Weil-Deligne Representations Study Group
Spring Term 2016
1
1.1
Weil Conjectures - David Loeffler
Motivation
Let Fq be a finite field, q = pn , X/Fq an irreducible, smooth, projective variety. We have
two questions:
• X(Fq ) is finite, but just how big is it?
• For any r, we have an extension of Fq of degree r. How does #X(Fqr ) grow with
r?
We have two easy examples:
• X = P1 , #X(Fqr ) = q r + 1,
• X = PN , #X(Fqr ) = 1 + q r + ⋯ + q N r .
1.2
Elliptic Curves
Theorem 1.1 (Hasse 1930s).
i. We have #E(Fqr ) = 1 + q r +”small error term”.
More precisely, if α, β are the roots of
X 2 − (q + 1 − #E(Fq ))X + q,
then #E(Fqr ) = q r + 1 − αr − β r .
1
ii. We have ∣α∣ = ∣β∣ =
√
q (Hasse’s Inequality).
Proof.
i. Recall that Gal(Fq /Fq ) is topologically generated by the the Frobenius
automorphism φ ∶ x ↦ xq .
#E(Fq ) = #E(Fq )φ=1 = #ker(1 − φ ∶ E → E) = deg(1 − φ) as separable.
The degree nap on End(E) is a quadratic form and every ψ ∈ End(E) has a dual
ψ̂ such that ψ ψ̂ = deg(ψ) ∈ Z and ψ + ψ̂ = deg(1 + ψ) − 1 − deg(ψ) ∈ Z.
Now we have φ2 − (φ + φ̂)φ + φφ̂ = 0 so φ satisfies the polynomial x2 − aq x + q
form before.
Now we can show by induction on r that φr + φ̂r = αr + β r for all r ≥ 0. Hence
deg(1 − φr ) = 1 + deg(φr ) − (φr + φ̂r ) = 1 + q r − (αr + β r ).
ii. deg(ψ) ≥ 0 for all ψ ∈ End(E) so for all m, n ∈ Z, deg(m + nφ) = m2 + mnaq +
√
qn2 ≥ 0, so the discriminant a2q + 4q ≤ 0, i.e. ∣aq ∣ ≤ 2 q.
∎
1.3
General Picture
We expect
#X(Fqr ) = ±λr1 ± λr2 ± ⋯ ± λrm , λi ∈ C.
Definition 1.2. We define the zeta function of X/Fq as:
∞
tr
Z(X/Fq , t) = exp(∑ #X(Fqr ) ).
r
r=1
Fact 1.3. Z(X/Fq , t) is a rational function if and only if a formula 1 holds.
Example 1.4.
∞
tr
Z(P1 /Fq , t) = exp(∑(1 + q r ) )
r
r=0
= exp(− log(1 − t) − log(1 − qt))
1
=
.
(1 − t)(1 − qt)
2
(1)
1.4
Weil Conjectures
i. For any X/Fq as above, Z(X, t) ∈ Q(t).
We have a functional equation for the Riemann zeta function ζ(1 − s) = (∗)ζ(s).
We expect Z(X/Fq , q −s ) to satisfy a functional equation relating s and dim(X) −
s.
ii. Functional Equation Z (X, q1d t ) = (∗)Z(X, t), where d = dim(X).
Generalisation of the Hasse Inequality: ±λr1 ± λr2 ± ⋯ ± λrm where the absolute
values are half-integer powers of q.
Definition 1.5. A q-Weil number of weight k is a number λ ∈ Q such that ∣λ∣ = q k/2
for every embedding Q ↪ C.
iii. Riemann Hypothesis All zeroes and poles of Z(X/Fq ) are q-Weil numbers. More
precisely,
P1 (t)⋯P2d−1 (t)
,
Z(X/Fq ) =
P0 (t)⋯P2d (t)
where the odd subscripts are in the numerator, even subscripts in the denominator,
and Pi has all zeroes of weight −i.
If we put ζ(X, s) = Z(X, q −s ), this says something about the real parts of zeroes
of ζ(X, s).
In 1949, Weil formulated these conjectures and proved them when X is a curve or an
abelian variety.
1.5
Dreaming of a Cohomology Theory
We want to count fixed points of φr on X(Fq ). If X/C, ψ ∈ Aut(X), then we can count
fixed points of ψ using cohomology.
2d
#fixed points = ∑ (−1)i trace(ψ ∗ acting on H i (X(C), Q)). This is the Lefschetz trace
i=0
formula.
Weil observed that if there exists a cohomolgy theory for algebraic varieties over Fq
with “reasonable” properties then i and ii of the Weil conjectures are automatic (a “Weil
cohomology theory”).
3
Fact 1.6. For any prime l, there exists a Weil cohomology theory with coefficients in
Ql . If l ≠ char Fq then we get étale cohomology (Grothendieck et al 1960s), whereas if
l = char Fq , then we get crystalline cohomolgy (Besttelot-Ogus 1980ish, after Dwork).
The Riemann Hypothesis part was proved by Deligne in 1972.
1.6
Addendum
Let X/Fq be a smooth projective variety, with #X(Fqr ) = ±λr1 ± λr2 ± ⋯ ± λrm , λi ∈ Q.
Claim 1. We can order the λi such that λi λm+1−i = q dim(X) .
Claim 2. The multiset of the λi is preserved by the map x ↦
q dim(X)
.
x
Note that (q d/2 , q d/2 , q d/2 , −q d/2 ), d = dim(X) shows that Claims 1 and 2 are not equivalent. In fact, Claim 1 is equivalent to Claims 2 and 3, where:
Claim 3. q d/2 and −q d/2 do not both appear to odd multiplicity.
Fact 1.7. Claim 2 holds for all varieties. Claim 1 holds for curves but not general
varieties.
1.6.1
Claim 1 for curves
Lemma 1.8. Let V be a vector space of even dimension 2g with an alternating, nondegenerate, bilinear form ⟨⋅, ⋅⟩.
If A ∶ V → V satisfies ⟨Ax, Ay⟩ = λ⟨x, y⟩ then det(A) = λg and AT JA = λJ where
J ∈ (Λ2 V )× (A acts as λ here) so J g ∈ (Λ2g V )× (A acts as λg here).
Now take
• V = Tl J(C),
• ⟨⋅, ⋅⟩ to be the Weil pairing,
• A to be Frobenius.
Then the product of Frobenius eigenvalues is q g hence −q d/2 appears with even multiplicity. This does not work in even dimension.
4
1.6.2
Counterexample for Surfaces
Take P2Fq and blow up some points (at most 7) in general position to get a del Pezzo
surface X.
H 2 (X, Ql ) ≅ Ql ⊗ N S(X)
as Gal(Fq /Fq )-modules up to a twist, where N S(X) is the Néron-Severi group (finitely
generated abelian group, defined as the group of divisors modulo algebraic equivalence).
N S(X) ≅ Z ⊕ Zr where the second term are the exceptional divisors.
Now take 1 point defined over Fq
⎛q
⎜ q
nius action is given by ⎜
⎜
0
⎝
q
2
and 2 conjugate points defined over Fq2 . The Frobe⎞
⎟
⎟, which has eigenvalues q, q, q, −q.
q⎟
0⎠
Representations of the Weil Group - Ariel Pacetti
Our main goal is to understand GQ = Gal(Q/Q) and we first approach this by trying to
understand its representations.
Finite dimensional ones: ρ ∶ GQ → GLn (C) continuous.
Exercise 2.1. All of them have finite image (open subgroups have finite index).
Our second approach is to take ρl ∶ GQ → GLn (Ql ).
Example 2.2. n = 1:
Qab
≅ ∏ Z×p → Z×l
p
ρl ∶ GQ → Q×l abelian representation.
where Qab = ⋃n Q(ζn ).
Q
5
Q
Qab
We get the cyclotomic character
Q
∏ Zp ×
→
p
Z×l ,
Id
Z×l Ð→ Z×l ,
Z×p → 1.
defined by σ ∈ GQ , σ(ζln ) = ζlan ∈ Z×l , a ∈ (Z/ln Z)× where σ(ζl ) = ζla1 and an ≡ an−1
mod ln−1 .
Q(ζln )
Q
Note Q(ζln ) is only ramified at l so if p ≠ l, then Frobp is well defined and χl (Frobp ) = p
(exercise).
2.1
Tate Module of an Elliptic Curve
Let E/Q be an elliptic curve, l prime, E[ln ] the Z/ln Z-module of ln torsion over Q. If
σ ∈ Gal(Q/Q), P ∈ E[ln ] then σ(P ) ∈ E[ln ].
E[ln ] ≅ Z/ln Z × Z/ln Z and choosing a basis we get a mod l representation
ρE,l ∶ GQ → GL2 (Z/ln Z) ⇢ GL2 (Zl ) ≅ Aut(Tl (E)).
We take Vl (E) = Tl (E) ⊗Zl Ql .
6
E[ln+1 ]
σ
×l
E[ln ]
E[ln+1 ]
×l
σ
E[ln ]
Definition 2.3. If p is a prime, we say ρ is unramified at p if ρl (Ip ) = 1.
Q
ker(ρl )
Q
p
Q
Theorem 2.4 (Néron-Ogg-Shafarevich). E/Q, ρE,l is unramified at the primes p ≠ l
where E has good reduction.
Recall Gal(Qp /Qp ) ⊂ Gal(Q/Q).
Q
Qp
where the inclusion Q ↪ Qp is non-canonical.
Q
Qp
Then we can restrict to studying representations of Gal(Qp /Qp ).
1 → Ip → Gal(Qp /Qp ) → Gal(Fp /Fp ) → 1,
where Gal(Fp /Fp ) ≅ lim Z/nZ = Ẑ ≅ ∏ Zp .
n
p
7
Qp
pro-p group
tame
Q
∏ Zq = ⟨τ ⟩
q≠p
un
Q
Ẑ = ⟨σ⟩
Qp
where σ is the Frobenius Frobp and Qp is the maximal tamely ramified extension. Note
στ σ −1 = τ p .
Definition 2.5. The Weil group WQp of Qp is (as an abstract group):
1
Ip
WQp
1
Ip
Gal(Qp /Qp )
r
Z
1
Gal(Fp /Fp )
1
where r(g) = σ n , n ∈ Z. The Weil group WQp = {σ n i ∣ n ∈ Z, i ∈ Ip } ≅ Ip ⋊ Z is the
preimage of Z. We give the Weil group the locally finite topology where Ip gets the
subgroup topology of Gal(Qp /Qp ) and we impose that it is open in the Weil group.
Theorem 2.6. Let E/K be an elliptic curve over a local field K with good reduction
Ẽ. Then
χ(ρl (FrobE,p ))(t) = t2 − ap t + p ∈ Z[t]
where ap = p + 1 − #Ẽ(Fp ).
GK
ρl
GL2 (Ql )
≀
where Ql ≅ C.
GL2 (C)
8
ρl
WK
GL2 (Ql )
≀
GL2 (C)
If ρl (Ip ) is finite, then the same trick works.
Theorem 2.7. If E is an elliptic curve, then ρl (Frobp ) is semisimple.
1 1
Remark 2.8. ρ ∶ WQp → GL2 (Ql ) is not semisimple: Ip → 0, σ ↦ (
).
0 1
Theorem 2.9 (Classification). Let ρl ∶ WQp → GLn (Ql ), l ≠ p, ρ(Ip ) finite, Frobenius
semisimple then ρl ≅ ⊕N
i=1 (ρi ⊗ χi ) where ρi are Artin irreducible representations and
χi are 1-dimensional characters.
3
Weil-Deligne Representations - Aurel Page
GK = Gal(K/K). Fix p, l ≠ p.
Recall GQp ⊂ GQ , charpol(Frobp ⟳ Tl E) is independent of l We aim to determine the
structure of ρ ∶ GK → GLn (Ql ), K/Qp finite using Weil-Deligne representations. This
has the advantage of being purely algebraic and we can make sense of the comparison
between l-adic representations for GK .
3.1
Ramification and the Tame Character
K/Qp finite. P ◁ I ◁ GK , where P is wild inertia, I is inertia.
⟨Frob⟩ =
GK
≅ Ẑ ≅ Gal(Fq /Fq )
I
K tame
K un
K
9
where
I
• K un = K = ⋃ K(ζn ),
p∤n
P
• K tame = K = ⋃ K un (π 1/n ),
p∤n
• Frob(ζn ) = ζnq , q = #residue field,
• π uniformiser in K.
K un (π 1/n )/K un is a Kummer extension so Gal(K un (π 1/n )/K un ) ≅ Z/nZ. FurtherI
more, = Gal(K tame /K un ) = ⟨τ ⟩ ≅ ∏ Zl .
P
l≠p
τ π 1/n = ζn π 1/n .
What is the action of conjugation by Frob? Choose Frob ∈ GK , and compute Frob τ Frob−1 .
Frob τ Frob−1 (π 1/n ) = ζnq π 1/n , where
Frob−1 (π 1/n ) = ωπ 1/n , for some ω ∈ µn (K un )
τ Frob−1 (π 1/n ) = ζn π 1/n .
Hence Frob τ Frob−1 = τ q .
Definition 3.1. The additive l-adic tame character tl is the map
tl (σ)(π
3.2
1/lk
tl ∶ I → Zl
k
t (σ)
) = ζlkl π 1/l .
Weil-Deligne Representations
1 tl (σ)
I → GL2 (Zl ), σ ↦ (
).
0
1
Recall the Weil group WK = {Frobn σ ∶ n ∈ Z, σ ∈ I} = I ⋊ Z. Extend the representation
10
to sp(2):
WK → GL2 (Zl ),
1 0
Frob ↦ (
),
0 q −1
1 tl (σ)
I ∋σ ↦ (
).
0
1
We can generalise this to get a representation to sp(n):
WK → GLn (Ql ),
⎛1
⎞
⎜ q −1
⎟
⎟ , is a diagonal matrix,
Frob ↦ ⎜
⎜
⎟
⋱
−n+1
⎝
⎠
q
⎛1 tl (σ) . . .
⎜
1
⋱
I ∋σ ↦ ⎜
⎜
⋱
⎜
⎝
N
tl (σ)n−1
(n−1)! ⎞
⋮
tl (σ)
1
⎟ t (σ)N
⎟=el
, is upper triangular, where
⎟
⎟
⎠
⎛0 1
⎞
⎜ ⋱ ⋱ ⎟
⎟ is an upper triangular nilpotent matrix.
= ⎜
⎜
⋱ 1⎟
⎝
0⎠
sp(n)(Frob) ⋅ N ⋅ sp(n)(Frob)−1 = qN .
Theorem 3.2 (Grothendieck’s Monodromy Theorem). Let ρ ∶ WK → GLn (Ql ) be continuous. Then there is a finite extension K ′ /K such that ρ ∣IK ′ is unipotent, i.e. there
exists N ∈ Mn (Ql ) nilpotent such that for all σ ∈ IK ′ , ρ(σ) = etl (σ)N .
Definition 3.3. Let E be a field of characteristic 0. An E-valued Weil-Deligne representation of WK is a pair (ρ, N ) where ρ ∶ WK → GLn (E) with ρ(I) finite and N ∈ Mn (E)
is nilpotent and ρ(Frob) ⋅ N ⋅ ρ(Frob)−1 = qN for all choices of Frob ∈ WK .
Theorem 3.4 (Deligne). Choose Frob ∈ WK . The map
⎧
n
n
⎪
⎪Frob σ ↦ ρ(Frob σ)etl (σ)N
(ρ, N ) ↦ ⎨
⎪
⎪
⎩WK → GL(V )
defines a bijection between isomorphism classes of Weil-Deligne representations of WK
over Ql and isomorphism classes of continuous l-adic representations of WK .
11
We have a dictionary:
(ρW D , N )
ρ
det(ρ)
det(ρW D )
⊕
⊕
WD
WD
ρ1 ⊗ ρ2
(ρ1 ⊗ ρ2 , N1 ⊗ Id + Id ⊗ N2 )
I
I
ρ = Vρ (fixed points)
(ker(N ))I
W
Finite image
N = 0 and ρ D (Frob) has finite order
Irreducible
N = 0 and ρW D is irreducible
N = 0 and ρW D is semisimple
Semisimple (direct sum of irreducibles)
Unramified
N = 0 and ρW D isunramified
ρW D is at most tamely ramified
At most tamely ramified
Potentially unramified
N =0
W
D
Semistable, i.e. ρ ∣I is unipotent
ρ
∣I is trivial, i.e. ρW D is unramified
Frobenius-semisimple, i.e. Frob acts semisimply
ρW D is semisimple
Theorem 3.5. Any Frobenius-semisimple continuous l-adic representation is a direct
sum of ρ ⊗ sp(n), with ρ(I) finite and ρ irreducible.
Example 3.6. Let E/K be an elliptic curve, ρE,l its l-adic representation.
• E has potentially good reduction if and only if N = 0 if and only if ρE , l is
irreducible or the direct sum of 2 characters.
0 1
• E has potentially multiplicative reduction if and only if N ∼ (
) if and only
0 0
if ρE,l ≅ χ ⊗ sp(2).
4
Weil-Deligne Representations and Elliptic Curves - Samuele
Anni
Notation 4.1.
• K/Qp finite, π uniformiser, vp valuation,
• k residue field, ∣k∣ = q,
• l ≠ p prime,
• E/K elliptic curve,
• Tl (E) = limE[ln ], Vl (E) = Ql ⊗Zl Tl (E),
←nÐ
12
• χl the l-adic cyclotomic character.
Definition 4.2. A Weil-Deligne representation for the Weil group WK with values over
F Ql is a pair (ρ, N ) such that
• ρ ∶ WK → GLn (F ),
• The image of inertia ρ(I) is finite,
• N ∈ Mn (F ) is nilpotent,
• ρ(Frob)N ρ(F rob)−1 = qN for all choices of Frob ∈ WK .
If a Weil-Deligne representation is Frobenius semisimple and indecomposable then it is
of the form τ ⊗ sp(n) where τ is irreducible and τ (I) is finite.
TODAY: E/K, ρE/K,l ∶ Gal(K/K) → GL(Vl (E)) ⊂ GL2 (Ql ).
We will look at the Weil-Deligne representation associated to ρE/K,l . For example, if
you fix an embedding i ∶ Ql Ð
→ C, then you get a complex Weil-Deligne representation
ρE/K,l,i . This is independent of the choice of embedding i and the prime l (Serre-Tate),
so we shall instead just write ρE .
Fact 4.3.
i. Néron-Ogg-Shafarevich (NOS): Let E/K, l ≠ p, ρE be as above. Then
E has good reduction if and only if ρE is unramified.
ii. E/K has potentially good reduction if and only if v(jE ) ≥ 0, where we set v(π) =
1.
iii. For p ≥ 5, if the reduction is potentially good, then it is good if and only if v(∆)
is a multiple of 12.
iv. Weil pairing: det ρE = χl .
v. If the reduction is potentially multiplicative, then the quadratic twist by −c6 has
split multiplicative reduction.
vi. Euler Factors ←→ Reduction type, i.e. point counting
L(E/K, s) = L(ρE , s)
L(ρE , s) ∶= P (q −s )−1 , where P (x) = det(1 − ρE (Frob−1 )x ∣Vl (E)I )
⎧
(1 − aq −s + q 1−2s )−1
⎪
⎪
⎪
⎪
L(E/K, s) = ⎨(1 − αq −s )−1
⎪
⎪
⎪
⎪
⎩1
if good reduction,
if multiplicative reduction,
if additive reduction,
13
1
where a = q + 1 − #Ẽ(k), α = {
−1
if split,
if nonsplit.
Proposition 4.4. E/K has potentially multiplicative reduction if and only if v(j) < 0,
if and only if ρE ≅ χ ⋅ χl ⊗ sp(2), where χ is the quadratic character associated to
√
K( −c6 )/K. Moreover, if χ is trivial (respectively unramified non-trivial, respectively
ramified), then this corresponds to E having split multiplicative (respectively nonsplit
multiplicative, respectively additive reduction).
Idea of Proof. BY NOS, the image of inertia is infinite by the classification since ρE is
Frobenius semi-simple and indecomposable so ρE ≅ ρ ⊗ sp(n), which implies that ρE ≅
(1-dim) ⊗sp(2). Now (1-dim)≅ χ ⋅ χl by the Weil pairing and det(ρE ) = χl .
2
So χl = det(ρE ) = χ2 ⋅ χ2l ⋅ χ−1
l so χ is trivial or quadratic. Looking at the Euler factor
we see χ is trivial if E has split multiplicative reduction. If χ is quadratic, then it is the
√
∎
quadratic character associated to Gal(K( −c6 )/K) (exercise).
If E/K has potentially multiplicative reduction, then there exists a quadratic character
χ such that E χ has split multiplicative reduction.
×
E(K) ⊗ χ ≅ E χ (K) ≅
K
as Gal(K/K)-modules.
qZ
×
K
Vl (E) ≅ Vl ( Z ) ⊗ χ,
q
ρE (g) = ρ(g)etl (i)N , g = Φm i ∈ WK , i ∈ I,
1 ∗
ρE (Φ) = ± (
)
0 q
Proposition 4.5. Suppose E/K has potentially good reduction. Set e as follows in the
below table.
v(∆) mod 12
e
0
1
6
2
4 or 8 3 or 9 2 or 10
3
4
6
Suppose p ≥ 5. Then ρE satisfies:
Case 1: K does not contain the eth roots of unity. Then ρE ≅ IndL/K χ, where L is a
quadratic extension of K.
14
Equivalently, ρE ≅ ρD2e ⊗ ψ where ρD2e is the faithful 2-dimensional representation of
the dihedral group with 2e elements and ψ is a 1-dimensional unramified character such
√
that ψ(Φ) = ± −q and D2e ≅ Gal(K(ζe , π 1/e )/K).
Case 2: If K contains the eth roots of unity, then ρE ≅ ηψ ⊕ (ηψ)−1 χl , where η is the
ramified character of Gal(K(π 1/e )/K) and ψ is unramified.
Proof. By NOS, the image of inertia is finite and by the the Weil pairing det(ρE (i)) = 1
for all i ∈ I.
Lemma 4.6. The characteristic polynomial of ρE (i) is independent of l for all i ∈ I, and
hence is a polynomial over Z. In our case, this implies that ord(ρE (i)) ∈ {1, 2, 3, 4, 6}.
E has potentially good reduction so since p ≥ 5, I = It (the tame inertia) so I ≅ Ce ,
where Ce ≅ C1 , C2 , C3 , C4 , C6 . By Fact iii, E has good reduction if and only if v(∆) is
a multiple of 12, so we have a correspondence of v(∆) with e. The Galois action factors
through K un (π 1/e ) and inertia has a generator acting with eigenvalues ζe , ζe−1 .
Case 1: ζe ∈/ K
F /K is nonabelian, L = K(ζe , π 1/e ) is a dihedral D2e extension. LEt ρD2e be the faithful
2-dimensional representation of D2e , then the Weil-Deligne representations are of the
form ρD2e ⊗ ψ where ψ is unramified.
√
By the Weil pairing, χl = det(ρD2e ) ⋅ ψ 2 = ψ 2 so ψ(Frob) = ± −q.
Case 2: ζe ∈ K
F /K is abelian, L = K(π 1/e ) is a cyclic totally ramified extension and the Weil-Deligne
representation voer F /K is faithful on inertia.
∎
5
What if l = p? - Marc Masdeu
15