NON-VANISHING RESULTS OF SPECIAL VALUES OF L-FUNCTIONS
ERIC URBAN
NOTES TAKEN BY PAK-HIN LEE
Abstract. Here are the notes I am taking for Eric Urban’s ongoing course on non-vanishing
results of special values of L-functions offered at Columbia University in Spring 2015 (MATH
G6675: Topics in Number Theory). As the course progresses, these notes will be revised. I
recommend that you visit my website from time to time for the most updated version.
Due to my own lack of understanding of the materials, I have inevitably introduced
both mathematical and typographical errors in these notes. Please send corrections and
comments to [email protected].
Contents
1. Lecture 1 (February 5, 2015)
2. Lecture 2 (February 10, 2015)
3. Lecture 3 (February 12, 2015)
2
6
9
Last updated: February 16, 2015.
1
1. Lecture 1 (February 5, 2015)
I want to talk about certain non-vanishing results for special values of L-functions. There
are a lot of features about this kind of statements. Some of the ones I am interested in
are obtained by algebraic and ergodic methods. Using this method we can even get nonvanishing in positive characteristic, which has a lot of applications in arithmetics. Let me
give a general picture of the L-functions we are interested in.
We will be studying motivic L-functions. Let M be a pure motive over Q, which will
always assumed to be irreducible. We can think of this as several realizations:
• `-adic realization: a Q` -vector space M` with action of GQ . This Galois action is
unramified outside finitely many primes containing `.
• de Rham realization: a Q-vector space MdR .
• Betti realization: a Q-vector space MB .
We have the comparison isomorphisms
MB ⊗ Q` ∼
= M`
and
MB ⊗ C ∼
= MdR ⊗ C.
The second isomorphism gives us periods, which are defined modulo rational numbers. But
if we fix a lattice that is stable under Galois, we can define periods up to `-adic units. These
will be the Deligne periods ΩM ⊂ C× /Q× .
The assumption that the motive M is pure implies the following. If we take a prime q 6= `
such that M` is unramified at q, then we can look at the characteristic polynomial
Y
P (Frobq , X) =
(X − αq,i )
i
where αq,i ∈ Q with a fixed embedding Q ,→ C. For M to be pure, we have
w
|αq,i | = q 2
where w is a fixed integer called the motivic weight of M . We can see that the polynomial
P belongs to E[x], where E is the field of coefficients, with degree equal to dimE MB . We
can also see that P is independent of `, and M` is called a compatible system of Galois
representations.
Once we have this, we can form an L-function
Y
L(M, s) =
Pq (M, q −s )
q
I
Frobq |M` q ),
with Pq (M, X) = det(1 − X
called the local factor at q. From the hypothesis
that M is pure, this Euler product L(M, s) is convergent if Re(s) > 1 + w2 . Moreover,
L(M, s) 6= 0 as long as Re(s) > 1 + w2 . This is just a formal argument:
Proof. Take the logarithm of the Euler product.
Let M 0 = M ∨ (w). Then M 0 is pure of the same weight. Conjecturally we have a functional
equation of the form
L(M 0 , 1 + w − s) = (M, s)L(M, s).
2
This epsilon factor is constructed at infinity and primes of ramification. The archimedean
factors are expressed in terms of gamma factors, which are expressed in terms of the Hodge
numbers of MdR . Using the functional equation, we know the behavior (vanishing or nonvanishing) at Re(s) < w2 . What is unknown and non-trivial is the vanishing or not of L(M, s)
for Re(s) ∈ [ w2 , w2 + 1], which is called the critical strip.
Example 1.1. Taking M to be the trivial motive (1-dimensional space with trivial Galois
action), the periods are powers of 2πi and L(M, s) is the Riemann zeta function ζ(s). The
result ζ(s) 6= 0 for Re(s) = 1 is already non-trivial.
There are two important cases to consider:
(1) the central value s = w+1
if this is a critical value (in particular it is an integer),
2
w
(2) s = 2 + 1 if this is an integer.
(1) and (2) cannot happen at the same time.
Let me give a list of examples that we will be considering. This kind of vanishing or
non-vanishing results is not known in many situations. We will consider only cases when
M is potentially self-dual or conjugate self-dual. “Potentially” is not standard terminology.
By “potentially self-dual” I mean there is an abelian extension F/Q such that M`0 ∼
= M` as
0 ∼
c
GF -modules, and “potentially conjugate self-dual” means M` = M` as GM -modules, where
M is defined over a CM field M0 and M/M0 is an abelian extension.
We will start with something self-dual or conjugate self-dual, and consider the twists by
finite order characters. For central values, we will always consider the case when M is
self-dual or conjugate self-dual.
In the self-dual case, the functional equation is
L(M, 1 + w − s) = (M, s)L(M, s).
We can define (M ) = (M, w+1
) = ±1. If (M ) = −1, then L(M, 1+w
) vanishes with odd
2
2
1+w
order. If (M ) = 1, then L(M, 2 ) vanishes with an even order, which could be zero in
which case we have non-vanishing.
What we will consider is a family of such self-dual or conjugate self-dual motives, and we
ask if we have non-generic vanishing in the second case.
I will continue by going into more examples.
Example 1.2.
(1) Dirichlet motives (Dirichlet L-functions): take a Dirichlet character χ and consider
Y
L(χ, s) =
(1 − χ(p)p−s )−1
p
with the understanding that χ(p) = 0 if p divides the conductor of χ.
(2) Hecke L-functions attached to CM fields: consider χ, a Hecke character of a CM field
M of type A0 (this is Weil’s terminology) given by
Y
χ((α)) =
σ(α)κσ
σ:M ,→Q
where κσ ∈ Z.
3
(3) Modular forms: let f be a cuspidal modular form of weight k and Nebentypus χ, and
consider its L-function L(f, s). The associated motive Vf is of weight k − 1, and we
0
have Vf,`
= Vf,` ⊗ χ−1 .
(4) Take two modular forms f and g. Consider M = Mf ⊗Mg . Then L(M, s) = L(f, g, s)
is the Rankin–Selberg L-function attached to f, g. We need to assume weight(f ) >
weight(g), and moreover g is CM. We have
L(f, g, s) = LK (f, χ, s)
where K is an imaginary quadratic field, and χ is an anti-cyclotomic Hecke character
of K.
Let us state the kind of results we have.
In the case of Dirichlet L-functions, we are interested in critical values. Suppose χ is a
Dirichlet character. Then L(χ, s) 6= 0 if Re(s) ≥ 1 (at s = 1 this is non-trivial). For n ≥ 1,
L(χ, 1 − n) =
where Bn,χ is the Bernoulli number defined by
X χ(a)teat
a
mod N
1 − et
Bn,χ
∈ Q(χ)
n
=
X
Bn,χ
n
tn
.
n!
By the functional equation, we get something for the positive integers, but we have to divide
by some power of π. Actually we don’t get any algebraicity result for the positive integers
because the gamma function has a pole. If χ is odd,
√ Bn,χ is zero for n odd.
If χ is the quadratic character attached to Q( −p)/Q, we have a formula
√
√
p
L(1, χ) = h(Q( −p))
π
√
for the class number of Q( −p).
In my original introduction I only mentioned about non-vanishing, but let me add this.
When s = w2 + 1 or w+1
is critical, we can look at the algebraic part
2
Lalg (M, s0 ) :=
L(M, s0 )
ΩDeligne
M,`
.
Deligne conjectured that this is an algebraic number, and if the `-adic unit is chosen appropriately this will even belong to Z` . Then we can look at the reduction modulo `.
Going back to the quadratic character above, the left hand side is Lalg (1, χ). The arithmetic meaning of vanishing modulo ` of this special value is the divisibility by ` of a class
number.
I only wrote the case of quadratic extensions, but we can do this for all Dirichlet characters. In particular, an interesting case is when χ varies in the set of Dirichlet characters of
conductor pn , for a fixed prime number p. The arithmetic meaning is that we are looking at
the class number h(Kn ), where Kn = Q(ζpn ). This is becoming Iwasawa theory. There are
two interesting cases:
(1) If ` 6= p, then we can look at ord` (h(Kn )) and we ask the behavior as n varies.
This is basically looking at ord` (Lalg (χ, 1)) modulo ` when χ is of conductor pn . For
4
almost all `, this is 0. The question is: what is the behavior of ord` (Lalg (χ, 1)) and
ord` (h(Kn )) when n → +∞?
(2) If ` = p, Iwasawa theory tells you that
en = ordp (h(Kn )) = µpn + λn + ν
when n 0. We have the Iwasawa algebra Λ = Zp [[T ]], and X is the Λ-module such
n
that X ⊗ Λ/((1 + T )p Q− 1) ∼
= Cl(Kn ). In
Qthis case we have the characteristic ideal
f (T ) ∈ Λ, where X ' Λ/(fi ) and f = fi .
Using the Weierstrass preparation theorem, we can write
f (T ) = pµ × P (T ) × u(T )
where P (T ) is a polynomial of degree λ and u(T ) ∈ Λ× . If we replace the class
group by the anti-cyclotomic part, f (T ) is the product of p-adic L-functions (Mazur–
Wiles). In characteristic p = `, the non-vanishing is equivalent to the vanishing of
the µ-invariant.
In case (1) and (2), we have the
Theorem 1.3 (Ferrero–Washington).
(1) If ` 6= p, then ord` (h(Kn )) is bounded when n → +∞. In other words, Lalg (1, χ) 6≡ 0
mod ` for almost all χ of conductor pn .
(2) If ` = p, then µ = 0. In other words, ordp (h(Kn )) = λn + ν when n 0.
I will not explain the proof, but I will state one key lemma which is used to prove this
theorem. We will see that ergodic theory is under the picture. If β ∈ Zp , we denote by
sn (β) ∈ Z ∩ [1, pn − 1] the number such that β ≡ sn (β) mod pn . and
xn (β) = p−n sn (β) ∈ [0, 1).
Lemma 1.4 (Key Lemma). Fix γ1 , · · · , γr ∈ Zp . Assume that γ1 , · · · , γr are Q-linearly
independent. Consider
Xn (β) = (xn (βγ1 ), xn (βγ2 ), · · · , xn (βγr )) ∈ [0, 1)r .
For almost all β ∈ Zp , as n → +∞, Xn (β) is equidistributed in [0, 1)r .
Let me finish by stating an equivalent form of this result. These numbers look like random
variables, in a strong way. There is a way to express this result in terms of one-parameter
subgroups. As an analogy,
Theorem 1.5 (Kronecker). Let γ1 , · · · , γr ∈ R be Q-linearly independent. Consider
(tγ1 , tγ2 , · · · , tγr ) ∈ (R/Z)r
when t ∈ R. Then the set {(tγ1 , tγ2 , · · · , tγr ) : t ∈ R} is dense in (R/Z)n . More generally,
for arbitrary
γi , the closure of {(tγ1 , tγ2 , · · · , tγr )} in (R/Z)r is a sub-torus of dimension
P
rankQ ( i Qγi ).
Next time we will see that there is another proof of Ferrero–Washington by Sinnott, whose
proof is more analogous with Kronecker’s result. Orbits being sub-torus is something very
general that we will see in results for next time.
5
2. Lecture 2 (February 10, 2015)
Last time I stated the following result of Ferrero–Washington.
Theorem 2.1.
(1) (Ferrero–Washington) ` = p. We have the vanishing of the µ-invariant of the Kubota–
Leopoldt p-adic L-functions.
(2) (Washington) ` 6= p. Fix a Dirichlet character λ of conductor N > 1, and consider
L(0, λχ) ∈ Fp when χ runs in the set of Dirichlet characters of conductor a power of
` such that χ factors through Z×
` /µ`−1 , i.e.,
χ
Z×
`
/
:
µ` ∞
Z×
` /µ`−1
where the image of χ is an `-power root of unity. Then L(0, λχ) 6= 0 ∈ Fp except for
finitely many such χ.
I said that one of the arguments in the proof involves some ergodic results. Recall that
we defined xn (β) ∈ [0, 1) for β ∈ Zp .
Lemma 2.2 (Key Lemma). Let γ1 , · · · , γr ∈ Zp be Q-linearly independent. Then the image
of
xn (β) = (xn (γ1 β), · · · , xn (γn β)) ∈ [0, 1)r
when n → ∞ and β ∈ Zp is dense in [0, 1)r .
I will not use this result, but will make an analogy with the
Theorem 2.3 (Kronecker). Let γ1 , · · · , γr ∈ R be Q-linearly independent. Then the image
(tγ1 , · · · , tγr ) ∈ (R/Z)r
for t ∈ Q is dense.
Here we are looking at orbits inside a torus, which is very general.
Another proof of Ferrero–Washington is by taking another analogy of Kronecker’s result.
Recall that for a ∈ Zp , the power series
X a
a
a
T = (1 + (T − 1)) =
(T − 1)n mod p ∈ Fp [[T − 1]]
n
r
makes sense.
Lemma 2.4 (Sinnott). Let γ1 , · · · , γr ∈ Zp be Q-linearly independent. Then the power series
T γ1 , · · · , T γr are algebraically independent in Fp [[T − 1]], i.e., the map Fp [X1 , · · · , Xr ] →
Fp [[T − 1]] given by Xi 7→ T γi is injective.
We can see this as saying that the induced image of the one-parameter formal torus inside
is dense.
Grm
6
Our goal now is to sketch Sinnott’s proof of Washington’s theorem for ` 6= p. Recall that
we want to look at the L-values as generalized Bernoulli numbers
X λ(a)teat X
tn
=
B
.
(1)
n,λ
Nt − 1
e
n!
a
We have
L(1 − n, λ) = −
Bn,χ
,
n
and in particular
L(0, λ) = −B1,λ .
From (1) we see that
B1,λ =
X λ(a)a
a
N
.
For a general λ 6= 1, we introduce the following power series
X
Φλ (t) =
λ(n)tn ,
n
where, of course, it is understood that if n is not prime to the level of λ, then λ(n) = 0. So
we can write this as
P
X X
λ(a)ta
a+kN
Φλ (t) =
λ(a)t
=
∈ OGm ,1 ,
N
1
−
t
k
(a,N )=1
which is the local ring of Gm over Fp (i.e., of Spec Fp [t, t−1 ]) at 1. (A priori we only know
P
Φλ (t) ∈ Fp [[t]].) Since a mod N λ(a) = 0, we can rewrite
P
λ(a)(ta − 1)
Φλ (t) =
1 − tN
and hence
P
− λ(a)a
Φλ (1) =
= −B1,λ = L(0, λ).
N
We can see the rational function Φλ ∈ Fp [[t]] as a function on µ`∞ = lim µ`n . Given any
−→
such function Φ, we can define the Hecke operator U` by
1X
Φ(tζ).
(Φ|U` )(t) =
` `
ζ =1
Then we have
Φλ |U` = λ(`)Φλ .
When we have such an eigenfunction Φ of U` , we can define a measure on Z` as follows.
Fix an isomorphism Z` ∼
= Z` (1), which basically means that for all n, we fix a primitive
n
`
` -th root of unity ζn such that ζn+1
= ζn . Thus a function on µ`∞ can be seen as a function
n
on Z` . Let C(Z/` Z, A) be the space of functions on Z/`n Z taking values in A, which will
be assumed to be an Fp -algebra. Then a measure on Z/`n Z is simply a map
C(Z/`n Z, A) → B
7
for any A-module B, so can be seen as
X
θn =
αx δx ∈ B · (Z/`n Z)
x∈Z/`n Z
where αx ∈ B and δx is the Dirac measure at x.
Now we want to construct a measure on Z` . This will be a map in Hom(C(Z` , A), B), where
C(Z` , A) is the space of continuous functions on Z` taking values in A. The construction of
a B-valued measure on Z` is given by the data of θn for all n such that θn+1 7→ θn under the
map πn : B · (Z/`n+1 Z) → B · (Z/`n Z).
If Φ is a function satisfying Φ|U` = aΦ, we can construct such a compatible sequence. If
we define
X
θΦ,n =
Φ(ζnx )δx ,
x∈Z/`n Z
then
πn (θΦ,n+1 ) = `aθΦ,n .
Hence if we choose
1
θΦ,n ,
(a`)n
we get a measure on Z` . We denote by dΦ this measure, and dΦ(t) the measure given by
X
θΦ,n (t) =
Φ(ζnx t)δx .
θ̂n =
x∈Z/`n Z
Of course dΦ(1) = dΦ.
If f is a continuous function on Z` taking values in Fp , then we have that
Z
X
1
f dΦ =
Φ(ζnx )f (x)
(a`)n
n
x∈Z/` Z
where n is such that f factors through Z` → Z/`n Z (such an n exists since f is continuous).
×
Let χ : (Z/`n Z)× → Z → Fp be the reduction mod p of a Dirichlet character. This can
P
be seen as a function on Z` with support contained in Z×
λ(m)tm ∈ Fp [[t]],
` . Since Φλ (t) =
we have
Z
X
1
χ dΦλ (t) =
Φλ (ζnx t)χ(x)
(λ(`)`)n
n
×
x∈(Z/` Z)
X
1
χ(x)λ(m)ζnxm tm
n
(λ(`)`) x,m
!
X X
1
=
χ(x)ζnxm λ(m)tm
(λ(`)`)n m
x
!
X
X
1
=
χ(x)ζnxm λ(m)tm
(λ(`)`)n
x
=
(m,`)=1
!
X
X
1
=
χ(m)−1
χ(xm)ζnxm λ(m)tm
(λ(`)`)n
x
(m,`)=1
8
=
G(χ) X
χ(m)−1 λ(m)tm
(λ(`)`)n
(m,`)=1
where G(χ) is the Gauss sum. Therefore,
Z
G(χ)
χ dΦλ (t) =
Φλχ−1 (t).
(λ(`)`)n
In particular, if we specialize at t = 1, we get
Z
G(χ)
L(0, χ−1 λ).
χ dΦλ =
n
(λ(`)`)
Here we can think of dΦλ as a measure on Z×
` . Recall that we are interested in the set of
−1
values L(0, χ λ) as χ varies among
χ
Z×
`
9
/
µ`∞
(2)
Γ = Z×
` /µ`−1
We want a measure on Γ ∼
= Z` . The question is whether we can find a Ψ ∈ Fp [[t]] such that
Z
Z
χ dΦλ
χ dΨ(t) =
Z×
`
Γ
∼
for all χ satisfying (2). For that we use the decomposition Z×
` = µ`−1 × Γ to get
X
Ψ(t) =
Φλ (t ).
∈µ`−1
−1
We want to show L(0, χ λ) 6= 0 for infinitely many χ. By looking at the values of Ψ(ζ)
for ζ ∈ µ`∞ , we would like to show that Ψ(ζ) 6= 0 except for finitely many ζ. If we assume
that this is zero for all ζ, then Ψ(t) is also zero. This is where the algebraic independence
statement of Sinnott’s key lemma comes in.
In summary, we have expressed L-values using rational functions. To prove Washington’s
theorem, there is an ergodic statement about algebraic independence.
3. Lecture 3 (February 12, 2015)
We were proving Washington’s theorem for ` 6= p, but we might only have time to do a
weak version of it.
Let N be an integer prime to p, and λ be Dirichlet character of level N with λ(−1) = −1.
Recall the parity condition λ(−1) = (−1)n for the non-vanishing of the special value
Bλ,n
L(1 − n, λ) = −
.
n
We introduce the function
PN −1
∞
a
X
n
a=1 λ(a)t
Φλ (t) =
λ(n)t =
1 − tN
n=1
and look at this as inside Fp [[t]] or Fp (t).
9
Fix ` 6= p, which will be assumed odd for simplicity. Generally, given a function Φ : µ`∞ →
Fp such that Φ|U` = aΦ for some a 6= 0, a ∈ Fp , we can associate a measure dΦ on Z` once
we have fixed an isomorphism Z` ∼
= Z` (1). This is the same as fixing a compatible system
`
= ζn . The measure is given by
of `-power roots of unity: ζn ∈ µ`n with ζn+1
Z
X
1
x
f dΦ =
f (x)Φ(ζM
)
M
(a`)
Z`
M
x∈Z/`
Z
where f factors through Z/`M Z. In particular, if you take Φ = Φλ in which case a = λ(`)
(which is non-zero since (`, N ) = 1) and χ a Dirichlet character of level `M , then
Z
G(χ)
χ dΦλ =
L(0, χ−1 λ)
M
×
(`λ(`))
Z`
where G(χ) is the Gauss sum, and L(0, χ−1 λ) is obtained from Φχ−1 λ (t).
Remark. It is well-known that G(χ) 6= 0 in Fp .
We are interested in those values for χ taking values in µ`∞ ⊂ Fp , i.e., for χ that factors
through
χ
Z×
`
/
9
Fp
Γ = Z×
` /µ`−1
where we have the decomposition Z×
` = µ`−1 × Γ, Γ = 1 + `Z` . (This is where the assumption
that ` is odd simplifies the situation.)
We want to construct a measure dΨ on Γ such that
Z
Z
χ dΦλ .
χ dΨ =
Z×
`
Γ
Note
Z
Z×
`
χ dΦλ =
X Z
∈µ`−1
where the integral on the right means
Z
Z
χ dΦλ =
χ1·Γ dΦλ =
·Γ
Z×
`
χ dΦλ ,
·Γ
1
(`a)M
X
x∈
Putting all these together, this means
Z
1
χ dΦλ =
×
(`a)M
Z`
x
χ(x)Φλ (ζM
).
1+`Z`
1+`M Z`
!
X
x∈Γ/Γ`M
χ(x)
X
x
Φλ (ζM
)
since we have assumed χ|µ`−1 = 1. Therefore we can take Ψ : µ`∞ → Fp to be given by
X
Ψ(t) =
Φλ (t ).
Remark. If ∈ µ`−1 is not ±1, then Φλ (t ) is not rational.
10
We can write
X
Ψ(t) =
Φλ (t ) + Φλ (t− ).
∈µ`−1 /{±1}
−1
Note that we have Φλ (t ) = Φλ (t) because λ(−1) = −1.
Today we want to show that there exists such a χ such that L(0, λχ−1 ) 6= 0 in Fp . This
is a weaker version of the theorem of Washington, which says that this happens for almost
all χ. It is not much more work to get the stronger version. To prove this, we assume that
L(0, λχ−1 ) = 0 for all χ : Γ → µ`∞ . We have seen that
Z
X
1
−1
x
0 = L(0, λχ ) = χ dΨ =
χ(x)Ψ(ζM
).
M
(λ(`)`)
Γ
M
x∈Γ/Γ`
This is true for all characters χ. By the orthogonality relations of characters, this implies
Ψ(ζ) = 0 for all ζ ∈ µ`∞ . (If we knew that Ψ is rational, it would be easy to derive a
contradiction. Now we need to work a little bit more.) From this, we will deduce that
Φλ (ζ ) + Φλ (ζ − ) = 0 for all ζ and ∈ µ`−1 /{±1}, which will give a contradiction.
Recall that we are studying Fp -valued functions on µ`∞ .
Lemma 3.1 (Sinnott). Take b1 , · · · , bn ∈ Fp not all zero, and a1 , · · · , an ∈ Z` pairwise
distinct. Consider the function
n
X
f (ζ) =
bi ζ ai ∈ Fp
i=0
for ζ ∈ µ`∞ . Then f has finitely many zeros.
Proof. Consider the finite field k ⊂ Fp containing b1 , · · · , bn . Let N0 be the integer such that
`N0 = #(µ`∞ ∩ k), and N1 be such that the ai ’s are all distinct modulo `N1 . Let N ≥ N0 + N1
and ζ ∈ µ`N be such that f (ζ) = 0 (which is possible if f had infinitely many zeros).
We know that
X
0 = ζ −aj f (ζ) = bj +
bi ζ ai −aj
i6=j
and take the trace trk(ζ)/k . Since ζ
ai −aj
∈
/ k, it has trace 0 and so
0 = [k(ζ) : k]bj .
Therefore we get that bj = 0 for all j, which is a contradiction.
It is convenient to introduce F, the ring of Fp -valued functions on µ`∞ , and N ⊂ F, the
ideal of functions which are almost equal to 0 (i.e., vanishing except for finitely many values).
Let F0 = F/N . Then the lemma says that
X
f (ζ) =
bi ζ ai
i
is a unit in F0 when ai 6= aj for all i 6= j and bi are not all zero.
Corollary 3.2. Take a1 , · · · , an ∈ Z` linearly independent over Z. Then the functions
ta1 , · · · , tan (in F0 ) are algebraically independent over Fp . In other words, the map
θ : Fp [X1 , · · · , Xn ] → F0
ai
sending Xi 7→ (ζ 7→ ζ ) is injective.
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Proof. Consider this map
X1k1 X2k2 · · · Xnkn 7→ (ζ 7→ ζ a1 k1 +a2 k2 +···+an kn ).
More generally, for any
f=
X
we have
θ(f )(ζ) =
β(k1 ,··· ,kn ) X1k1 X2k2 · · · Xnkn ,
X
β(k1 ,··· ,kn ) ζ a1 k1 +···+an kn .
Let us call α(k1 , · · · , kn ) = a1 k1 + · · · + an kn . If (k1 , · · · , kn ) 6= (k10 , · · · , kn0 ), then
α(k1 , · · · , kn ) 6= α(k10 , · · · , kn0 ) because the ai ’s are linearly independent.
If f is in the kernel, then it has infinitely many zeros, so the lemma forces all the β(k1 ,··· ,kn )
to be zero. Therefore f = 0.
Proposition 3.3. Let k be a field, and Y1 , · · · , Yr be monomials in X1 , · · · , Xn
Y c
Yi =
Xj ij
j
such that
Yia
=
Yjb
implies either a = b = 0 or i = j. Let Pi (z) ∈ k(z) be such that
P1 (Y1 ) + P2 (Y2 ) + · · · + Pr (Yr ) = 0
(3)
then P1 (z), · · · , Pr (z) ∈ k.
Sketch proof. Let R = k[X1 , · · · , Xn , X1−1 , · · · , Xn−1 ], which is a factorial ring with units
given by constants and monomials. The condition implies that if f (z), g(z) ∈ k[z] are
relatively prime, then f (Yi ) and g(Yi ) are also relatively prime.
We can easily reduce to the case r = 2 by induction. Writing Pi = fgii , (3) becomes
f1 (Y1 )g2 (Y2 ) = −f2 (Y1 )g1 (Y2 ).
We can deduce that Pi (z) ∈ k[z, z −1 ]. From (3) again, this forces Pi (z) ∈ k.
Using this proposition and the previous corollary, we will consider for r =
Φ(t1 ) + · · · + Φ(tr ) = 0
and deduce that Φ ∈ Fp . We will come back to this next time.
12
`−1
,
2