Computing zeta functions with p-adic · PDF fileComputing zeta functions with p-adic...
Transcript of Computing zeta functions with p-adic · PDF fileComputing zeta functions with p-adic...
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Computing zeta functions with p-adiccohomology
Jan Tuitman
University of Oxford
August 15, 2012
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Zeta functions
Let Fq be the finite field with q = pn elements, and let X be analgebraic variety over Fq.
Definition
Z (X ,T ) := exp(∞∑i=1
|X (Fqi )|T i
i !).
RemarkNote that a priori this is just an element of Q[[T ]]. However, muchmore is known by the famous Weil Conjectures.
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The Weil Conjectures
TheoremIf X is proper smooth of dimension d, then
Z (X ,T ) =p1p3 . . . p2d−1p0p2p4 . . . p2d
,
where for all i :
1. pi =∏
j(1− αi ,jT ) ∈ Z[T ],
2. the transformation t → qd/t maps the roots of pi to theroots of p2d−i (preserving their multiplicities),
3. |αi ,j | = qi/2 for all j and every embedding Q → C.
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Computing zeta functions
ProblemHow to compute the zeta function Z (X ,T ) (fast)?
MotivationWhen X is a projective nonsingular curve, the DiscreteLogarithm Problem on Jac(X ) is easy if |Jac(X )(Fq)| does nothave a large prime factor. Now we have
Z (X ,T ) =χ(T )
(1− T )(1− qT )
and one can show that
|Jac(X )(Fq)| = χ(1).
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Computing zeta functions
RemarkA naive method for computing Z (X ,T ) is to compute
|X (Fq)|, |X (Fq2)|, . . .
up to some point by trying all possible values for the coordinates.However, this takes time at least q which is exponential in log q.
Better ways to compute Z (X ,T ) are:
1. Schoof (Pila) type algorithms using l-adic etale cohomology,
2. Kedlaya, Lauder, etc. type algorithms using p-adic (or rigid)cohomology.
We will focus on the p-adic algorithms in this talk.
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Rigid cohomology
Let Qq denote the unramified extension of degree n of the field ofp-adic numbers Qp and Zq its ring of integers.
To X one can associate a sequence of finite dimensionalvectorspaces H0
rig (X ), . . . ,H2drig (X ) over Qq, together with an
action F ∗p of the p-th power Frobenius map Fp on X (that sendsevery coordinate to its p-th power), such that the followingLefschetz formula holds:
Z (X ,T ) :=2d∏i=0
det(1− qd(F ∗np )−1T |H irig (X ))(−1)
i+1.
To compute Z (X ,T ) it is enough to compute the maps F ∗p on the
H irig (X ) with high enough p-adic precision.
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Rigid cohomology for affine varietiesWe will explain the construction of rigid cohomology for affinevarieties.
Let X be a smooth affine variety over Fq with coordinate ring A.Choose a lift of A to Zq i.e., a smooth Zq algebra
A = Zq[x1, . . . , xl ]/(f1, . . . fm),
such thatA ∼= A⊗Zq Fq.
DefinitionThe overconvergent functions in x1, . . . , xl are the formal powerseries that have radius of convergence strictly greater than 1:
Qq〈x1, . . . , xl〉† = ∑
aI xI |aI ∈ Qq, ∃ρ > 1 : lim
|I |→∞aIρ|I | = 0.
where I = (i1, . . . , il) and |I | = i1 + . . .+ il .
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Rigid cohomology for affine varieties
DefinitionThe weak completion of A is defined as:
A† = Qq〈x1, . . . , xl〉†/(f1, . . . , fm).
TheoremThe p-th power Frobenius map Fp : A→ A can be lifted to a ringhomomorphism σ : A† → A†.
DefinitionThe de Rham complex of A† is defined as:
0 −−−−→ A†d−−−−→ Ω1
A†d−−−−→ Ω2
A†d−−−−→ . . .
Ω1A† =
Adx1 ⊕ . . .⊕Adxl(df1, . . . , dfm)
, ΩiA† = Λi (Ω1
A†) for i > 1.
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Rigid cohomology for affine varieties
DefinitionThe rigid cohomology spaces H i
rig (X ) are the cohomology spaces
(ker d/im d) of the de Rham complex of A†. The action F ∗p ofFrobenius on these Qq vector spaces is induced by σ.
RemarkThe H i
rig (X ) do not depend on the choices made in theirconstruction, are contravariantly functorial and can be defined in asimilar (but slightly more complicated) way for all separatedschemes of finite type over Fq.
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Hyperelliptic curves
Let p be an odd prime, q = pn a power of p and
f = y2 − Q(x),
for some monic Q ∈ Fq[x ] of degree 2g + 1 with distinct roots.Then f defines a hyperelliptic curve X of genus g .
Let U ⊂ X be the open affine (x , y) ∈ A2Fq|f (x , y) = 0, y 6= 0,
so that the coordinate ring of U is
A = Fq[x , y , y−1]/(f ).
We can now choose a monic lift Q ∈ Zq[x ] of Q, put f = y2 −Q,and define
A = Zq[x , y , y−1]/(f),
so that we getA† = Qq〈x , y , y−1〉†/(f).
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Hyperelliptic curves: cohomology
If A(x) = Q(x)B(x) +Q′(x)C (x) with A,B,C ∈ Qq[x ], then inH1rig (U) we have the relation
A(x)dx
y s≡(
B(x) +2C ′(x)
s − 2
)dx
y s−2 ,
which follows from
d
(C (x)
y s−2
)≡ 0.
This relation can be used to reduce any differential to one withs = 1, 2.
Similarly, powers of x greater than 2g (2g − 1 when s = 1) can beeliminated.
When applied to overconvergent differentials, these reductionprocedures can be shown to converge.
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Hyperelliptic curves: cohomology
TheoremA basis for H1
rig (U) over Qq is given by
x0 dx
y, x1 dx
y, . . . , x2g−1 dx
y ∪ x0 dx
y2, . . . , x2g dx
y2.
Moreover, H1rig (X ) is naturally contained in H1
rig (U), with basisgiven by
x0 dx
y, x1 dx
y, . . . , x2g−1 dx
y.
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Hyperelliptic curves: Frobenius
We need to compute a lift σ of Fp to A†. Note that sinceGal(Qq/Qp) ∼= Gal(Fq/Fp), the lift σ is uniquely defined on Qq.For the images of the variables, we choose:
x 7→ xp
y 7→ yp(1 +Qσ −Qp
y2p)1/2
RemarkThe image of y can be computed to any required p-adic precisionby expanding the square root.
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Kedlaya’s algorithm
1. Compute σ(y) to ‘enough’ p-adic precision,
2. Apply σ to the differentials x0 dxy , x
1 dxy , . . . , x
2g−1 dxy ,
3. Reduce the results from the previous step back to a linearcombination of the basis elements x0 dx
y , x1 dx
y , . . . , x2g−1 dx
y to obtain the matrix of F ∗p ,
4. Compute F ∗q = (F ∗p )n and its (rounded, reverse) characteristicpolynomial χ(T ).
Then
Z (X ,T ) =χ(T )
(1− T )(1− qT ), |Jac(X )(Fq)| = χ(1).
RemarkWe have skipped over some minor details: computation of σ(y)−1,Newton lift instead of expanding square root, p-adic precision loss(analysis), semilinearity of F ∗p , etc.
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Complexity and implementation
Cost of Kedlaya’s algorithm:
1. time : O(pn3g4)
2. space: O(pn3g3)
So this is not polynomial in log q, unless p is fixed. Therefore, thealgorithm is limited to small p.
There is an improvement of the algorithm(by D. Harvey) thatimproves the dependence on the characteristic to
√p and therefore
allows for bigger p.
Kedlaya’s algorithm is implemented in MAGMA and Harvey’sversion in SAGE.
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Families of hyperelliptic curves
Consider Q ∈ Fq[x , t] monic of degree 2g + 1 in x . Let
r(t) = Resx(Q,dQ
dx),
denote the resultant of Q with respect to x . Now,
f = y2 − Q(x , t)
defines a family X/S of hyperelliptic curves of genus g overS = Spec V , where V = Fq[t, 1/r(t)].
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Lifting the family
Let Q(x , t) be a lift of Q to Zq[x , t] and denote
r(t) = Resx(Q, dQdx
).
Then the polynomial
f = y2 −Q(x , t)
defines a family X/S of hyperelliptic curves of genus g overS = Spec V with V = Zq[t, 1/r(t)], that lifts the family X/S .
Define the family
U/S = Spec Zq[t, 1/r(t)][x , y , y−1]/(f)
of open affines of X/S, let W be its coordinate ring and U/S itsreduction modulo p.
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Relative rigid cohomology
DefinitionLet
A† = Qq〈t, 1/r(t)〉†
be the ring of overconvergent functions on S and σ : A† → A† alift of the p-th power Frobenius on A.
The relative rigid cohomology space H1rig (U/S) is defined as the
A† moduleW†dx ⊕W†dy
df.
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Relative rigid cohomology
Theorem
1. H1rig (U/S) is a free A† module with basis
x0 dx
y, x1 dx
y, . . . , x2g−1 dx
y ∪ x0 dx
y2, . . . , x2g dx
y2,
2. carries a natural Gauss-Manin connection
∇ : H1rig (U/S)→ H1
rig (U/S)⊗ Ω1A† ,
3. comes with a Frobenius isomorphism
F ∗p : σ∗H1rig (U/S)→ H1
rig (U/S),
that is horizontal with respect to ∇.
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Relative rigid cohomology
DefinitionH1rig (X/S) is the A† submodule of H1(U/S) generated by
x0 dx
y, x1 dx
y, . . . , x2g−1 dx
y.
RemarkThe connection ∇ and the Frobenius F ∗p both restrict toH1rig (X/S).
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The differential equation
Suppose that Φ ∈ M2g×2g (A†) is the matrix such that
F ∗p (x j−1 dx
y) ≡
2g∑i=1
Φij(x i−1 dx
y),
and let N ∈ M2g×2g (A†) be the matrix such that
∇(x j−1 dx
y) ≡
r∑i=1
Nij(x i−1 dx
y).
Then F ∗p being horizontal with respect to ∇ is equivalent to
dΦ
dt=
(dtσ
dt
)ΦNσ − NΦ.
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Deformation algorithm
The idea of the deformation method (for hyperelliptic curves) is asfollows:
1. Choose a family X/S of hyperelliptic curves, for which theaction of F ∗p on H1
rig (X0) is known or easy to compute.
2. Solve the differential equation
dΦ
dt=
(dtσ
dt
)ΦNσ − NΦ,
using the initial value from the previous step, to obtain Φ as amatrix with entries in A†.
3. Specialize Φ at some τ ∈ S(Zq) for which σ(τ) = τ (aTeichmuller lift) to obtain the matrix of F ∗p on H1
rig (Xτ ) andcompute Z (Xτ ,T ).