Collusions in Teichm\"uller expansions
Trevor Hyde

TL;DR
This paper investigates recurring patterns in Teichmüller expansions within cyclotomic extensions, developing a unified theory based on affine group dynamics to explain these phenomena.
Contribution
It identifies three patterns in Teichmüller expansions and introduces a new group action framework to unify and explain these observations.
Findings
Identified three recurring patterns in Teichmüller expansions.
Developed a unifying theory using affine group actions.
Explained the patterns through the dynamics of group actions.
Abstract
If is a prime ideal over in the th cyclotomic extension of , then every element of the completion has a unique expansion as a power series in with coefficients in called the Teichm\"uller expansion of at . We observe three peculiar and seemingly unrelated patterns that frequently appear in the computation of Teichm\"uller expansions, then develop a unifying theory to explain these patterns in terms of the dynamics of an affine group action on .
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAlgebraic Geometry and Number Theory · Analytic Number Theory Research · Homotopy and Cohomology in Algebraic Topology
Collusions in Teichmüller expansions
Trevor Hyde
Dept. of Mathematics
University of Michigan
Ann Arbor, MI 48109-1043
(Date: April 25th, 2017)
Abstract.
If is a prime ideal over in the th cyclotomic extension of , then every element of the completion has a unique expansion as a power series in with coefficients in called the Teichmüller expansion of at . We observe three peculiar and seemingly unrelated patterns that frequently appear in the computation of Teichmüller expansions, then develop a unifying theory to explain these patterns in terms of the dynamics of an affine group action on .
1. Introduction
Let be a prime, a power of , and let be a primitive th root of unity. For any prime ideal over , we have an isomorphism . Let be the reduction modulo map. We call a section of a lift. Given a lift , there is a unique way to expand any element of the completion as a power series in with coefficients in .
While there are many lifts of , there is a unique multiplicative lift called the Teichmüller lift [2, Chp. XII, Exer. 16]. Let denote the multiplicative group of th roots of unity. Since is coprime to , the reduction map restricts to an injective group homomorphism . Extending the restriction to include 0 we see that is an isomorphism of multiplicative monoids. The Teichmüller lift is defined as the inverse of this isomorphism, hence is multiplicative. Because is the unique multiplicative lift, can be seen as a canonical set of coefficients with which to expand elements of . We refer to the expansion of an element with respect to as the Teichmüller expansion of .
The extension of complete local rings has degree where and is unramified. There is a unique unramified extension of of each degree up to isomorphism [5, Thm. 3], hence provides a model. Teichmüller expansions give a formal way to construct this unique extension of in terms of alone; these are the -typical Witt vectors [5, Chp. 2 §6]. More generally, Teichmüller lifts are essential to the construction of the big Witt vectors for any commutative ring [1]. Thus one reason to be interested in Teichmüller expansions is to understand the ring structure of Witt vectors . The elements of may be written as power series in with coefficients in . Our coefficients are closed under multiplication–this is the characteristic property of the Teichmüller lift –but are not closed under addition. The additive structure is complicated by “carrying.” Hence we need to compute the Teichmüller expansions of in order to do arithmetic in .
Teichmüller expansions are laborious to compute by hand and in the case are less convenient than the usual expression of elements as power series in with integral coefficients . The difficulty is circumvented with the help of a machine. Peculiar patterns frequently arise as one computes Teichmüller expansions. In this paper we observe three seemingly unrelated phenomena and develop a unifying theory to explain them in terms of the dynamics of an affine group action on the global ring .
We collect our observations in Section 2, followed by a review of cyclotomy in Section 3. We obtain results in Section 4 and apply them to explain our examples in Section 5.
2. Observations
Recall that is a prime, is a power of , and is a primitive th root of unity. Given an element and a prime ideal over , the Teichmüller expansion of at is the unique series
[TABLE]
such that for each . The are called Teichmüller coefficients of at .
To compute explicit Teichmüller expansions we must first choose a prime over in . The Kummer-Dedekind theorem [4, Chp 1, 8.3] says that
[TABLE]
where is congruent modulo to an irreducible factor of the cyclotomic polynomial in . There are such irreducible factors of degree .
For the moment, let and let be a th root of unity. The cyclotomic polynomial
[TABLE]
factors into a product of degree 4 irreducible polynomials over whose reductions modulo 2 are:
[TABLE]
Let .
Below are examples of Teichmüller expansions at of sums of two distinct roots of unity.
[TABLE]
No apparent patterns emerge in the sequence of Teichmüller coefficients for an individual expansion. However, we do see some striking relationships between the expansions of different elements. First notice the conspicuous [math] appearing as the coefficient of in each expansion. Continuing the expansions this phenomenon persists:
[TABLE]
Looking closer we see the coefficients of and match in each expansion
[TABLE]
suggesting these expansions may actually be the same under a permutation of the coefficients. The table below supports this claim, showing distributions of the 16 digits in each of the four expansions up to 500 terms.
[TABLE]
The rows are in fact permutations of one another with enough distinct entries to almost determine the bijection between them. Notice that the permutations appear to fix zero. We call this phenomenon the permutation conspiracy: seemingly unrelated elements of having the same Teichmüller expansion up to a permutation of the coefficients fixing zero. We explain the permutation conspiracy in Section 5.
Not every Teichmüller expansion of at is a permutation of one seen above. Here are examples of periodic expansions:
[TABLE]
Note that the exponents on the left hand side differ by a multiple of 5 in each case.
The following expansions are related by a permutation conspiracy but also have restricted coefficients taken from the set .
[TABLE]
The exponents on the left hand side differ by multiples of 3 in each case.
How do we account for these special expansions with periodic or restricted coefficients? Can we predict when such phenomena will occur and what coefficients will appear? An affirmative answer is provided in Section 5.
Still working with , we now compare the Teichmüller expansions of an element at both of the primes over .
[TABLE]
In each example, the product is independent of whenever its nonzero. The values of the products are respectively (recall that is a 15th root of unity.) We refer to this relationship between the Teichmüller coefficients of at different primes as prime collusion.
To get a better sense of prime collusion, let us consider examples when . Then is a 63rd root of unity. The polynomial factors into degree 6 irreducible polynomials in .
[TABLE]
Let \mathfrak{p}_{i}=\big{(}2,g_{i}(\zeta)\big{)}. Each element has 6 expansions, for instance:
[TABLE]
Again we have a constant product:
[TABLE]
for all when the product is nonzero. However, observe that for ,
[TABLE]
These pairwise products refine the collusion noted between all 6 primes. This may lead us to expect collusions to come in pairs, but the next example shows collusion in triples of primes:
[TABLE]
Here we have constant triple products for and :
[TABLE]
Furthermore, gives another example of the restricted coefficient phenomenon, since are the only coefficients appearing in any of these expansions. Note that in all cases we have seen of the restricted coefficients phenomenon, the total number of permissible Teichmüller representatives has been a power of .
The remainder of the paper is divided into three sections. Section 3 reviews background on cyclotomic fields. Section 4 develops theory which we use in Section 5 to explain the permutation conspiracy, restricted coefficients phenomenon, and prime collusion. All three are related to the dynamics of an affine group action on established in Theorem 3.
3. Background
We review the basic theory of cyclotomic fields. Proofs may be found in Lang [3, Chp. IV]. To begin, the polynomial factors in as
[TABLE]
where is an irreducible polynomial of degree called the th cyclotomic polynomial. Recall that is Euler’s totient function. The roots of are primitive th roots of unity: algebraic integers such that and for any proper divisor . If is a primitive th root of unity and is an integer coprime to , then is again a primitive th root of unity. Furthermore, every primitive th root of unity may be expressed as for some with . Therefore splits completely in and is a Galois extension of degree with Galois group canonically isomorphic to . We use this isomorphism frequently without further comment, writing to denote the action of but also viewing as a unit modulo as in .
If is a prime, then is typically not irreducible in . By Hensel’s lemma [4, Lem. 4.6], the factorization is determined by the orbits of Frobenius on the primitive th roots of unity. If , then and the size of the orbits of Frobenius on the primitive th roots of unity is the same as the multiplicative order of modulo . That is, if is minimal such that , then factors into irreducible degree polynomials in . The case of interest to us is when . With written in this form it is clear that and is the multiplicative order of modulo .
The affine group is defined by
[TABLE]
where the elements are considered as linear functions in the formal variable and the group operation is composition of functions:
[TABLE]
Another way to view is as a semidirect product
[TABLE]
If is a subgroup, then is the subgroup
[TABLE]
There is a -linear action of on given by
[TABLE]
In particular, if , then .
If , let be the subgroup generated by . If is a prime in over , then for each since is the decomposition group of each over .
In Section 4 we are interested in the number of fixed points of an element in . We determine this in Proposition 2. First, a lemma.
Lemma 1**.**
There exist polynomials such that
[TABLE]
It follows that for any integer we have
[TABLE]
Proof.
If with , then
[TABLE]
Thus we can follow the usual Euclidean algorithm to get the desired polynomial identity (1). Dividing (1) by we have
[TABLE]
in . Evaluating at we deduce
[TABLE]
Multiplying by yields the identity (2). ∎
Proposition 2**.**
The element has fixed points in iff , and in that case it has precisely fixed points in . Since [math] is always fixed, it follows that the total number of fixed points in is .
Proof.
If is a fixed point of , then hence in . Let . Then by Lemma 1. Reducing modulo we conclude that .
Supposing , the linear equation has a unique solution modulo which lifts to distinct solutions modulo . Thus has a total of fixed points in . ∎
4. Theory
The Teichmüller expansion of at is defined locally as an infinite sum which does not converge in the global ring . Nevertheless, Theorem 3 shows that the global Galois group acts nicely on Teichmüller expansions.
Recall our definition of the affine group and its subgroup assuming ,
[TABLE]
Theorem 3**.**
Let be a primitive th root of unity. Suppose is a prime over , and . If , then
[TABLE]
In other words, if
[TABLE]
is the Teichmüller expansion of at , then
[TABLE]
is the Teichmüller expansion of at .
Proof.
Let be the sum of the first terms of the Teichmüller expansion of at . Then
[TABLE]
is the unique element of which may be written as a polynomial in of degree less than with coefficients in such that . If , then . Since is fixed by we have
[TABLE]
which is a polynomial in of degree less than with coefficients in . Hence
[TABLE]
by uniqueness. This implies that for each . ∎
To summarize Theorem 3, the affine group acts coordinatewise on Teichmüller expansions while permuting the primes over .
Next we deduce three results using Theorem 3 to explain the permutation conspiracy, restricted coefficient phenomenon, and prime collusion in Section 5. Proposition 4 applies to the permutation conspiracy.
Proposition 4**.**
If and , then
[TABLE]
Hence the Teichmüller expansion of at is the same as that of at up to a permutation of the coefficients fixing 0.
Proof.
By Theorem 3 we have
[TABLE]
since fixes (see Section 3). Note that every element of fixes [math] and the rest of the permutation claim follows from being closed under the action of . ∎
Proposition 5 helps us understand the restricted coefficients phenomenon.
Proposition 5**.**
If is invariant under , then the Teichmüller coefficients of at any prime over are fixed points of . If , then and there are fixed points of in .
Proof.
Theorem 3 implies that for ,
[TABLE]
so the Teichmüller coefficients of at are fixed points of . If , then there is some nonzero Teichmüller coefficient. Hence has fixed points in . Proposition 2 tells us and that the total number of fixed points in is . ∎
Finally, Proposition 6 explains prime collusion.
Proposition 6**.**
Suppose is invariant under an element of order . Then for any and prime over ,
- (1)
If , then for all . 2. (2)
If , then
[TABLE]
where are given by
[TABLE]
Alternatively, are the unique elements such that for each maximal prime power divisor ,
- (a)
If , then
[TABLE] 2. (b)
If then
[TABLE]
where and denotes the normalized -valuation of .
Proof.
Let denote the th iterate of in . Then Theorem 3 implies that for each
[TABLE]
since is fixed by . So (1) follows from [math] being fixed by .
Now suppose . Then
[TABLE]
where
[TABLE]
Note that may be 0 in which case we interpret the action of as simply multiplication by . It does not seem possible to find simple evaluations for these sums modulo , but we can do so modulo the maximal prime power divisors and then use the Chinese Remainder Theorem to show that these local computations uniquely determine and .
If , then the sums simplify to well-known values,
[TABLE]
If , then
[TABLE]
which implies with .
Next we compute
[TABLE]
where the first equality results from switching the order of summation and reindexing. Multiplying by yields
[TABLE]
Multiplying by again we have
[TABLE]
So . ∎
5. Application
We revisit the examples from Section 2, applying the results from Section 4 to explain the conspiracies and collusion.
Permutation conspiracy
Let and let be a th root of unity. Proposition 4 implies that the Teichmüller expansion at of any element in the orbit of is a permutation of the Teichmüller expansion of at fixing 0.
Recall these Teichmüller expansions at \mathfrak{p}_{1}=\big{(}2,\zeta^{4}+\zeta+1\big{)} when from Section 2:
[TABLE]
The permutation conspiracies are consequences of the following calculations:
[TABLE]
It’s important to note that each element of above has the form .
The permutations are determined explicitly by the linear functions; applying to the exponents of the Teichmüller coefficients in yields the expansion of below it. The group has order and the element has a trivial stabilizer, thus an orbit with 60 elements. Therefore, of the sums of the form with , approximately of them will be permutations of the expansion of . More generally for any , always has trivial stabilizer under , hence the proportion of which are permutations of is
[TABLE]
We saw two periodic expansions
[TABLE]
which are related by . The periodic expansion itself is special and can be understood by summing the geometric series which converges locally:
[TABLE]
This identity is equivalent to , telling us is a primitive 3rd root of unity.
Restricted coefficients
Our last example of the permutation conspiracy also exhibited the restricted coefficient phenomenon.
[TABLE]
The permutations follow from
[TABLE]
To see why the coefficients all belong to , notice that is invariant under . Proposition 4 says the coefficients of the Teichmüller expansions of at both primes and are invariant under . The fixed points of in are precisely . This set has elements, as predicted, since and .
Prime collusion
We observed that the product of the Teichmüller coefficients of over certain groupings of primes were often independent of the index and always restricted. For example, with we had
[TABLE]
Then appears to be true for each when the product is not zero. To verify this, notice that is invariant under the order two element . Proposition 6 tells us
[TABLE]
where and . Hence,
[TABLE]
Since , the products are independent of . Proposition 7 shows this is always the case for .
Proposition 7**.**
If with , then is invariant under . Let when \sigma=\scalebox{0.6}[1.0]{-}1. Then
[TABLE]
Proof.
Verifying the invariance of under is straightforward. We apply Proposition 6 with \sigma=\scalebox{0.6}[1.0]{-}1 and . In this simple case we may evaluate the summations for and directly: and . We conclude that when ,
[TABLE]
∎
With we saw collusion in triplets of primes together with restricted coefficients.
[TABLE]
The element is invariant under the order three subgroup generated by . The fixed points of are , hence the restricted coefficients in the expansions above by Proposition 5. The collusion in the triplets is caused by the invariance of under the order three element . Using Proposition 6 we compute
[TABLE]
The value of is irrelevant since . Hence
[TABLE]
Since for we conclude that
[TABLE]
whenever .
Conclusion
Permutation conspiracies, restricted coefficient phenomena, and prime collusions are three readily apparent and seemingly unrelated patterns emerging in the Teichmüller expansions of sums of roots of unity. All three are consequences of the linear action of the affine group on : permutation conspiracies occur between elements in the same orbit under an element of ; restricted coefficients occur for elements fixed under some element of ; and prime collusions occur for elements fixed under a general element of .
Acknowledgements
The author thanks Bob Lutz for helpful feedback on this manuscript.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] Hazewinkel, Michiel. “Witt vectors. part 1.” Handbook of algebra 6, 2009. 319-472.
- 2[2] Lang, Serge. Algebra. Vol. 211. Springer Science & Business Media, 2002.
- 3[3] Lang, Serge. Algebraic number theory. Vol. 110. Springer Science & Business Media, 2013.
- 4[4] Neukirch, Jürgen. Algebraic number theory. Vol. 322. Springer Science & Business Media, 2013.
- 5[5] Serre, Jean-Pierre. Local fields. Vol. 67. Springer Science & Business Media, 2013.
