Number Systems

estimates, which we already know to be elements ofZp[ζq−1], are in fact elements ofZp. It would be a good idea for readers to skip this section, and return to it only if they want to understand the use of the Frobenius action in the proofs of Theorems 4.18, 5.12, 6.13, and 7.14. Section 2.8 includes notations we use with multivariable polynomials. There we also include a basic fact about polynomials, which we use to prove that certain of our bounds on p-adic valuations of weights are sharp.

We consider certain algebraic extensions of Qp whose behavior is similar.

Proposition 2.2 (Unramified Extensions of the p-Adics). Qp(ζ_pⁿ−1) is a degree n Galois extension of Qp, and the Galois group of Qp(ζ_pⁿ−1) over Qp is the cyclic group of order n generated by the automorphism Fr which takes ζpⁿ−1 to ζ_p^pn−1. The elements of Qp(ζpⁿ−1) that are integral over Zp form the ring Zp[ζpⁿ−1]. Zp[ζpⁿ−1] is a discrete valuation ring of characteristic 0 with the unique nonzero prime ideal generated by p, and Qp(ζpⁿ−1) is the field of fractions of Zp[ζpⁿ−1]. Thus each nonzero element of Qp(ζpⁿ−1) can be written uniquely as p^mu with m ∈ Z and u a unit in Zp[ζ_pⁿ−1], where the nonzero elements of Zp[ζ_pⁿ−1] are precisely such elements with m ≥ 0, and the units in Zp[ζ_pⁿ−1] are precisely such elements with m = 0. Qp(ζ_pⁿ−1) and Zp[ζ_pⁿ−1] are complete in the topology defined by this valuation. The set consisting of zero and the powers of ζ_pⁿ−1 is a set of representatives of the equivalence classes modulo p in Zp[ζ_pⁿ−1]. The quotient modulop of Zp[ζpⁿ−1]is the finite field Fpⁿ, whose cyclic group of units is generated by the reduction modulo pof ζpⁿ−1. The automorphismFr onQp[ζpⁿ−1]induces an automorphism (which we shall also call Fr) on Fpⁿ; this induced automorphism takes each element to its pth power and it generates the Galois group of order n of Fpⁿ over Fp. Furthermore, Qp(ζ_pⁿ1−1)∩Qp(ζ_pⁿ2−1) =Qp(ζ_pgcd(n1,n2)−1) andZp[ζ_pⁿ1−1]∩Zp[ζ_pⁿ2−1] =Zp[ζ_pgcd(n1,n2)−1].

Thus, Qp(ζ_pⁿ1−1) ⊆Qp(ζ_pⁿ2−1) if and only if n₁ |n₂, and Zp[ζ_pⁿ1−1]⊆ Zp[ζ_pⁿ2−1] if and only ifn₁ |n₂.

Proof. This follows from Proposition 16 (along with Corollary 1) in Chapter IV of [47], assuming the theory developed in that book up until that point, most particularly Propo- sitions 3 and 8 of Chapter II.

Let us examine in more detail the valuation mentioned in Propositions 2.1 and 2.2 above.

For a nonzero element a∈ Qp(ζpⁿ−1), the unique integer m such that a = p^mu for some unit u ∈ Zp[ζ_pⁿ−1] is called the p-adic valuation of a. If a also lies in Qp(ζ_pn0

−1), then it is not hard to show that the p-adic valuation ofa in this other field is precisely the same as its p-adic valuation in the Qp(ζ_pⁿ−1). Thus no reference to the field containing a is necessary, and thep-adic valuation ofais simply denotedv_p(a). We definev_p(0) =∞, and we consider∞strictly greater than any integer and set anything plus∞to∞. We say that

two elementsaandbarecongruent modulop^mto mean thatvp(a−b)≥m. Thus the notion of equivalence of elements modulo powers ofpis independent of which unramified extension of Qp we regard as the ambient field. With these conventions, we have the following easily verified properties of thep-adic valuation:

Lemma 2.3 (Properties of vp). For any a, b∈Qp(ζpⁿ−1), we have the following:

(i) v_p(a) =∞ if and only if a= 0.

(ii) v_p(ab) =v_p(a) +v_p(b).

(iii) v_p(a+b)≥min{v_p(a), v_p(b)}, with equality when v_p(a)6=v_p(b).

For a ∈ Qp(ζpⁿ−1), we also define the p-adic absolute value of a, denoted |a|_p, to be p^−vp(a), where p^−∞ is considered to be 0. The properties of vp translate into properties of

|·|_p as follows:

Lemma 2.4 (Properties of |·|_p). For any a, b∈Qp(ζ_pⁿ−1), we have the following:

(i) |a|_p = 0 if and only if a= 0.

(ii) |ab|_p =|a|_p|b|_p.

(iii) |a+b|_p ≤max{|a|_p,|b|_p}, with equality when |a|_p 6=|b|_p.

Thus the p-adic absolute value provides a metric on Qp(ζ_pⁿ−1), where the distance between a and b is |a−b|_p. This metric defines a topology on Qp(ζ_pⁿ−1) that we call the p-adic topology. It is this topology that is discussed in Propositions 2.1 and 2.2 above.

In Proposition 2.2, we saw that the finite fields of characteristic p can be obtained as quotients modulo p of rings of algebraic integers in unramified extensions of Qp. We shall also be interested in the quotients of these rings modulo powers of p. The Galois ring of characteristic p^m and order p^mn, denoted GR(p^m, n) is the quotient modulo p^m of Zp[ζ_pⁿ−1]. Note that GR(p, n) is the finite fieldFpⁿ of orderpⁿ. Also note that GR(p^m,1) is the integer residue ringZ/p^mZ. The ring GR(p^m, n) contains Z/p^mZas a subring and can be thought of as an extension ofZ/p^mZobtained by adjoining a root of unity of orderpⁿ−1.

Furthermore, the statements regarding intersections and containments of extensions of Qp

andZp in Proposition 2.2 imply that GR(p^m, n1)∩GR(p^m, n2) = GR(p^m,gcd(n1, n2)), and therefore GR(p^m, n1)⊆GR(p^m, n2) if and only ifn1|n2. In this case, GR(p^m, n2) is a free GR(p^m, n1)-module. Note that GR(p^m, n) is a principal ideal ring withm+1 ideals, namely p^jGR(p^m, n₂) forj = 0,1, . . . , m. Herep⁰GR(p^m, n₂) is the entire ring andp^mGR(p^m, n₂) is the zero ideal. For more information on Galois rings, the reader should consult the book of McDonald [34].

It does not make sense to consider the ring GR(p^m1, n) as a subring of GR(p^m2, n) when m₁ < m₂, since the two rings have different characteristics. However, we do have a way of relating elements of the one ring to elements of the other. Since m1 < m2, reduction modulop^m1 furnishes an epimorphism from GR(p^m2, n) to GR(p^m1, n). Thus form1 ≤m2, if a ∈ GR(p^m2, n), we define πm1(a) ∈ GR(p^m1, n) to be the reduction modulo p^m1 of a.

Also, if a∈ Zp[ζpⁿ−1], we define πm1(a) ∈GR(p^m1, n) to be the reduction modulo p^m1 of a. Since ζ_pⁿ−1 is a root of unity of order pⁿ−1 over Qp and since π₁(ζ_pⁿ−1) is a root of unity of orderpⁿ−1 in Fpⁿ (see Proposition 2.2 above), we know that π_m(ζ_pⁿ−1) is a root of unity of order pⁿ−1 in GR(p^m, n) for every positive integer m. As a convention, we defineπ∞ to be the identity map onZp[ζ_pⁿ−1].

Now we wish to define a map from GR(p^m1, n) to GR(p^m2, n) whenm1 < m2. Proposi- tion 2.2 tells us that each element a ∈ Zp[ζpⁿ−1] can be written uniquely as P∞

i=0aipⁱ, where each ai is either zero or a power of ζpⁿ−1. We call this the canonical expan- sion of a ∈ Zp[ζpⁿ−1]. This implies that when m is a positive integer, each element a of GR(p^m, n) can be written uniquely as Pm−1

i=0 a_ipⁱ, where each a_i is either zero or a power of π_m(ζ_pⁿ−1). We likewise call this the canonical expansion of a ∈ GR(p^m, n).

For m₁ ≤ m₂ and a ∈ GR(p^m1, n), with canonical expansion a = Pm1−1

i=0 a_ipⁱ, we de- fine τ_m2(a) to be an element b ∈ GR(p^m2, n) with canonical expansion b = Pm1−1

i=0 b_ipⁱ, where b_i = 0 whenever a_i = 0 and b_i = π_m2(ζ_pⁿ−1)^j whenever a_i = π_m1(ζ_pⁿ−1)^j. We also define τ∞(a) = Pm1−1

i=0 cipⁱ, where ci = 0 whenever ai = 0 and ci = ζ_p^jn−1 when- ever ai = πm1(ζpⁿ−1)^j. For each positive integer m, we call πm the Teichm¨uller lift to characteristic p^m and call τ∞ theTeichm¨uller lift to characteristic 0.

Form1 ≤m2inZ+∪ {∞}, we haveπm1◦πm2 =πm1 andτm2◦τm1 =τm2. Furthermore,

ifa∈GR(p^m1, n), πm1(τm2(a)) =a, but if a∈GR(p^m2, n), then it is not always true that τm2(πm1(a)) =a.

To summarize the relationships between the various unramified extensions ofQp and the various Galois rings, we have the following commutative diagrams, where unmarked arrows are inclusion maps:

Qp(ζpⁿ−1) ←−−−− Zp[ζpⁿ−1] −−−−→^πm GR(p^m, n) −−−−→^π1 Fpⁿ

x





x





x





x



 Qp ←−−−− Zp

πm

−−−−→ Z/p^mZ −−−−→^π1 Fp

and

Qp(ζ_pⁿ−1) ←−−−− Zp[ζ_pⁿ−1] ←−−−−^τ∞ GR(p^m, n) ←−−−−^τm Fpⁿ

x





x





x





x



 Qp ←−−−− Zp

τ∞

←−−−− Z/p^mZ ←−−−−^τm Fp.

In each diagram, the two rows coincide when n = 1, and the last two columns coincide when m= 1.

We transplant the notion ofp-adic valuation from the unramified extensions ofQpto the Galois rings. The p-adic valuation of a nonzero elementa∈GR(p^m, n), denoted v_p(a), is defined to be the greatestksuch thata∈p^kGR(p^m, n). We definev_p(0) =∞in GR(p^m, n).

Thus we have defined v_p: GR(p^m, n) → {0,1, . . . , m−1,∞}. Note that for any m₁ ∈ Z, m₂ ∈Z+∪ {∞} with m₁ ≤m₂, anda∈GR(p^m1, a), we have v_p(τ_m2(a)) =v_p(a). On the other hand, ifm1≤m2 are positive integers anda∈GR(p^m2, n), thenvp(πm1(a)) =vp(a) if vp(a) < m1 or vp(a) = ∞, but vp(πm1(a)) = ∞ when m1 ≤ vp(a) < ∞. Likewise, if a ∈ Zp[ζpⁿ−1] and m1 is a positive integer, then vp(πm1(a)) = vp(a) if vp(a) < m1 or vp(a) =∞, butvp(πm1(a)) =∞when m1≤vp(a)<∞.

In Proposition 2.2, we used Fr to denote the field automorphism of Qp(ζ_pⁿ−1) that fixes Qp pointwise and takes ζ_pⁿ−1 to ζ_p^pn−1. Note that Fr restricted to Zp[ζ_pⁿ−1] is an automorphism of rings. We also used Fr to denote the field automorphism it induces on Fpⁿ through reduction modulo p; this automorphism takes every element to its pth power.

Because the automorphism Fr of the field Qp(ζpⁿ−1) takes p^m top^m, it permutes the ideal p^mZp[ζpⁿ−1] of the ringZp[ζpⁿ−1], and thus induces an automorphism of the ring GR(p^m, n)

for each positive m through reduction modulo p^m. This automorphism fixes pointwise the subring GR(p^m,1) =Z/p^mZof GR(p^m, n) and maps the elementπm(ζpⁿ−1) toπm(ζpⁿ−1)^p. Throughout the thesis we call all of these automorphisms theFrobenius automorphismand denote them all by Fr.

It is not difficult to show that Fr commutes withπ_mandτ_m for allm∈Z+∪ {∞}. That is, we have the commutative diagrams

Qp(ζpⁿ−1) ←−−−− Zp[ζpⁿ−1] −−−−→^πm GR(p^m, n) −−−−→^π1 Fpⁿ Fr





y ^Fr





y ^Fr





y ^Fr



 y Qp(ζ_pⁿ−1) ←−−−− Zp[ζ_pⁿ−1] −−−−→^πm GR(p^m, n) −−−−→^π1 Fpⁿ

and

Qp(ζpⁿ−1) ←−−−− Zp[ζpⁿ−1] ←−−−−^τ∞ GR(p^m, n) ←−−−−^τm Fpⁿ Fr





y ^Fr





y ^Fr





y ^Fr



 y Qp(ζ_pⁿ−1) ←−−−− Zp[ζ_pⁿ−1] ←−−−−^τ∞ GR(p^m, n) ←−−−−^τm Fpⁿ, where unlabeled arrows are inclusion maps.

Recall thatQp(ζ_pⁿ1−1)⊆Qp(ζ_pⁿ2−1) if and only ifn₁ |n₂, and recall that GR(pⁿ¹, m)⊆ GR(pⁿ², m) if and only if n₁ | n₂. Suppose n₁ | n₂. Since Fr generates the Galois group Gal(Qp(ζ_pⁿ2−1)/Qp) of order n₂ and the Galois group Gal(Qp(ζ_pⁿ1−1)/Qp) of order n₁, we see that Frⁿ¹ generates the Galois group Gal(Qp(ζ_pⁿ2−1)/Qp(ζ_pⁿ1−1)) of ordern₂/n₁. Thus an element of Qp(ζ_pⁿ2−1) is in Qp(ζ_pⁿ1−1) if and only if it is fixed by Frⁿ¹. Likewise, for any positive integer m, an element of GR(p^m, n2) is in GR(p^m, n1) if and only if it is fixed by Frⁿ¹. This can be checked by writing the canonical expansion of an arbitrary element of GR(p^m, n2) and applying Frⁿ¹. If any coefficient is neither zero nor a power ofπm(ζpⁿ¹−1), the element will not be fixed by Frⁿ¹; otherwise the element will be fixed.

If n₁ | n₂, we define the trace map Trⁿ_n²₁: Qp(ζ_pⁿ2−1) → Qp(ζ_pⁿ1−1) by Trⁿ_n²₁(a) = P(n2/n1)−1

j=0 Fr^n1j(a). Since Fr commutes with π_m for all positive integers m, it induces a trace map on the Galois rings, which we express with the same notation, i.e., we write Trⁿ_n²₁: GR(p^m, n₂) → GR(p^m, n₁). That this trace map is a surjective GR(p^m, n₁)-linear map follows from the fact that Trⁿ_n²₁:Qp(ζpⁿ²−1) → Qp(ζpⁿ¹−1) is a surjective Qp(ζpⁿ¹−1)- linear map. Of course Trⁿ_n²₁: GR(p, n2) → GR(p, n1) is just the usual trace from Fpⁿ² to

Fpⁿ¹. Since Fr commutes with the maps πm and τm for all m ∈Z+∪ {∞}, we also know that Trⁿ_n²₁ commutes with πm and τm for all n1, n2 ∈Z+ and m ∈Z+∪ {∞}. In terms of commutative diagrams, we have

Qp(ζ_pⁿ2−1) ←−−−− Zp[ζ_pⁿ2−1] −−−−→^πm GR(p^m, n₂) −−−−→^π1 Fpⁿ² Trⁿn²1





y ^Tr

n2 n1





y ^Tr

n2 n1





y ^Tr

n2 n1



 y Qp(ζ_pⁿ1−1) ←−−−− Zp[ζ_pⁿ1−1] −−−−→^πm GR(p^m, n₁) −−−−→^π1 Fpⁿ¹

and

Qp(ζ_pⁿ2−1) ←−−−− Zp[ζ_pⁿ2−1] ←−−−−^τ∞ GR(p^m, n₂) ←−−−−^τm Fpⁿ² Trⁿn²1





y ^Tr

n2 n1





y ^Tr

n2 n1





y ^Tr

n2 n1



 y Qp(ζpⁿ¹−1) ←−−−− Zp[ζpⁿ¹−1] ←−−−−^τ∞ GR(p^m, n1) ←−−−−^τm Fpⁿ¹, where unlabeled arrows are inclusion maps.