Formal group exponentials and Galois modules in LubinTate extensions
Abstract
Explicit descriptions of local integral Galois module generators in certain extensions of adic fields due to Pickett have recently been used to make progress with open questions on integral Galois module structure in wildly ramified extensions of number fields. In parallel, Pulita has generalised the theory of Dwork’s power series to a set of power series with coefficients in LubinTate extensions of to establish a structure theorem for rank one solvable padic differential equations.
In this paper we first generalise Pulita’s power series using the theories of formal group exponentials and ramified Witt vectors. Using these results and LubinTate theory, we then generalise Pickett’s constructions in order to give an analytic representation of integral normal basis generators for the square root of the inverse different in all abelian totally, weakly and wildly ramified extensions of a padic field. Other applications are also exposed.
Introduction
The main motivation for this paper came from new progress in the theory of Galois module structure. Indeed, explicit descriptions of local integral Galois module generators due to Erez [5] and Pickett [16] have recently been used to make progress with open questions on integral Galois module structure in wildly ramified extensions of number fields (see [18] and [23]). Precisely, let be a prime number. Pickett, generalising work of Erez, has constructed normal basis generators for the square root of the inverse different in degree extensions of any unramified extension of . His constructions were obtained by using special values of Dwork’s power series. Moreover, they have recently been used by Pickett and Vinatier [18] to prove that the square root of the inverse different of is free over under certain conditions on both the decomposition groups of and the base field , when is a finite odd degree Galois extension with group . In parallel, Pulita has generalised the theory of Dwork’s power series to a set of power series with coefficients in LubinTate extensions of in order to classify rank one adic solvable differential equations [19].
Our main goal was to generalise Erez and Pickett’s construction in order to give explicit descriptions of integral normal basis generators for the square root of the inverse different in all abelian totally, weakly and wildly ramified extensions of a adic field. In this paper, our goal is totally achieved using a combination of several tools : formal group exponentials, LubinTate theory, and the theory of ramified Witt vectors. This leads us to generalise Pulita’s formal power series to power series with coefficients in LubinTate extensions of any finite extension of . At the same time, we also get explicit generators for the valuation ring over its associated order, in maximal abelian totally, weakly and wildly ramified extensions of any adic field.
Notation. Let be a rational prime, and let be the field of adic numbers, be a fixed algebraic closure of and be the completion of with respect to the adic absolute value. We let and be the normalised adic valuation and absolute value on such that and . As and are completely determined by each other, either can be used in the statement of results; we will use the valuation as this is the convention in the literature on Galois modules in LubinTate extensions. Throughout this paper, for any extension considered we will always assume is contained in and we will denote by , and its valuation ring, maximal ideal and residue field respectively. We identify the residue field of with the field of elements, . For any , we denote by the group of th roots of unity contained in .
Presentation of the paper. Let be a root of the polynomial . Dwork’s exponential power series with respect to is defined as
where the right hand side is the composition of the two power series and . Dwork’s power series is overconvergent, in the sense that it converges with respect to on an open disc for some ([14], Chap. 14, §2, Remark after Lem. 2.2); it also has the property that is equal to a primitive th root of unity in ([14], Chap. 14, §3, Thm 3.2).
Dwork’s power series was recently generalised by Pulita [19] to a set of power series with coefficients in LubinTate extensions of : Let be some LubinTate polynomial with respect to the uniformising parameter , i.e.,
Let be a coherent set of roots associated to , namely a sequence of elements of such that , and ; we refer to as an th LubinTate division point with respect to . For , Pulita defines the exponentials
This generalises Dwork’s power series as when . For all choices of and , the power series is overconvergent and has the property that is a primitive th root of unity in . Comparing degrees then shows us that for all and all choices of . We remark that this result is also a consequence of basic LubinTate theory, see [15] or [20] for details of this theory.
In this paper, we first generalise Pulita’s exponentials to power series with coefficients in LubinTate extensions of any finite extension of , in particular by combining Fröhlich’s notion of a formal group exponential ([7], Chap. IV, §1) with the theory of ramified Witt vectors. Note that we impose no other restrictions on our base field and no restrictions on the uniformising parameter used to construct the LubinTate extensions of . Inspired by the methods of Pulita, we prove the following core result of the paper :
Theorem 1
Let be a finite extension of , with valuation ring, maximal ideal and residue field denoted by , and respectively. Let for some power of .
Let and be two uniformising parameters for , and let be LubinTate polynomials with respect to and respectively, i.e.,
We write for the unique formal group that admits as an endomorphism and for the unique power series such that
Let be a coherent set of roots associated to .

For every , the formal power series
lies in , and is overconvergent if .

Moreover, we have the congruence .
To compare to Pulita’s result, we remark that if , , and , then .
We then apply Theorem 1 to give two explicit results in LubinTate theory. For each integer , we denote by the th LubinTate extension of with respect to . This extension is abelian and totally ramified, with degree and conductor . Let be a LubinTate polynomial with respect to . The extension is generated by any primitive th LubinTate division point with respect to , i.e., any element such that whereas .
As a first application of Theorem 1, we give an analytic representation of LubinTate division points as values of the power series for all . Precisely, keeping the same notation, we prove :
Proposition 1

If , then is a primitive th LubinTate division point with respect to , for all integers with .

If , then is a coherent set of roots associated to .
Another application of Theorem 1 is concerned with an explicit description of the action of the Galois group over the LubinTate extension for all . Indeed, since is a primitive th LubinTate division point with respect to , from standard theory (see [11], §67, specifically Theorem 7.1) we know that the elements of are those automorphisms such that , where runs over a set of representatives of — here is a specific power series in , see Section 1.1 for full details. Therefore, the following proposition gives a complete description of in terms of values of our power series.
Proposition 2
Let . For , let with . Then,
Note that, when , this proposition implies the relation
This is a generalisation of ([13], Chap. 14, Thm. 3.2).
Finally, we shall prove our second main result, as a consequence of Theorem 1 and Proposition 1 : we use the power series to construct explicit normal basis generators in abelian, weakly and wildly ramified extensions of any adic field. In this manner, we generalise the constructions of Erez and Pickett, and give support towards the resolution of open questions in Galois module structure theory :
Theorem 2
Let be a finite extension of , with valuation ring , residue field and residue cardinality . Fix a uniformising parameter of and let be some LubinTate polynomial of degree with respect to . Let be a maximal abelian totally, weakly and wildly ramified extension of . Let denote the unique fractional ideal in whose square is equal to the inverse different of (see Section 3.3).
Let be another uniformising element of , with ; and let be a LubinTate polynomial with respect to . If , then

the trace element is a uniformising parameter of and a generator of the valuation ring of over its associated order in the extension ;

if is odd, then the elements
are both generators of over .
Furthermore, Part 2 of this theorem will enable us to give explicit integral normal basis generators for the square root of the inverse different in every abelian totally, weakly and wildly ramified extension of any adic field (see Corollary 3.2).
We also remark that in this second part, the first element seems the more natural, however the second is in fact the generalisation of Erez’s basis generator for the square root of the inverse different. If these basis generators can be used in local calculations in a similar way to those of Erez and Pickett, it should be possible to solve the case of whether is free over whenever the decomposition groups at wild places are abelian and in particular whenever itself is abelian. We hope this will be possible in the future, however, so far these calculations have eluded us.
Organisation of the paper. This paper is organised into three sections. In Section 1, we give the background to the theory we need to prove our results. Precisely, we first introduce LubinTate formal groups in their original setting and also in terms of Hazewinkel’s so called functional equation approach. We also introduce Fröhlich’s notion of formal group exponentials and logarithms. Following work of Ditters, Drinfeld, and Fontaine and Fargues, we finally introduce the theory of ramified Witt vectors needed to generalise Pulita’s methods. In Section 2, we study the properties of the power series and prove Theorem 1. We also improve on Fröhlich’s original bound of the radius of convergence of any formal group exponential coming from a LubinTate formal group over a nontrivial extension of . In Section 3, we explore the applications described above and prove Propositions 1 and 2, as well as Theorem 2.
1 Background
1.1 Formal groups
Let be a commutative ring with identity.
Definition 1.1
We define a dimensional commutative formal group over to be a formal power series such that
Throughout, all the formal groups we consider will be dimensional commutative formal groups. For brevity we will now refer to these simply as formal groups.
Properties 13 can be used to prove that there exists a unique such that (see Appendix A.4.7 of [9]). This means that the formal group endows , among other sets, with an abelian group structure.
Notation 1.2
When considering a set endowed with such a group structure we write the group operation as . We will also use the notation to determine this group operation composed with the group inverse, for example
Definition 1.3
Let and be two formal groups over . A homomorphism over the ring , , is a formal power series such that
Moreover, we say that the homomorphism is an isomorphism if there exists a homomorphism such that .
LubinTate formal groups
We now describe a special type of formal group, due originally to Lubin and Tate. Such formal groups are used in local class field theory to construct maximal totally ramified abelian extensions of a adic field . For full details see, for example, [15] or [20].
Let be a finite extension of , fix a uniformising parameter of and let be the cardinality of the residue field of .
Definition 1.4
We define as the set of formal power series over such that
Such power series are called LubinTate series with respect to .
For each there exists a unique formal group which admits as an endomorphism. Such formal groups are known as LubinTate formal groups. For each and each , there exists a unique formal power series, , such that and
Further, the map is an injective ring homomorphism and for any , the formal groups and are isomorphic over .
Let . For and , the formal power series and converge to limits in when evaluated at elements of . We can thus use the abelian group operation and the injective ring homomorphism to endow with an module structure. For every , we then let
be the set of torsion points of this module and refer to it as the set of the th LubinTate division points with respect to . If if and only if , then we say is a primitive th division point.
We let
The set depends on the choice of the polynomial but the field depends only on the uniformising parameter . The extensions are totally ramified, abelian and of degree . We have and , where and are the maximal abelian and unramified extensions of respectively.
Hazewinkel’s approach to LubinTate formal groups
We now describe a different approach to the construction of LubinTate formal groups due to Hazewinkel. This approach enables us to use Hazewinkel’s functional equation lemma to prove the integrality of various power series relating to formal groups, which will be essential in the sequel. For full details see [9] (Chap. I, §2).
Recall that is a rational prime, is a finite extension of , is a fixed uniformising parameter of and is the cardinality of the residue field of . For any series we construct a new power series by the recursion formula (or functional equation) :
We denote by the unique power series such that
Note that if and the coefficient of in is invertible in , then , see [9, A.4.6].
We now state two parts of the functional equation lemma for this special setting. For the full statement in all generality and its proof, see [9, §2.22.4].
Theorem 1.5 (Hazewinkel, [9, 2.2])
Let and suppose that the coefficient of in is invertible in . Then,

.

.
It is routine to check that is a formal group and from part 1. of the previous theorem we know that it has coefficients in . In fact, it is a LubinTate formal group and every LubinTate formal group can be constructed in this manner. This link is described in the following proposition.
Proposition 1.6 (Hazewinkel, [9, 8.3.6])
Let with
Then,

.

If we let , then .

These relations give a one to one correspondence between the LubinTate formal groups obtained from power series and power series with .
We also observe that for any , substituting into part of Theorem 1.5 then gives us , and so if , then
Formal group exponentials
Hazewinkel’s power series and can be thought of as special formal group isomorphisms which were first studied by Fröhlich in ([7], Chap. IV, §1).
Let be any field of characteristic , let be a formal group over and let be the additive formal group. There exists a unique isomorphism over such that loc. cit., Prop. 1). The inverse of is known as the formal group exponential and is denoted by ; we note that necessarily we also have, . , known as the formal group logarithm (
Now let be a LubinTate formal group for as in Prop. 1.6. We then have
(1) 
and these power series are uniquely determined by the following equivalent identities:
(2) 
We also observe that,
(3) 
Remark 1.7
The reason these power series are referred to as formal group exponentials and formal group logarithms is that if , then and , the multiplicative formal group. We then have and where and are the standard logarithmic and exponential power series.
1.2 Witt vectors
This section is concerned with the notion of ramified Witt vectors, generalising the classical theory of Witt vectors introduced by Witt in his original paper [26]. This notion was first developed independently by Ditters [3] and Drinfeld [4], and then by Hazewinkel [10] from a formal group approach in a more general setting. The reader is also referred to Section 5.1 of the current preprint [12] of Fontaine and Fargues.
Standard Witt vectors
We first briefly recall the construction of “standard” Witt vectors. Let be a prime number, and let be a sequence of indeterminates. The original Witt polynomials are defined by :
The standard Witt vectors can be constructed as a functor from the category of commutative rings to itself. Precisely, if is a commutative ring, we first define as the set of infinite sequences . The elements of are called Witt vectors, and to each Witt vector , one can attach a sequence whose coordinates are called the ghost components of and are defined by the Witt polynomials : , for all .
The set is then uniquely endowed with two laws of composition that satisfy the axioms of a commutative ring, in such a way that the ghost map becomes a ring homomorphism.
Under this functor, any ring homomorphism is sent to the ring homomorphism which is defined componentwise, i.e., . See ([1], Chap. IX) for more details.
Ramified Witt vectors
Let be a prime number. Let be a finite extension of , with valuation ring and residue field . We fix a uniformising parameter of , and write with . Ramified Witt vectors over are constructed as a functor from the category of algebras to itself, starting with generalised Wittlike polynomials and then proceeding along the lines of the construction of the usual Witt vectors. For convenience, as well as to collect some useful properties of the ramified Witt vectors, we shall briefly describe this functor.
In the case of ramified Witt vectors, the relevant polynomials are :
Let be an algebra. We first define as the set as the set of infinite sequences over . We shall use the notation for elements in , and for elements in .
If , we define its ghost components as for all . The sequence is called the ghost vector of . This defines a map, that we shall denote by or simply by when the setup is explicit, called the ghost map of :
The following lemma is essential for what follows.
Lemma 1.8
Let be an algebra with no torsion. Then, the ghost map is injective. If, moreover, there exists an algebra endomorphism such that for all , then the image of is the subalgebra of the product algebra given by
Proof. We proceed along the lines of ([1], No 2, Sect. 1, Par. 1 & 2), replacing multiplication by by multiplication by , and replacing by in the exponents. The first assertion is a consequence of the equivalence
Therefore, for every sequence , there exists at most one element such that .
The second assertion is a consequence of the following relation that can easily be proved in the same way as Lemma 1 of ([1], No 2, Sect. 1), since :
In particular, for and for any sequence , this implies that
Therefore, according to , we can prove by iteration on that a sequence is in the image of if and only if for all .
In particular, the algebra , endowed with the endomorphism given by and , satisfies the above lemma and is such that is bijective because the relation is satisfied for all . Therefore, the map transfers the structure of an algebra to . Moreover, for all and all , this defines polynomials and in , and in and in such that
Now, for any arbitrary algebra , we endow the set with laws of composition given by
Moreover, if is any homomorphism of algebras, we define the map componentwise, i.e., . This map commutes with the previous laws of composition.
Next, for a fixed algebra , we consider the algebra which satisfies the assumptions of Lemma 1.8 with . In particular, the ghost map induces a bijection between and some subalgebra of that respects the previous laws of composition, which gives the structure of an algebra. Now, the surjective homomorphism yields a surjective map , which endows with the structure of an algebra as well, thereby proving the following:
Proposition 1.9 ([12], Lemme 5.1)
The setvalued functor given by factors through a unique algebravalued functor
such that, for any algebra , the ghost map is a homomorphism of algebras.
In particular, is a algebra with Witt vector as the zero element, and Witt vector as the identity element.
An important remark is that, if is another uniformising parameter, there exists a unique isomorphism of functors, , between and , that commutes with the ghost maps (see Section 5.1 of [12]). In particular, this is the reason why elements of are simply called ramified Witt vectors.
Let be an algebra. There are three maps that play a crucial role in . The first is the Teichmüller lift , which is multiplicative and given by :
The second is the Frobenius map , defined uniquely by the polynomials introduced above. Precisely, as a consequence of Lemma 1.8, one can prove that this is the unique endomorphism of the algebra that satisfies :
As noticed in [12], these two maps do not depend on , in the sense that they commute with the isomorphism for any other uniformising element .
The last map is the Verschiebung map and it is additive :
Contrary to the others, this map depends on the choice of . Precisely, . Note also the relation .
These maps satisfy the following properties, most of which can be proved after being translated to ghost components using Lemma 1.8 :