MATH G9905 RESEARCH SEMINAR IN NUMBER THEORY
(FALL 2014)
LECTURE 7 (OCTOBER 21, 2014)
GIOVANNI ROSSO
INTRODUCTION TO SPECIAL VALUES OF L-FUNCTIONS AND
THEIR p-ADIC INTERPOLATION, II
NOTES TAKEN BY PAK-HIN LEE
Abstract. The aim of these talks is to give an introduction to the theory of L-functions
from an arithmetic point of view and to explain how to construct their p-adic avatars. We
shall begin by explaining the values taken by Riemann’s zeta function at integers. Subsequently, we shall try to give a conjectural description of the values at certain integers for
other L-functions, such as the one of an elliptic curve.
In the second part, we study how the values of an L-function vary p-adically and we
explain what the conjectural p-adic L-function should look like. We conclude with some
general conjectures on the properties of p-adic L-functions, with special emphasis on the
so-called trivial zeros.
Last time we saw that the L-functions L(M, s) are very nice, and for certain integers n ∈ Z
(depending on M ) we can find an almost canonical complex number c+ (M, n) such that
L(M, n)
∈ Q.
c+ (M, n)
Today we will study the p-adic properties of these rational numbers.
The values of the Riemann zeta function at the negative integers are
Bn+1
ζ(−n) = −
∈ Q.
n+1
Since N ⊂ Zp is dense, we may ask the following question: does there exist ζp (s) (s ∈ Zp )
such that ζp (n) = ζ(−n) for all n ∈ N? The answer is almost yes.
Recall f : Zp → Qp is continuous if and only if for every M > 0, there exists N > 0 such
that
f (x) ≡ f (y) (mod pM )
when x ≡ y (mod pN ). This is just the usual -δ definition translated into congruences. We
n+1
want to prove the congruence properties for Bn+1
. Consider the Taylor expansion
X
text
tn
=
B
(x)
n
et − 1 n≥0
n!
with the following properties:
Last updated: October 23, 2014. Please send corrections and comments to [email protected].
1
• Bn (0) = Bn is the n-th Bernoulli number;
• Bn (x) is a polynomial of degree n;
• B1 (x) = x − 12 .
We will use these to study the congruence properties.
Let LC(Zp ) be the space of locally constant functions Zp → Qp .
Definition
1. µ : LC(Zp ) →PQp is a distribution if for all compact open U of Zp with
`
U = i Ui , we have µ(U ) =
µ(Ui ). µ is a measure if it is a bounded distribution, i.e.
|µ(U )|p ≤ C for all U .
If µ is a measure and f ∈ LC(Zp , Qp ), we define
Z
X
f (aU )µ(U ).
f dµ = lim
−→ `
Zp
U =Zp
aU ∈U
Let
N
µn (a + p Zp ) = p
N (n−1)
Bn
a
pN
.
Proposition 2. For all n ≥ 1 and a ∈ Zp ,
X
µn (a + pN Zp ) =
b≡a
µn (b + pN +1 Zp )
(mod pN +1 )
This shows that µn is a distribution, but it is not a measure. The problem is that this
function is not bounded at all. Since Bn is a polynomial of degree n,
n
a
a
Bn
∼
pN
pN
so µn can have arbitrarily large p-adic norm.
Take α ∈ Z×
p \{n}, and let
µn,α (a + pN Zp ) = µn (a + pN Zp ) − α−n µn (α(a + pN Zp )).
For example,
a
1
1 −1 a − α−1 [αa] 1
−1 [αa]
µ1,α (a + p Zp ) = N − − α
+ α =
− (1 − α−1 )
N
N
p
2
p
2
p
2
N
where [αa] ∈ Z is such that 0 ≤ [αa] < pN and [αa] ≡ αa (mod pN ).
Theorem 3. For all n ≥ 1 and α ∈ Z×
p \{1}, µn,α is bounded. Moreover, we have dn ∈ N
such that dn µn,α has values in Zp , and
dn µn,α (a + pN Zp ) ≡ nan−1 dn µ1,α (a + pN Zp )
(mod pN ).
This means we can more or less recover these measures from the first one.
Corollary 4.
Z
Z
dµn,α = n
Zp
Zp
2
xn−1 dµ1,α (x).
Note
Z
x
Z×
p
n−1
Z
dµ1,α =
x
n−1
Z
dµ1,α −
xn−1 dµ1,α
pZp
Zp
Bn (0)
Bn (0)
−n Bn (0)
n−1
−n Bn (0)
−α
−p
−α
=
n
n
n
n
Bn
= (1 − α−n )(1 − pn−1 )
n
= −(1 − α−n )(1 − pn−1 )ζ(1 − n).
This tells us that to p-adically interpolate the Riemann zeta function, we have to remove
the Euler factor at p.
∼
Now we introduce the Mellin transform. Recall there is an isomorphism 1 + pZp → Zp :
every x ∈ 1 + pZp can be written as x = (1 + p)γ with γ ∈ Zp . If µ is a measure,
!
Z
X Z γ (1 + T )γ dµ(γ) =
dµ(γ) T n ∈ Qp ⊗ Zp [[T ]]
n
Zp
Zp
n
. Mahler’s theorem says that the binomial polynimals give a basis
where nγ = γ(γ−1)···(γ−n+1)
n!
of locally constant functions. Thus we have an isomorphism
∼
Meas → Qp ⊗ Zp [[T ]] : µ 7→ Fµ (T ).
If 1 + T = (1 + p)s where s ∈ Zp , then
Z
Z
s
γs
Fµ ((1 + p) − 1) =
(1 + p) dµ(γ) =
xs dµ(x).
1+pZp
Zp
This is the p-adic Mellin transform. We say that a function G(s) is an Iwasawa function if
G(s) = F ((1 + p)s − 1)
with F (T ) a formal series with coefficients in Zp .
×
× ∼
Recall we have a map ω : Z×
p → µp−1 which factors through Fp → µp−1 , and every x ∈ Zp
can be written as ω(x)hxi where hxi ∈ 1 + pZp .
The measure µ1,α gives p − 1 Iwasawa functions: for i ∈ [0, · · · , p − 2],
Z
α
ω(x)1−i hxis−1 dµ1,α (x)
ζp,i (s) =
Z×
p
for s ∈ Zp .
Theorem 5. For all n ≡ i (mod p − 1) and n ≥ 0, we have
α
(n) = (1 − α1−n )(1 − p−n )ζ(−n).
ζp,i
For i 6≡ 0,
α (s)
ζp,i
1−α1−s
ζ α (s)
p,0
is holomorphic. 1−α
1−s has a simple pole at s = 0 with residue
α
ζp,0 (s)
Ress=0
= 1 − p−1 = (1 − p−1 ) Ress=0 ζ(s).
1 − α1−s
Thus we can call this function the p-adic avatar of the Riemann zeta function.
3
Corollary 6. If : 1 + pZp → Qp has finite order, then
Z
(x)ω 1−i (x)hxin−1 dµ1,α = (1 − αn−1 )(1 − (ω 1−i )0 (p)pn )L(−n, ω n−i )
Z×
p
where L(−n, ω n−i ) =
P
m
ω n−i (m)
.
ms
This takes care the p-adic L-functions for Dirichlet characters.
For a motive M (MB , MdR and M` for all ` prime), we get an L-function L(M, s). For m
critical we get c+ (M, n).
Conjecture 7.
L(M, n)
∈ Q.
c+ (M, n)
Can we do a p-adic interpolation of L(M, n)? The first problem is we do not have many
critical integers, by comparing Γ(s − p) and Γ(1 − s + i − p). So we need to let the motive
M vary.
×
Let : 1 + pZp → Q be of finite order. Then we can form the twist M ⊗ . If
X am
,
L(M, s) =
s
m
m
then
L(M ⊗ , s) =
X am (m)
m
How does
L(M ⊗,n)
c+ (M,n)
ms
.
vary p-adically, i.e. if 1 ≡ 2 , then is
L(M ⊗ 2 , n)
L(M ⊗ 1 , n)
≡
?
+
c (M, n)
c+ (M, n)
Example 8. Let E/Q be an elliptic curve with good ordinary reduction at p. Thus p - ap (E),
and the Hecke polynomial X 2 − ap (E)X + p = (X − αp )(X − βp ) with vp (αp ) = 0 and
vp (βp ) = 1.
Theorem 9. For each choice of δ = α or β, we have Lδp (E, s) such that
Lδp (E, (1 + p) − 1) = G δ −M
L(E ⊗ , 1)
Ω+
E
where has conductor pM . In particular, if we take the trivial character,
L(E, 1)
Lδp (E, 0) = (1 − δ −1 )2
.
Ω+
E
We know that the L-function has a complex representation
Z i∞
L(E, s) =
f (y)y s dy.
0
We can do something similar p-adically and obtain Lδp (E, s).
We have two p-adic L-functions for E, because we can make two different choices of
“eigenvalues at p”.
4
Let us know return to motives. We consider the Galois representation
Mp : Gal(Q/Q) → GLn (Qp ).
Suppose Mp is semistable (á la Fontaine). For a fixed integer d ∈ N (depending on M )
choose d “good” eigenvalues D = {α1 , · · · , αd } of the Frobenius at p for the representation
Mp . D can be thought of as the p-adic equivalent of Fil• MdR .
Conjecture 10 (Perrin-Riou, Coates). Associated to M and D we have a p-adic L-function
LD
p (M, s) such that
L(M ⊗ , n)
.
LD
p (M, (1 + p) − 1) = G CD
c+ (M, n)
In the final part, I will give another way of constructing p-adic L-functions which does not
directly use distributions.
Let f ∈ Sk (Γ0 (N )) and
Y
L(f, s) =
[(1 − α` `−s )(1 − β` `−s )]−1 .
`
We want to interpolate
Y
[(1 − α`2 `−s )(1 − α` β` `−s )(1 − β`2 `−s )]−1 .
L(Sym2 f, s) =
`
Fix a character . Then
L(Sym2 f ⊗ , 1) = hf, Θ()Ek ()i
where
Θ() =
X
2
(n)q n
Z
and
Ek () =
X
L(0, σn )P (n)q n
n
√
where σn is the quadratic character associated with Q( n)/Q. For 1 and 2 such that
1 |1+pN Zp = 2 |1+pN Zp , we have
Θ(1 )Ek (1 ) ≡ Θ(2 )Ek (2 )
(mod pN ).
Let


!
 X s n2  X
ΘEk (s) = 
nq 
Lp (s, σn )Pn (ns )q n ,


n
n∈Z
(n,p)6=1
which is a formal series in q with Iwasawa functions as coefficients.
Define `f : Mk (Γ0 (N )) → C by
hf, gi
g 7→
.
hf, f i
`f is defined over a number field K, so we have `f : Mk (Γ0 (N ), K) → K. Then `f extends
to a linear form Mk (Γ0 (N ), Kp ) → Kp where Kp is the local field of K at a p-adic place.
5
Hypothesis. f is of finite slope for Up if f has level divisible by p and
X
f |Up =
anp (f )q n = αf
for some α 6= 0.
We can now define the p-adic L-function
Lαp (Sym2 f, s) = `f (ΘEk (s)).
For s = (p + 1) − 1,
Lαp (Sym2 f, (1 + p) − 1) = G α−2 cond We have
L(Sym2 f ⊗ , 1)
.
hf, f i
hf, f i
∈ Q.
f, 1)
This construction also works in families, where we get a big L-function in two variables.
c+ (Sym2
6