MATH G9905 RESEARCH SEMINAR IN NUMBER THEORY
(FALL 2014)
LECTURE 6 (OCTOBER 14, 2014)
GIOVANNI ROSSO
INTRODUCTION TO SPECIAL VALUES OF L-FUNCTIONS AND
THEIR p-ADIC INTERPOLATION, I
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.
This talk will be mostly complex-analytic in nature. p-adic L-functions will be introduced
in the next talk.
Why do we study L-values?
Let (a, b) ∈ Z2 . What is the probability that gcd(a, b) = 1? Since 2 cannot divide both a
and b, we have
1
P (2 | a & 2 | b) =
4
and similarly for 3, 5, · · · , so
1
1
1
6
P (gcd(a, b) = 1) = 1 −
1−
··· =
= 2.
4
9
ζ(2)
π
Recall the Riemann zeta function
−1
Y
1
.
ζ(s) =
1− s
p
Theorem 1. If n > 0, then
(−1)n+1 B2m (2π)2n
ζ(2n) =
2(2n)!
where B2m ∈ Q is the Bernoulli number, given by
∞
X
t
tn
=
B
.
n
et − 1 n=0
n!
Last updated: October 28, 2014. Please send corrections and comments to [email protected].
1
For the odd positive integers , we know ζ(3) ∈
/ Q and at least one of ζ(5), ζ(7), ζ(9), ζ(11) ∈
/
Q. In general, we don’t know much about ζ(2n + 1).
For the negative integers, we have
Bn+1
ζ(−n) = −
∈ Q,
n+1
where B2n+1 = 0.
To explain the factor of 2πi, we need a bit of geometry. Let
Gm = Spec(Z[X, Y ]/(XY − 1)).
If ` is a prime number, we let
Z` (1) = lim Gm (Q)[`n ] = lim µ`n .
←−
←−
Then we will see that the corresponding L-function is
Y
L(Z` (1), s) =
(1 − q −s−1 ) = ζ(s + 1).
q
In general, for Z` (m), we have
L(Z` (m), s) = ζ(s + m).
Recall the real gamma function
ΓR (s) = π −s/2 Γ(s/2).
Then we have the functional equation
ζ(s)ΓR (s) = ΓR (1 − s)ζ(1 − s)
for all s ∈ C − {0, 1}.
Remark. If s = −2n where n > 0, then ΓR (−2n) = ∞, so ζ(−2n) = 0.
Consider Gm (C) = C× . If we consider C as a topological space, we have the singular
homology
H1 (C× ) ' Z
and de Rham cohomology
1
HdR
(C× ) ' Z
1
upon fixing a generator dz
. These two groups are related by the isomorphism HdR
(C× ) '
z
1
(H1 (C× ))∨ via HdR
(C× ) × H1 (C× ) → C given by
Z
hω, γi = ω.
γ
Note
Z
dz
= 2πi.
S1 z
The complex conjugation F∞ acts on H1 (C× ) via S 1 → −S 1 .
We want to generalize this to ζ(n) = L(Z` (n), 0). Let
H1 (C× (n)) := (2πi)n H1 (C× )
with action of F∞ given by (−1)n , and
1
1
HdR
(C× (n)) := HdR
(C).
2
If n is even, F∞ is trivial on H1 (C× (n)); if n is odd, F∞ is −1 on it.
We are ready to explain the 2πi in ζ(n) for n positive even:
∼
1
(C× (n)) .
(2πi)n = det H1 (C× (n))∨,F∞ =1 → HdR
For ζ(n) with n negative odd, we have
1
1 = det H1 (C× (n))∨,F∞ =1 → HdR
(C× (n))
because H1 (C× (n))∨,F∞ =1 = 0.
To generalize this to higher-dimensional L-functions,
Definition 2. A motive M is the data of
• M
space over Q with F∞ involution and a Hodge decomposition MB ⊗Q C '
LB a vector
p,q
such that F∞ (H p,q ) = H q,p ,
p,q∈Z H
• MdR a vector space over Q with filtration Fili MdR on MdR ⊗ C,
• for all prime `, M` a Galois representation of GQ = Gal(Q/Q) over a Q` -vector space,
which are compatible in the following sense:
L
0
• MB ⊗ C ' MdR ⊗ C such that p0 >p H p ,q → Filp MdR ;
• MdR ⊗Q C ' M` ⊗Q` Q` for a fixed C ' Q` ;
• the M` ’s form a compatible system: for all q - ``0 ,
I
I
det(1 − T Frq |M` q ) = det(1 − T Frq |M`0q ) ∈ Z` [T ]
q
where the geometric Frobenius Frq is given by Fr−1
q (x) ≡ x (mod q).
This is quite abstract. Let X be an algebraic, projective, smooth variety over Q. We
define
MB = Hi (X(C))
with Hodge structure,
i
MdR = HdR
(X)
and
i
M` = Hét
(XQ` , Q` ).
Example 3. Let E be the elliptic curve defined by y 2 = x3 + ax + b, where a, b ∈ Q. Then
E(C) ' C/(Z + γZ) for some γ ∈ C\R. This is a torus. We have
H1 (E(C)) ' 1Z ⊕ γZ
and
H 1 (E) = Z
dx
dx
⊕ Zx .
y
y
Then
H1 (E(C)) ⊗ C ' H 1,0 ⊕ H 0,1 = C(1 + γ) ⊕ C(1 + γ).
The filtration is
1
Fil1 (HdR
) = H 0 (E, Ω1 ) = Z
Finally, we need the `-adic realization. The `n -torsion is
E[`n ] ' (Z/`n Z)2
3
dx
.
y
and the Tate module is
V` E := lim E[`n ] ' (Z` )2
←−
n
n
with Galois action, since the ` -torsion points have Q-coefficients. Set T` E = V` E ⊗ Q` .
Deligne gave a recipe to construct from a motive an L-function. Consider
I
Pq (T ) = det(1 − T Frq |M` q )
and
L(M, s) =
Y
Pq (q −s )−1 .
q
Definition 4. M is pure of weight i if
MB ⊗ C =
M
H p,q .
p,q
p+q=i
If M is pure of weight i, we can think of it as a “piece” of H 1 (X), for a certain variety X.
Conjecture 5. If M is pure of weight i and M` is unramified at q for q - `, then the roots
i
of Pq (T ) αq ’s are Weil numbers of weight i, i.e. for all σ : Q ,→ C, |αq |σ,C = q 2 .
This conjecture implies that if M is pure, then L(M, s) converges absolutely for Re(s) >
+ 1.
1
We now want to extend L(M, s) for all s ∈ C. Denote hp,q = dimC H p,q , and h 2 ,± =
i i
dimC (H 2 , 2 )F∞ =±1 . If i is odd, this is defined to be 0. Define
h 2i ,−
h 2i ,+
Y
i
i
p,q
ΓC (s − p)h ΓR s −
Λ∞ (M, s) =
ΓR s − + 1
2
2
p<q
i
2
where the complex gamma function is ΓC (s) = 2(2π)−s Γ(s) (and the indices p and q are not
necessarily prime!). The L-function satisfies the functional equation
Λ∞ (M, s)L(M, s) = (s)Λ∞ (M, i + 1 − s)L(M, i + 1 − s).
Definition 6. We say that n ∈ Z is critical for M if neither Λ∞ (M, n) nor Λ∞ (M, i + 1 − n)
has a pole.
Example 7. For an elliptic curve, s = 1 is the only critical point.
Proposition 8. If n is critical for M , then there is an isomorphism
∼
i+n
I∞ : F i+1−n MdR → (MB ⊗ C)F∞ =(−1)
Here F i M is the filtration on de Rham cohomology, coming from the spectral sequence
p+q
H p (X, Ωq ) ⇒ HdR
(X).
Fix Q-bases ωi and γi of the left hand side and right hand side respectively, and define
c+ (M, n) = det(I∞ )
with respect to (ωi ) and (γi ).
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Conjecture 9 (Deligne).
L(M, n)
∈ Q.
c+ (M, n)
∼
1
1
E
Example 10. If i = n = 0, then F 1 HdR
→ MBF∞ =1 . For an elliptic curve E, F 1 HdR
dx
1
is ωE ( y is the invariant differential, which is a global section of Ω ). Using the modular
π
parametrization H → X0 (N ) → E and modular form fE ↔ E, set
ωE = (2πi)fE (z)dz
and
We have
γ = Im(π∗ (i∞ − 0)) ∈ H1 (E(C))F∞ .
Z ∞
2πifE (iy)dy.
L(f, 1) =
0
This formula proves the conjecture.
Fix a prime p. Suppose the Deligne conjecture is true. Then what are the p-adic properties
of cL(M,n)
+ (M,n) ∈ Q? For all m > 0, ζ(−m) ∈ Q. If m becomes a p-adic variable, then does ζ(−m)
make sense?
The references are:
• Deligne, Valeurs de fonctions L et périodes d’intégrales 1;
• Schneider, Introduction to the Beilinson conjectures 2.
1Available
2Available
at http://publications.ias.edu/sites/default/files/33_Valeursde.pdf.
at http://wwwmath.uni-muenster.de/u/pschnei/publ/beilinson-volume/Schneider.
pdf.
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