζ −1 q , ψ ′ , ϕ A (p) f ) then it suffices to show that over the extension R ∞ = R[q 1/p ∞ , ζ 1/p ∞ ], there are liftsx andỹ of x and y to I 1 CS (R ∞ ) and a liftζ of ζ in µ p ∞ ∼ (R ∞ ) such thatζ ·x =ỹ. The desired lifts are given by the splittings 1/p n → (q 1/p n , 1/p n ) and 1/p n → ζ −1/p n q 1/p n of E q and E ζ −1 q , respectively, andζ = (ζ 1/p n ) n . Thatζ ·x 1 =x 2 then follows from commutativity of the following diagram mod ζ − 1: 3. Action on Serre-Tate coordinates. We now compute the local expansion in the formal neighborhood of an F p -point of I Katz . So, fix . It follows from the work of Serre-Tate (as described in [7]) that the formal neighborhood of this point is SpfR for R a smooth complete 2-dimensional local ring over W (F p ). The data attached to the induced SpfR → I Katz identifies E with the universal deformation of E 0 , and the level structures with their unique deformations.
Moreover, the Serre-Tate coordinate q ∈ 1 + m R = G m (R) of the induced extension where it is claimed that the local Serre-Tate coordinates q extend to a function on all of I Katz (or sometimes a rigid analytic incarnation thereof). These claims seem to be based on a misapplication of a theorem of Serre-Tate (cf. Katz [7]) classifying extensions of Q p /Z p by G m over Artin local rings to the more general setting of rings where p is nilpotent, a setting in which the classification no longer holds. In fact, the claim itself is incorrect, as we now explain. Already in characteristic p the existence of such a global coordinate boils down to the claim that for E univ the universal curve over I Katz,Fp , the canonical extension . This is not the case: if it were, then E Euniv, ϕuniv,ϕ and, if we fix a compatible system of prime-to-p roots of unity ζ N , we obtain a basis (ζ N ) N , (q 1/N ) N for T Z (p) Tate(q) over R and thus a trivialization ϕ A (p) f of the prime-to-p adelic Tate module. The cusps of I Katz are the R-points in the GL 2 ( Z (p) )-orbit of For g ∈ V Katz,A and a cusp c, we call the element c(g) ∈ A ⊗R the q-expansion of g at c. We find Corollary 6.5.1. If c is a cusp of I Katz and g ∈ V Katz,A has q-expansion at c then, for ζ ∈ G m (A), ζ · g := (ζ −1 ·) * g has q-expansion at c (6.5.1.1) Proof. The q-expansion in 6.5.1.1 is the image of c(g) under the the map Thus, because the action of G m commutes with the action of GL 2 ( Z (p) ) (because the latter is in the kernel of det ur ), we may assume our cusp is given by the triple Tate(q), ϕ can , ϕ (ζN )N ,(q 1/N )N .
Remark 6.5.2. Using Corollary 6.5.1 over A = Z p [ǫ], we find that if we differentiate the G m -action along t∂ t in the sense of 5.3, the induced operator on q-expansions is −q∂ q = −θ (we get a minus sign because to get the derivation in 5.3 we did not compose with an inverse as we have to obtain the natural left action on functions).

Fourier transform and the algebra action
We now explain how the G m -action induces an action of Cont(Z p , Z p ) on V Katz via p-adic Fourier theory. We then explain the relation between our approach and a classical construction of the algebra action (which is in fact equivalent to the G m -action) due to Gouvea [4] in terms of the exotic isomorphisms of Katz [9]. 7.1. The action of Cont(Z p , R). For any p-adically complete ring R, the action map for the G m -action is described by a continuous map The natural left action of G m (R) on V Katz,R is by ζ · g = (ζ −1 ·) * g, and we can express this using the action map: if we consider ζ ∈ G m (R) as the map R[[T ]] → R given by T → ζ, and write ι for the inverse map G m → G m , then ζ · g is the image of (ι × Id) * a * g under the induced map More generally, if we identify R[[T ]] with the continuous R-linear dual of Cont(Z p , R) via the Amice transform, we obtain an R-linear map If we let χ ζ be the R-valued character of Z p given by χ ζ (a) = ζ a , viewed as an element of Cont(Z p , R), we find That we have an action of G m to begin with is equivalent to this being an algebra action of Cont(Z p , R) on V Katz,R .
More generally, we find (using the notation for cusps as in 6.5 above): Theorem 7.1.1. If g ∈ V Katz,R has q-expansion at a cusp c then the q-expansion of f · g is Proof. Because for a general R, where the first map is the isomorphism induced by diag(p n , 1) and the second map is the projection. In fact, this map is the canonical Frobenius lift (sending E to E/µ p n with the induced level structures), and the isomorphism that factors it is one of Katz's exotic isomorphisms (cf. [9, Lemma 5.6.3]). The action of µ p n ⊂ G m on I Katz is identified through the exotic isomorphism with the action of µ p n = U • p /p n U • p on p n U • p \I CS . The latter space has a moduli interpretation relative to I Katz as parameterizing splittings of the canonical extension µ p n → E[p n ] → 1/p n Z p /Z p , and the action of µ p n is just by changing the splitting. In particular, one can compute the action on q-expansions using this relative moduli interpretation to recover the action of all finite order characters, which, combined with the q-expansion principle, is enough to produce the full algebra action of Theorem 7.1.1. This is essentially what is done by Gouvea [4, III.6.2], who first constructed this action (which he thought of as a twisting measure, much in the spirit of our application in the next section).
Remark 7.2.1. The moduli problem I 1 CS is the inverse limit of the moduli problems parameterizing splittings at level p n over I Katz , and thus is the inverse limit along the canonical Frobenius lift of I Katz . As observed by Caraiani-Scholze [1], this implies that V 1 CS is isomorphic to the Witt vectors of the perfection of V Katz,Fp .

Eisenstein measures
In a series of papers ( [8,10,9]), Katz introduced increasingly general Eisenstein measures with values in V Katz interpolating Eisenstein series. These Eisenstein measures specialize at the cusps and ordinary CM points to p-adic L-functions interpolating L-values attached to characters of the idèle class group of Q or an imaginary quadratic extension.
The papers [8,10] are concerned with single variable p-adic L-functions, whereas [9] gives two variable L-functions by interpolating not just holomorphic Eisenstein series but also certain real analytic Eisenstein series.
In this section, we explain how "half" of the two-variable measure can be produced by a type of convolution of the single-variable measure with the G m -action. To keep the exposition clear, we work at level K p = GL 2 ( Z (p) ) away from p. Remark 8.0.1. The real analytic Eisenstein series are related to the holomorphic Eisenstein series by iterated application of the differential operator θ -thus, we can summarize the difference between our approach and that of Katz by saying that instead of applying θ and then interpolating, we have first interpolated θ and then applied this interpolated operator to the holomorphic Eisenstein measure. 8.1. Measures. For R a p-adically complete Z p -algebra and X a profinite set, an R-valued measure on X is an element µ ∈ Hom Zp (Cont(X, Z p ), R).
Note that such a µ is automatically continuous for the p-adic topology on Cont(X, Z p ) and R. In fact, the stronger basic congruence property holds: if f ≡ g mod p n , then µ(f ) ≡ µ(g) mod p n -this observation is at the heart of the application of measures to p-adic L functions.
Remark 8.1.1. An R-valued distribution is an R-valued functional on the space of locally constant functions on X, C ∞ (X, Z p ). The space C ∞ (X, Z p ) is dense in Cont(X, Z p ), thus when R is p-adically complete a distribution automatically completes to a measure, and the two notions are equivalent. We will use this below.

Measures on products.
Proposition 8.1.3. Let X and Y be profinite sets, and R a p-adically complete Z p -algebra. If ( , ) is an R-valued Z p -bilinear pairing on Cont(X, Z p )×Cont(Y, Z p ), then there is a unique R-valued measure µ on X × Y such that for f ∈ Cont(X, Z p ) and g ∈ Cont(Y, Z p ), (8.1.3.1) µ(f g) = (f, g).
Proof. By Remark 8.1.1, it suffices to construct a functional on C ∞ (X × Y, Z p ) satisfying (8.1.3.1), and then verify that (8.1.3.1) holds for any continuous f and g and the unique extension of that distribution to a measure. So, suppose we have constructed a measure µ such that (8.1.3.1) holds for f and g locally constant.
Then, for any continuous f and g and n ∈ Z >0 , pick f n and g n locally constant such that f ≡ f n mod p n and g ≡ g n mod p n . Then f g ≡ f n g n mod p n and thus µ(f g) ≡ µ(f n g n ) ≡ (f n , g n ) ≡ (f, g) mod p n and we conclude µ(f g) = (f, g).
Thus it remains to construct the distribution and show that it is unique. The bilinear pairing ( , ) induces a functional on C ∞ (X, Z p ) ⊗ Zp C ∞ (Y, Z p ), thus to conclude, it suffices to show that the product map is an isomorphism: For any profinite set W , C ∞ (W, Z p ) is the colimit over finite coverings U = {U 1 , ..., U n } of W by disjoint compact opens of C ∞ U (W, Z p ), the space of functions constant on each of the U i . In particular, if U = {U 1 , ..., U n } is such a cover of X and V = {V 1 , ..., V m } is such a cover of Y then is such a cover of X × Y , and the covers of this form are cofinal for covers of X × Y by disjoint compact opens. Considering the basis of characteristic functions, we find that the product map induces an isomorphism C ∞ U (X, Z p ) ⊗ Zp C ∞ V (Y, Z p ) → C ∞ U ×V (X × Y, Z p ) and passing to the colimit over covers U and V, we conclude.
Comparing with (8.2.5.1), we see that the measure ν interpolates the same Eisenstein series as ϕ * µ (a,1) , although with a different normalizing factor (recall that this normalizing factor removes the powers of p in the denominator of the constant term ζ(1 − k) of G k when k ≡ −1 mod p − 1). This type of construction may be useful for studying special values of families of automorphic forms and their images under differential operators on other Shimura varieties where explicit computations with q-expansions are not always available.