Leopoldt's Conjecture for Abelian and non-Abelian cases

50 Pages·2012·0.37 MB·English

Leopoldt’s Conjecture for Abelian and
non-Abelian cases
Eric Stansifer
Master’s Thesis, 2012 December 5
Advised by Prof. Massimo Bertolini
ALGANT Erasmus Mundus
Universiteit Leiden
Universita´ degli Studi di Milano
Contents
1 Introduction 3
1.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Notation and definitions 7
2.1 Construction of C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
p
2.2 p-adic analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 p-adic exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Leopoldt’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 p-adic Schwarz’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.6 Alternative asymptotic notation . . . . . . . . . . . . . . . . . . . . . 14
2.7 Bounds on nonzero algebraic elements . . . . . . . . . . . . . . . . . . 14
2.8 Vandermonde determinant . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Brumer’s Theorem on Q-relations of logarithms 17
3.1 Construction of a function with many zeros . . . . . . . . . . . . . . 17
3.2 Finding more zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Brumer’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4 Leopoldt’s Conjecture for abelian extensions 24
5 Leopoldt defect for non-abelian extensions 29
6 Masser’s Theorem 30
6.1 General results from commutative algebra . . . . . . . . . . . . . . . 30
6.2 Preliminary lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.3 Central lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6.4 Upper bound on the number of zeros of a polynomial . . . . . . . . . 35
7 Leopoldt defect for non-abelian extensions, II 37
7.1 Generalized Dirichlet exponents µ and χ . . . . . . . . . . . . . . . . 37
7.2 Bound on χ for polynomials with many zeros . . . . . . . . . . . . . . 38
7.3 Construction of a polynomial with many zeros . . . . . . . . . . . . . 41
7.4 Bound on the Leopoldt defect . . . . . . . . . . . . . . . . . . . . . . 43
Page 2 of 50
1 Introduction
The Dirichlet L-series L(s,χ) is a generalization of the Riemann-zeta function,
∞
(cid:88) 1
ζ(s) =
ns
n=1
given by
∞
(cid:88) χ(n)
L(s,χ) =
ns
n=1
for a Dirichlet character χ : Z → C (that is, χ(ab) = χ(a)χ(b) and for some positive
n, χn = 1 except where χ is zero). In both cases, the series converges for complex
numbers s with real part greater than 1, but can be extended meromorphically to
the whole complex plane, with a simple pole at s = 1 in ζ(s) = L(s,1).
The behavior of these functions near s = 1 is of central importance, and a
classical result of number theory gives that
L(1,χ) (cid:54)= 0
for χ (cid:54)= 1. A corollary of this is the famed Dirichlet’s Theorem, that for relatively
prime integers a and n, there are infinitely many prime numbers congruent to a
modulo n. The value of L(1,χ) also encodes information related to the regulator of
a number field K, and L(1,χ) (cid:54)= 0 implies that the regulator is nonzero, which gives
certain linear independence results on the units of the number field K.
Given the importance of the Dirichlet L-series, Heinrich-Wolfgang Leopoldt
defined a p-adic analog L (s,χ) to the Dirichlet L-series and raised the question in
p
1962 [7, p. 57] of whether one still has L (1,χ) (cid:54)= 0. Similarly, the value of L (1,χ)
p p
is related to the p-adic regulator of a number field K, and one has L (1,χ) (cid:54)= 0 if
p
and only if the p-adic regulator is nonzero.
In this document we will concern ourselves with an equivalent form of the
question that Leopoldt raised, namely whether a collection of linear independent
units in a number field K will also be Z -linearly independent in the p-adic
p
completions of K. Leopoldt’s Conjecture states that this always holds true:
that linearly independent units in K will always be Z -linearly independent in the
p
p-adic completions.
This conjecture was settled in the affirmative for abelian extensions K of Q by
Armand Brumer in 1967 [4], by completing a proof of James Ax in 1965 [2] that the
p-adic regulator of such K is nonzero. While the abelian situation is the only case
considered by Leopoldt, as that is the only situation in which the value of L (1,χ)
p
is relevant, the equivalent forms of Leopoldt’s Conjecture are meaningful for any
number field K of Q and it is natural to ask whether Leopoldt’s Conjecture holds
in these situations.
Page 3 of 50
So far no general proof for Leopoldt’s Conjecture has been found. We can
measure the degree of failure of Leopoldt’s Conjecture for a particular number field
withtheLeopoldt defect, andseveralresultshavebeenfoundthatgiveboundsonthe
Leopoldt defect. One substantial result was given by D.W. Masser [8] and Michel
Waldschmidt [10] in 1981.
This thesis then follows in two main movements. In our first part, sections 3 and
4, we prove that Leopoldt’s Conjecture is true for abelian extensions of Q (Theorem
22). We continue this proof in section 5 with a variation for non-abelian galois
extensions, which gives a weak bound on the Leopoldt defect (Theorem 29). In our
second part, sections 6 and 7, we prove the bound on the Leopoldt defect found by
Masser and Waldschmidt (Theorem 48).
1.1 Outline
Part of our motivation is to avoid dependencies on deep external results (except that
in section 6 we will need some facts of commutative algebra which we quickly review
in section 6.1), so we begin in section 2 with introducing the conventions that we
will use in the rest of the paper, as well as proving a number of general results that
will be needed.
The field C , which is the p-adic equivalent of the complex numbers C, will play
p
a key role throughout this paper, so in sections 2.1, 2.2, and 2.3 we develop the
basic notions of p-adic analysis, which is closely analogous to ordinary analysis on
C, and in particular investigate the p-adic exponential and logarithmic functions.
This culminates in section 2.5 with a p-adic form of Schwarz’s Lemma, which is
nicer to state than its analog for C.
In section 2.4 we state Leopoldt’s Conjecture and define the Leopoldt defect.
Several times in this paper we will be interested in the growth behavior of
functions f(x) for arbitrarily large x ∈ R; in particular, we can show that certain
algebraic numbers are zero if they are the values of functions that get arbitrarily
small, using Theorem 12 (section 2.7) to give a lower bound on the size of a non-
zero algebraic number. The standard notation for analyzing asymptotic behavior is
big-Oh notation, but to avoid problems with that notation we introduce a variation
on big-Oh notation in section 2.6.
We begin our proof of Leopoldt’s Conjecture in the abelian case with the proof
of Brumer’s Theorem (Theorem 19), which is the goal of section 3. This theorem
is a p-adic variation of Baker’s Theorem, proven by Baker in 1966 [3] (for which he
received the Fields Medal), which states that for algebraic numbers α ,...,α , if
1 n
thelogα areQ-linearlyindependent, thentheyareQ-linearlyindependent. Brumer
i
proved the analogous result with p-adic logarithms. While his proof closely follows
theoriginalbyBaker, Brumerwasabletomakeanumberofsimplificationsashewas
(cid:12) (cid:12)
not trying to establish a computable lower bound on (cid:12)β log α +···+β log α (cid:12),
1 p 1 n p n
but only to demonstrate that it is nonzero. Our proof presented here follows again
Page 4 of 50
the general form of Brumer’s proof very closely. Our Lemma 15 is a stronger form of
Brumer’sLemma1, whichhestateswithoutproof(byanalogytoBaker’sLemma2);
we found that by strengthening Lemma 15, certain later steps for proving Brumer’s
Theorem could be skipped altogether.
GivenBrumer’sTheorem,insection4weproveLeopoldt’sConjectureforabelian
extensions K of Q. The key step of this proof is Lemma 26, which is based on a
technique by James Ax [2, Lemma, p. 2]. An alternative approach, also using
Brumer’s Theorem, is due to Washington [11, Theorem. 5.25]. In both proofs
the essential step requires representation theory of abelian groups. As Ax proves
Leopoldt’s Conjecture in the form of the p-adic regulator being nonzero, which is
not the form that we stated the conjecture, we need Lemmas 23 and 24 to prove
that these two forms of the conjecture are equivalent (we will need these again for
Theorem 48). The direction of the equivalence that we require is trivial, but the
converse can be found in [11, Theorem. 5.31].
In [2, p.587], Ax alludes to extending his work to non-abelian cases, writing
“[This method] would also give information even when the Galois group G of K/Q
is not abelian.”. As we have isolated the requirement that G is abelian to a single
step of Lemma 26, we follow his advice in section 5 to give a Lemma 28, a trivial
variation of Lemma 26 that assumes no knowledge of the irreducible representations
of G of dimension greater than 1. This immediately gives us Theorem 29, a weak
bound on the Leopoldt defect for galois extensions K/Q.
In the second half of this paper we pursue a theorem of Waldschmidt, which uses
a deep result of Masser to give a bound on the Leopoldt defect (which is in fact
never weaker than the bound we found in Theorem 29).
In section 6 and section 7.2 we very closely follow the work of Masser in [8]
with the goal to prove Theorem 39. We start in section 6.1 by stating a number of
general results of commutative algebra, which can be found in [1] or [12]: namely,
the essential results of primary decompositions and their behavior in localizations;
Krull’s height theorem; and the Hilbert polynomial. We also need a few facts related
to the leading coefficient of the Hilbert polynomial proven in [5]. We then continue
in the remainder of section 6 and in section 7.2 closely mimicking the technique of
Masser.
This results in Masser’s Theorem (Theorem 39), which we will use in section
7. Given a polynomial of exponential functions with zeros of a particular form,
this theorem shows a relationship between the number of such zeros, the total
degree of the polynomial, and the distribution of the zeros. In section 7.3, we
construct(followingtheapproachofWaldschmidt[10])suchapolynomialwithmany
zeros and low degree, proving that the coefficient χ(X,Y) is small, which encodes
the fact that the zeros are not highly concentrated in low-dimensional subspaces.
The construction of the polynomial closely resembles the similar construction we
performed in section 3.1; both use the pigeon-hole principle in a very similar way
to show that the desired degree of the polynomial is high enough to find one that is
Page 5 of 50
small in many places, and uses Theorem 12 to show that, as the values are algebraic
numbers, they are so small that they must be zero.
We put together these facts in Theorem 47, loosely following the approach of
Waldschmidt, to show that the dimension of the C -space spanned by these zeros is
p
at least half the rank of the zeros. Then in Theorem 48, we show that these zeros
can be chosen to be the locations of the p-adic logarithms of the units of a number
field K. While this does not show that these zeros are C -linearly independent, we
p
do find that the dimension of the C -space spanned by them is at least half of their
p
Z-rank, which gives us our bound on the Leopoldt defect.
Page 6 of 50
2 Notation and definitions
2.1 Construction of C
p
Throughout this document we will fix a prime p. We denote by Q the field of
p
p-adic numbers (i.e. the completion of Q under the p-adic valuation) and by Q
p
the algebraic closure of Q . The valuation on Q can be uniquely extended to Q ,
p p p
and we denote by C the completion of Q under that valuation. Then we naturally
p p
obtain a valuation z (cid:55)→ |z| on C , whose normalization we fix by
p
1
|p| = .
p
The logarithm of the p-adic valuation gives the p-adic order v : C → Q with
p
v(p) = 1.
The field Q also carries the ordinary absolute value, which can be non-uniquely
extended to all of C . Let z (cid:55)→ |z| be the ordinary absolute value on C . The field
p ∞ p
C under|·| isisomorphictothefieldofcomplexnumbersasafield-with-valuation,
p ∞
therefore let C = C . We will use the symbol C when we wish to emphasize our
p
interest in the absolute value |·| as opposed to the valuation |·|.
∞
Let Q be the algebraic closure of Q in C, and let A be the integral closure of Z
in Q, i.e., the ring of algebraic integers. We say a number field is a finite extension
of Q in C, and a local field is a finite extension of Q in C . Then for every number
p p
field K and local field L we have inclusions
Q ⊂ K ⊂ Q ⊂ C
Q ⊂ L ⊂ Q ⊂ C .
p p p
For a number field K, let Θ be the set of all embeddings of K into C ; then
K p
|Θ | = [K : Q]. We write Θ for Θ when K is clear. Every θ ∈ Θ induces a
K K
valuation on K that extends the p-adic valuation on Q, so it must induce the p-adic
valuation for some prime p of K lying above p. Conversely, every p-adic valuation
on K is induced by some θ ∈ Θ. For each prime p of K lying above p, fix a choice of
θ ∈ Θ which induces the p-adic valuation on K. Let K be the completion of θ (K)
p p p
in C , and let K be the compositum of the K for all p above p. In particular, K
p p p p
and K are local fields.
p
Let U = {z ∈ C |z| = 1}. For reals 0 < r ≤ 1, define
p
U = {z ∈ C |z −1| < r} ⊂ U.
r p
Each u ∈ U can be written uniquely as u = ζ·u where ζn = 1 for some n relatively
1
prime to p and u ∈ U .
1 1
For a number field K let O = K ∩A be the integral closure of Z in K; for a
K
local field L let O = {z ∈ L |z| ≤ 1} be the integral closure of Z in L.
L p
Page 7 of 50
2.2 p-adic analytic functions
We give Cn the ultrametric norm; for z = (z ) ∈ Cn, let
p i p
|z| = max|z |.
i
i
Then for real r > 0, let
B−(r) = (cid:8)z ∈ Cn |z| < r(cid:9),
n p
B (r) = (cid:8)z ∈ Cn |z| ≤ r(cid:9).
n p
We will use multi-index notation in this document; for κ ∈ Zn, we write
zκ = zκ1···zκn,
1 n
(cid:107)κ(cid:107) = κ +···+κ ,
1 n
κ! = κ !···κ !,
1 n
where z ∈ Rn for some ring R (generally C ).
p
We give Zn the usual partial order (coordinate-wise comparison), and when
indexing over κ we shall always implicitly mean only κ ∈ Zn such that κ ≥ 0 (i.e.,
each coordinate non-negative). Furthermore, for nonnegative real numbers x we
write Zn(x) to mean
Zn(x) = {a ∈ Zn 0 ≤ a ≤ x for each i}.
i
Definition 1. A function f : B (r) → C is analytic (on B (r)) if there exist
n p n
f ∈ C such that
κ p
(cid:0) (cid:1)
lim |f |r(cid:107)κ(cid:107) = 0
κ
(cid:107)κ(cid:107)→∞
and f(z) = (cid:80) f zκ for all z ∈ B (r). We say f : B−(r) → C is analytic (on
κ≥0 κ n n p
B−(r)) if it is analytic on B (r(cid:48)) for all 0 < r(cid:48) < r.
n n
If f is analytic on B (r ) then the coefficients f of its power series are uniquely
n 0 κ
defined (and do not depend on r ). Let the radius of convergence R of f be the
0
(cid:0) (cid:1)
supremum of the r such that lim |f |r(cid:107)κ(cid:107) = 0; then we can extend f via its
(cid:107)κ(cid:107)→∞ κ
power series to all of B−(R). We will not need to deal with the case of the radius
n
of convergence being infinite.
Now if f = (cid:80)f zκ is analytic on B (r), we write
κ n
(cid:0) (cid:1)
|f| = sup |f |r(cid:107)κ(cid:107) .
r κ
κ
If L is a complete subfield of C containing all of the f , then the restriction of f to
p κ
B (r)∩Ln takes values in L. In this case we say that f is L-analytic (on B (r));
n n
similarly for B−(r).
n
Page 8 of 50
As a first application, we define exp and log via their power series. Let
p p
ε = p−1/(p−1),
the radius of convergence of exp . Now define
p
(cid:88)∞ zk
log (1−z) = − ,
p k
k=1
(cid:88)∞ zk
exp z = .
p k!
k=0
TheformerisQ -analyticonB−(1), andthelatteronB−(ε). Infact, z (cid:55)→ log (1−z)
p 1 1 p
and z (cid:55)→ 1−exp (z) are two-sided inverses from B−(ε) to B−(ε). Furthermore, the
p 1 1
definition of log (z) can be extended uniquely to all of C× via log p = 0 and
p p p
log ζ = 0 for all ζn = 1 with n relatively prime to p. These properties can be found
p
in Propositions 5.4, 5.5, and 5.7 of [11].
Finally, we define a differential operator ∆ for f analytic on B (r), via
1
(cid:88)
∆f = (k +1)f zk.
k+1
k≥0
Then ∆f is analytic on B (r), and satisfies the expected properties. For integers
1
m ≥ 0, let ∆m be the iterated differential operator. One sees that f has a zero at
z ∈ B (r) of multiplicity at least k if and only if (∆if)(z ) = 0 for all 0 ≤ i < k.
0 1 0
2.3 p-adic exponentiation
We will define the notation uz for u ∈ U and z ∈ B−(r ) for some r > 1 as an
ε 1 u u
extension of the ordinary definition for z ∈ Z. Before we can give this definition, we
will need to investigate the convergence properties of uk for u ∈ U ,k ∈ Z.
1
Lemma 2. For u ∈ U , we have
1
(cid:12) (cid:12) 1
(cid:12)log u(cid:12) ≤ .
p −e·log|u−1|
(cid:12) (cid:12)
If further u ∈ U , then (cid:12)log u(cid:12) = |u−1|.
ε p
Proof. Write u = 1+z, with |z| < 1. We have
(cid:18) (cid:18)zpa(cid:19)(cid:19)
v(log u) ≥ min v
p a≥0 pa
= min(pav(z)−a)
a≥0
Page 9 of 50
Regarding pav(z)−a as a function of a ∈ R, we find it has a unique global minimum
a ∈ R, given by pa0 = 1/(v(z)logp) or a = −log(v(z)logp)/logp. Thus
0 0
1 log(v(z)logp) log(−e·log|z|)
v(log u) ≥ pa0v(z)−a = + = ,
p 0 logp logp logp
which gives us the desired result. If additionally u ∈ U , then we find that a < 1,
ε 0
so it suffices to consider only a = 0 and a = 1. Comparing shows that the minimum
for pav(z)−a is achieved at a = 0, so v(log u) = v(z).
p
Lemma 3. For u ∈ U and nonnegative k ∈ Z, we have
1
(cid:12) (cid:12) 1
(cid:12)uk −1(cid:12) ≤ |k|· .
−e·log|u−1|
(cid:12) (cid:12)
If further u ∈ U , then (cid:12)uk −1(cid:12) = |k(u−1)|.
ε
Proof. Write u = 1+z, with |z| < 1. For nonnegative integers m,n, we have
v(cid:0)umpn −1(cid:1) = v(cid:0)(1+z)mpn −1(cid:1)
(cid:18) (cid:18)mpn(cid:19) (cid:19)
= v mpnz + z2 +···+zmpn
2
≥ minv(cid:0)pn−azpa(cid:1)
a≥0
= n+min(pav(z)−a).
a≥0
(This calculation required use of the p-adic valuation of binomial coefficients.)
Proceeding as in the previous lemma, we eventually find that for nonnegative k ∈ Z,
(cid:12) (cid:12) (cid:12) (cid:12) 1
(cid:12)u±k −1(cid:12) = (cid:12)uk −1(cid:12) ≤ |k|· .
−e·log|u−1|
If u ∈ U , we continue as in the previous lemma. In general, for k ∈ Z with |k|
ε
(cid:12) (cid:12) (cid:12) (cid:12)
sufficiently small, we have (cid:12)uk −1(cid:12) = (cid:12)klog u(cid:12), which can be derived in other ways
p
(e.g. [11, Lemma 5.5]).
Corollary 4. For every positive real r,r(cid:48) < 1, we have (U )pk ⊂ U for sufficiently
r(cid:48) r
large k. In particular, for every u ∈ U , we have upk ∈ U for sufficiently large k.
1 r
(cid:12) (cid:12)
For u ∈ U , let r = ε/(cid:12)log u(cid:12) = ε > 1. Then the function
ε u p |u−1|
exp (zlog u)
p p
is analytic on z ∈ B−(r ). Now if k ∈ B−(r )∩Z, we have
1 u 1 u
exp (klog u) = exp (log (uk)) = uk
p p p p
Page 10 of 50