Computing zeta functions in families of Ca,b curves using
Transcript of Computing zeta functions in families of Ca,b curves using
Weil ConjecturesMonsky-Washnitzer Cohomology
Ca,b curvesRelative Cohomology and Deformation
Computing zeta functions in families of Ca,b
curves using deformation
Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren
December 2007 - Berlin
Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren Computing zeta functions in families of Ca,b curves using deformation
Weil ConjecturesMonsky-Washnitzer Cohomology
Ca,b curvesRelative Cohomology and Deformation
Weil Conjectures
Monsky-Washnitzer Cohomology
Ca,b curves
Relative Cohomology and Deformation
Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren Computing zeta functions in families of Ca,b curves using deformation
Weil ConjecturesMonsky-Washnitzer Cohomology
Ca,b curvesRelative Cohomology and Deformation
The Zeta Function and Weil Conjectures
Let C be smooth projective curve over Fq, then zeta function ofC is
Z (t) = Z (C; t) = exp
( ∞∑r=1
Nrt r
r
)with Nr the number of points on C with coordinates in Fqr .Weil Conjectures:
I Z (t) is rational function over Z: P(t)(1−t)(1−qt)
I P(t) =∏2g
i=1(1− αi t) with g genus of C and |αi | =√
q
I P(t) =∑2g
i=0 ai t i with a0 = 1, a2g = qg and ag+i = qiag−i
I Nr = qr + 1−∑2g
i=1 αri and P(1) is the order of Jac(C/Fq)
Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren Computing zeta functions in families of Ca,b curves using deformation
Weil ConjecturesMonsky-Washnitzer Cohomology
Ca,b curvesRelative Cohomology and Deformation
Computational Approaches
I l-adic compute zeta mod small primes l + CRT.I Need explicit description of l-torsion of abelian varietyI Practical for genus 1 and somewhat for genus 2 curves
(Schoof, Pila, A-H, . . . )
I p-adic compute zeta mod high power of pI Canonical Lift / AGM:
I Ordinary abelian varieties admitting lift of FrobeniusI Elliptic curves over Fpn : Satoh, Mestre, Harley, . . .I Hyperelliptic curves over F2n : Mestre, Lercier-Lubicz, . . .
I p-adic Cohomology
Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren Computing zeta functions in families of Ca,b curves using deformation
Weil ConjecturesMonsky-Washnitzer Cohomology
Ca,b curvesRelative Cohomology and Deformation
Algebraic de Rham Cohomology
I Let A be a ring, e.g. the coordinate ring of a curveI The module of Kaher differentials D1(A) isI Generated over A by symbols da with a ∈ A with rules
d(a + b) = da + db
d(a · b) = adb + bda
I Elements of dA are called exact
Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren Computing zeta functions in families of Ca,b curves using deformation
Weil ConjecturesMonsky-Washnitzer Cohomology
Ca,b curvesRelative Cohomology and Deformation
Algebraic de Rham Cohomology
I X smooth affine curve over field K with coordinate ring
A = K[x , y ]/(f (x , y))
I Since f (x , y) = 0 get ( ∂f∂x dx + ∂f
∂y dy) = 0, so
D1(A) =(A dx + A dy)
(A( ∂f∂x dx + ∂f
∂y dy))
I First algebraic de Rham cohomology group is
H1DR(A) =
D1(A)
dA
Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren Computing zeta functions in families of Ca,b curves using deformation
Weil ConjecturesMonsky-Washnitzer Cohomology
Ca,b curvesRelative Cohomology and Deformation
Monsky-Washnitzer Cohomology
I C smooth affine curve over Fq with coordinate ring
A = Fq[x , y ]/(C(x , y))
I Let Qq be unramified extension of Qp with valuation ring Zq
I Lift C(x , y) to C(x , y) ∈ Zq[x , y ]
I Dagger ring A† of A := Zq[x , y ]/(C(x , y)) is
A† := Zq〈x , y〉†/(C(x , y)) ,
I Zq〈x , y〉† consists of power series∑
ri,jx iy j ∈ Zq[[x , y ]]
∃δ, ε ∈ R, ε > 0,∀(i , j) : ordpri,j ≥ ε(i + j) + δ.
Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren Computing zeta functions in families of Ca,b curves using deformation
Weil ConjecturesMonsky-Washnitzer Cohomology
Ca,b curvesRelative Cohomology and Deformation
Monsky-Washnitzer Cohomology
I Note: Zq〈x , y〉† is closed under integrationI Monsky-Washnitzer cohomology:
D1(A†)d(A†)
⊗Zq Qq,
I Lefschetz-fixed point theorem: exists a Zq-algebraendomorphism Fq : A† → A† such that
ZC(T ) =det(
I− qF∗−1q
∣∣∣ H1MW (A/Qq)
)1− qT
.
I Since q = pn, have F = Fσn−1
p · Fσn−2
p · · ·Fσp · Fp.
Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren Computing zeta functions in families of Ca,b curves using deformation
Weil ConjecturesMonsky-Washnitzer Cohomology
Ca,b curvesRelative Cohomology and Deformation
Ca,b curves
I Ca,b curve C over finite field Fq,
C(x , y) = ya + cb,0xb +∑
ai+bj<ab
ci,jxiy j (cb,0 6= 0).
I Absolutely irreducible and genus is g = (a−1)(b−1)2 .
I Unique degree 1 place Q at infinity and vQ(x) = −a,vQ(y) = −b.
I Various subclasses of Ca,b curves:I Hyperelliptic curves: a = 2 and b = 2g + 1I Superelliptic curves: ci,j = 0 for j > 0
Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren Computing zeta functions in families of Ca,b curves using deformation
Weil ConjecturesMonsky-Washnitzer Cohomology
Ca,b curvesRelative Cohomology and Deformation
Ca,b curves - Monsky-Washnitzer
I The affine curve C has coordinate ring A := Fq[x , y ]/(C).I Take any lift C(x , y) with same Newton polygon as C(x , y)
I Let A† be the dagger ring of A := Zq[x , y ]/(C)
I Elements of A† can be represented as∑a−1
l=0∑+∞
k=0 ak ,lxky l
and the valuation of ak ,l grows linearly with k .
I Basis for H1(A/Qq)
xky l dx for k = 0, . . . , b − 2 and l = 1, . . . , a − 1
I Reduction formulae to express any differential on this basis
Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren Computing zeta functions in families of Ca,b curves using deformation
Weil ConjecturesMonsky-Washnitzer Cohomology
Ca,b curvesRelative Cohomology and Deformation
Ca,b curves - Reduction Formulae
I Can always exchange dy-differentials into dx-differentialsby partial integration and differentiation
I Note: x r ysdx for 0 < s < a totally ordered by pole atinfinity for r ≥ b − 1
I The differential
ωr ,s = x r−(b−1)ysdC−d
x r−(b−1)
aa + s
ya+s +∑
ai+bj<ab
jci,j
s + jx iys+j
is exact and has pole order determined by λx r ysdx andλ 6= 0
Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren Computing zeta functions in families of Ca,b curves using deformation
Weil ConjecturesMonsky-Washnitzer Cohomology
Ca,b curvesRelative Cohomology and Deformation
Ca,b curves - Frobenius on A†
I The necessary conditions on the Frobenius σ on A† are
xσ ≡ xp mod p and yσ ≡ yp mod p and Cσ(xσ, yσ) = 0
I Main idea: lift Frobenius on x and y simultaneously suchthat denominator in the Newton iteration is invertible in A†.
I Let Z ∈ A† such that xσ = xp + αZ and yσ = yp + βZ , then
Cσ(xσ, yσ) = Cσ(xp + αZ , yp + βZ ) = 0 and Z ≡ 0 mod p
I Since C non-singular, then can compute α, β, γ ∈ A with
1 = α
(∂C∂x
)p
+ β
(∂C∂y
)p
+ γC
Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren Computing zeta functions in families of Ca,b curves using deformation
Weil ConjecturesMonsky-Washnitzer Cohomology
Ca,b curvesRelative Cohomology and Deformation
Ca,b-curves - Zeta Function
I The action of Fp on a differential form xky ldx is given by
F∗p (xky l dx) ≡ (xσ)k (yσ)l dxσ.
I Substituting power series for xσ and yσ, we can writeF∗
p (xky ldx) on basis of H1(A/Qq) using the reductionformulae.
I This gives matrix Fp which is an approximation of theaction of F∗
p on H1(A/Qq).
I The polynomial χ(t) := t2gP(1/t) can then beapproximated by the characteristic polynomial ofFpFσ
p · · ·Fσn−1
p .
Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren Computing zeta functions in families of Ca,b curves using deformation
Weil ConjecturesMonsky-Washnitzer Cohomology
Ca,b curvesRelative Cohomology and Deformation
Relative CohomologyI C(x , y , t) ∈ Fq[t ][x , y ] family of smooth curves over Spec S
with S = Fq[t , r(t)−1].
t0, r(t0) 6= 0
t1, r(t1) = 0
Spec A
Spec At0
SpecS
Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren Computing zeta functions in families of Ca,b curves using deformation
Weil ConjecturesMonsky-Washnitzer Cohomology
Ca,b curvesRelative Cohomology and Deformation
Relative Cohomology
I Let A = S[x , y ]/(C(x , y , t)) and for all t0 ∈ Fq withr(t0) 6= 0 write At0
= A/(t − t0)
I Aim: describe how the action of Frobenius onH1
MW (At0/Qq) alters as t0 varies.
I Let C(x , y , t) and r(t) be lifts of C(x , y , t) and r(t)I Define S† = Zq〈t , r(t)−1〉† = Zq〈t , z〉†/(zr(t)− 1) along
with the S†-module
A† =Zq〈t , r(t)−1, x , y〉†
(C(x , y , t))
Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren Computing zeta functions in families of Ca,b curves using deformation
Weil ConjecturesMonsky-Washnitzer Cohomology
Ca,b curvesRelative Cohomology and Deformation
Relative Cohomology
I Construction of relative MW cohomology group
H1MW (A/S†Qq
) =D1
t (A†)dt(A†)
⊗Zq Qq
I With dt exterior derivation on A† over S†, i.e. no derivationwith respect to t (t considered constant)
I Can define Zq-algebra endomorphism Fq on A† that liftsthe Frobenius action Fq on A
I Action on t can be chosen Fq(t) = tq
Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren Computing zeta functions in families of Ca,b curves using deformation
Weil ConjecturesMonsky-Washnitzer Cohomology
Ca,b curvesRelative Cohomology and Deformation
Relative Cohomology
I Fq induces map F∗q on H1
MW (A/S†Qq)
I Let t0 ∈ Fq be a non-zero of r(t) and let t0 ∈ Zq beTeichmuller lift, i.e. unique root of X q − X ∈ Zq[X ] thatreduces to t0 mod p.
I H1MW (At0
/Qq) equals H1MW (A/S†Qq
)/(t − t0)
I F∗q induces map on H1
MW (At0/Qq) equal to Frobenius
action of before
Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren Computing zeta functions in families of Ca,b curves using deformation
Weil ConjecturesMonsky-Washnitzer Cohomology
Ca,b curvesRelative Cohomology and Deformation
Deformation
I Looked at action of Frobenius on all fibers at once, but nothow the action on different fibers is related!
I Need to take into account derivation with respect to tI Construct MW-complex on surface
0 → A† d→ D1(A†) d→ D2(A†) → 0
I Gauss-Manin connection
∇ : H1MW (A/S†Qq
) → H1MW (A/S†Qq
) : ω 7→ ∇(ω)
Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren Computing zeta functions in families of Ca,b curves using deformation
Weil ConjecturesMonsky-Washnitzer Cohomology
Ca,b curvesRelative Cohomology and Deformation
Gauss-Manin Connection
I Gauss-Manin connection
∇ : H1MW (A/S†Qq
) → H1MW (A/S†Qq
) : ω 7→ ∇(ω)
I Let ω be represented by ω ∈ D1(A†), then can rewrite
d(ω) = ϕ ∧ dt
I ϕ modulo dt and dt(A†) then is ∇(ω)
Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren Computing zeta functions in families of Ca,b curves using deformation
Weil ConjecturesMonsky-Washnitzer Cohomology
Ca,b curvesRelative Cohomology and Deformation
Gauss-Manin Connection
I Theorem: ∇ ◦ F∗q = qtq−1 ◦ F∗
q ◦ ∇I Proof: commutativity of diagram
D1(A†)d- D2(A†)
D1(A†)
F∗q
?d- D2(A†).
F∗q
?
Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren Computing zeta functions in families of Ca,b curves using deformation
Weil ConjecturesMonsky-Washnitzer Cohomology
Ca,b curvesRelative Cohomology and Deformation
Differential Equation
I Let s1, . . . , sd be an S†Qq-basis of H1
MW (A/S†Qq)
I Let F , N be (d × d)-matrices with entries in S†Qqsuch that
F∗q (sj) =
d∑i=1
Fi,jsi , ∇(sj) =d∑
i=1
Ni,jsi
for j = 1, . . . , d .I Theorem of Gauss-Manin connection then gives rise to
first-order differential equation
N · F +ddt
F = qtq−1 · F · N(tq).
Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren Computing zeta functions in families of Ca,b curves using deformation
Weil ConjecturesMonsky-Washnitzer Cohomology
Ca,b curvesRelative Cohomology and Deformation
Differential Equation
I Similar differential equation for matrix Fp
N · Fp +ddt
Fp = ptp−1 · Fp · Nσ(tp)
I Solve this starting from matrix Fp (t0) of easy fiber
I Note: Fp matrix over S†Qqso overconvergent
I Matrix Fp of different fiber then simply is Fp (ti)
Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren Computing zeta functions in families of Ca,b curves using deformation
Weil ConjecturesMonsky-Washnitzer Cohomology
Ca,b curvesRelative Cohomology and Deformation
Algorithm in Practice
I Two applications:I compute zeta function of given Ca,b curveI generate Ca,b curve with nearly prime Jacobian
I Given curve C1(x , y) embed this in family
C(x , y , t) = tC1(x , y) + (1− t)C0(x , y)
with C0 easy fiber:I C0(x , y) = ya + xb + 1 for p - a, bI C0(x , y) = ya + xb + y for p | bI C0(x , y) = ya + xb + x for p | a
I Computing Fp(0) easy since can use superellipticalgorithm
Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren Computing zeta functions in families of Ca,b curves using deformation
Weil ConjecturesMonsky-Washnitzer Cohomology
Ca,b curvesRelative Cohomology and Deformation
Example: Elliptic Curves
I Family: C(x , y , t) = y3 − x3 − x − t over F5n
I Basis of H1MW (A/S†Qq
) is {ydx , xydx}I Matrix Fp(0) modulo 520(
60982419361512 00 83213137279115
)I Connection matrix N over Q is(
5t6(t2+4/27)
−79(t2+4/27)
527(t2+4/27)
7t6(t2+4/27)
)
Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren Computing zeta functions in families of Ca,b curves using deformation
Weil ConjecturesMonsky-Washnitzer Cohomology
Ca,b curvesRelative Cohomology and Deformation
Example: Elliptic Curves
I Solve diff equation N · Fp + ddt Fp = ptp−1 · Fp · Nσ(tp)
I Giving matrix over S†Qq= 〈t , 1/(t2 + 4/27)〉
I Frobenius matrix of any fiber t i simply evaluate Fp at tiI Note: t i can be in extension field of Fp
I Example: evaluate at 1 given Fp(1) of y2 − x3 − x − 1(9076222447517 76647025390324
26756651910595 15618474818105
)
Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren Computing zeta functions in families of Ca,b curves using deformation
Weil ConjecturesMonsky-Washnitzer Cohomology
Ca,b curvesRelative Cohomology and Deformation
Results of quick hack in Magma
equation t ∈ Fpn genus g precomp time/curveY 3 + X 4 + (t + 1)XY + 1 259 3 553s 14.5sY 3 + X 5 + X 2 + t + 1 243 4 135s 6.5sY 3 + X 4 + (t + 1)XY + 1 337 3 1064s 13sY 3 + X 5 + XY + tY + 1 329 4 4128s 22sY 3 − X 4 + tX 2 + t − 1 523 3 30.5s 2sY 3 − X 5 − X 2 + tX − 1 519 4 837s 20sY 3 + X 4 + tX − 1 5200 3 515s 538s
I Ref: C-implementation of Kedlaya version for first 2 linesgives 5000 and 7000 seconds per curve
Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren Computing zeta functions in families of Ca,b curves using deformation
Weil ConjecturesMonsky-Washnitzer Cohomology
Ca,b curvesRelative Cohomology and Deformation
Conclusion
I Deformation very useful for curves tooI Easier to use than extending Kedlaya to more general
classesI Also works for non-degenerate curvesI Faster than Kedlaya both for random curves and certainly
to generate Ca,b curves with nearly prime order
Wouter Castryck, Hendrik Hubrechts & Frederik Vercauteren Computing zeta functions in families of Ca,b curves using deformation