Computational Aspects of L-functions
Computational Aspects of L-functions
International Workshop ”Probability, Analysis and Geometry”
Lomonosov Moscow State University and Ulm University
Michel Börner, Inst. for Pure Mathematics
Ulm University
October 3, 2014
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
1 / 21
Computational Aspects of L-functions
Introduction
Introduction
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
2 / 21
Computational Aspects of L-functions
Introduction
Situation
Smooth projective curve Y over number field,
genus g(Y)
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
3 / 21
Computational Aspects of L-functions
Introduction
Situation
Smooth projective curve Y over number field,
genus g(Y)
↓
L-series L(Y, s) =
an
∑ ns = ∏ Lp (Y, s)
n
p
conductor N
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
3 / 21
Computational Aspects of L-functions
Introduction
Properties
Conjectured Properties
analytic continuation to meromorphic function
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
4 / 21
Computational Aspects of L-functions
Introduction
Properties
Conjectured Properties
analytic continuation to meromorphic function
functional equation
Λ(Y, s) = ±Λ(Y, 2 − s),
where Λ(Y, s) := Ns/2 · (2π )−gs · Γ(s)g · L(Y, s).
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
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Computational Aspects of L-functions
Introduction
Special case
Special case: Y hyperelliptic of genus g ≥ 2 over Q.
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
5 / 21
Computational Aspects of L-functions
Introduction
Special case
Special case: Y hyperelliptic of genus g ≥ 2 over Q.
Conductor
N=
∏ pf
p
,
p
fp = 0 ⇐= Y has good reduction at p .
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
5 / 21
Computational Aspects of L-functions
Introduction
Special case
Special case: Y hyperelliptic of genus g ≥ 2 over Q.
Conductor
N=
∏ pf
p
,
p
fp = 0 ⇐= Y has good reduction at p .
Y : y2 + h(x) · y = g(x)
⇐⇒ Y : y2 = f (x) := 4g(x) + h2 (x)
deg g = 2g(Y) + 1, deg h ≤ g(Y).
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
5 / 21
Computational Aspects of L-functions
Introduction
Computation
Use sage package developed by Tim Dokchitser:
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
6 / 21
Computational Aspects of L-functions
Introduction
Computation
Use sage package developed by Tim Dokchitser:
Conductor N, genus g(Y)
↓
Dokchitser package (sage)
↓
bound M (M ∼
√
N)
↓
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
6 / 21
Computational Aspects of L-functions
Introduction
Computation
Use sage package developed by Tim Dokchitser:
Conductor N, genus g(Y)
↓
Dokchitser package (sage)
↓
bound M (M ∼
√
N)
↓
calculate an with n ≤ M
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
6 / 21
Computational Aspects of L-functions
Introduction
Functional equation
calculate an with n ≤ M
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
7 / 21
Computational Aspects of L-functions
Introduction
Functional equation
calculate an with n ≤ M
↓ + bad Lp , fp
Dokchitser: Check conjectured functional equation
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
7 / 21
Computational Aspects of L-functions
Introduction
Functional equation
calculate an with n ≤ M
↓ + bad Lp , fp
Dokchitser: Check conjectured functional equation
Based on
I. Bouw, S. Wewers: Computing L-functions and
semistable reduction of superelliptic curves,
arXiv:1211.4459.
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
7 / 21
Computational Aspects of L-functions
Introduction
What we do
Aims:
Toolbox in sage to calculate L(Y, s) =
for large class of examples
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
an
∑ ns
n
October 3, 2014
8 / 21
Computational Aspects of L-functions
Introduction
What we do
Aims:
Toolbox in sage to calculate L(Y, s) =
for large class of examples
an
∑ ns
n
Verify functional equation for many examples
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
8 / 21
Computational Aspects of L-functions
Algorithm
Algorithm
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
9 / 21
Computational Aspects of L-functions
Algorithm
Local factor
Euler product L(Y, s) =
∏ Lp (Y, s)
p
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
10 / 21
Computational Aspects of L-functions
Algorithm
Local factor
Euler product L(Y, s) =
1
∏ Lp (Y, s) = ∏ Pp (p−s ) .
p
p
Case I: Good reduction at p.
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
10 / 21
Computational Aspects of L-functions
Algorithm
Local factor
Euler product L(Y, s) =
1
∏ Lp (Y, s) = ∏ Pp (p−s ) .
p
p
Case I: Good reduction at p.
P(T ) = 1 + c1 T + c2 T2 + . . . + cg Tg + . . . + c2g T2g
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
10 / 21
Computational Aspects of L-functions
Algorithm
Local factor
Euler product L(Y, s) =
1
∏ Lp (Y, s) = ∏ Pp (p−s ) .
p
p
Case I: Good reduction at p.
P(T ) = 1 + c1 T + c2 T2 + . . . + cg Tg + . . . + c2g T2g
Computations:
c1 , . . . , cg by point counting over Fp1 , . . . , Fpg
cg+1 , . . . , c2g from c1 , . . . , cg via mirror rule
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
10 / 21
Computational Aspects of L-functions
Algorithm
Assembling L-series
Use c2 , c3 , . . . to calculate ap2 , ap3 , . . . (pk ≤ M)
and amn = am · an (gcd(m, n) = 1) to assemble an :
∞
L(Y, s) =
apk
p k =0
Michel Börner, Inst. for Pure Mathematics
an
∏ ∑ pks = ∑ ns
Ulm University
( Computational Aspects of L-functions
)
n
October 3, 2014
11 / 21
Computational Aspects of L-functions
Algorithm
Assembling L-series
Use c2 , c3 , . . . to calculate ap2 , ap3 , . . . (pk ≤ M)
and amn = am · an (gcd(m, n) = 1) to assemble an :
∞
L(Y, s) =
apk
an
∏ ∑ pks = ∑ ns
p k =0
n
Complexity for L-series coefficients an for n ≤ M :
Point counting: O(pk ) for all pk ≤ M
ap and an using the ci
Lp for bad p
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
11 / 21
Computational Aspects of L-functions
Algorithm
Bad reduction
Case II: Bad reduction at p.
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
12 / 21
Computational Aspects of L-functions
Algorithm
Bad reduction
Case II: Bad reduction at p.
Find ’bad primes’. Recall Y : y2 = f (x)
disc(f ) =
∏ pi
ei
i
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
12 / 21
Computational Aspects of L-functions
Algorithm
Bad reduction
Case II: Bad reduction at p.
Find ’bad primes’. Recall Y : y2 = f (x)
disc(f ) =
∏ pi
ei
i
and if Ȳ/F2 singular
2 ∈ {bad primes} .
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
12 / 21
Computational Aspects of L-functions
Algorithm
Bad factor p 6= 2: Two curves
Case IIa: p 6= 2.
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
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Computational Aspects of L-functions
Algorithm
Bad factor p 6= 2: Two curves
Case IIa: p 6= 2.
For each bad p 6= 2, write f ≡ f̄ mod p
Ȳ : y2 = f̄ (x) := r̄(x)2 · s̄(x)
Michel Börner, Inst. for Pure Mathematics
Ulm University
,
( Computational Aspects of L-functions
)
s̄ squarefree .
October 3, 2014
13 / 21
Computational Aspects of L-functions
Algorithm
Bad factor p 6= 2: Two curves
Case IIa: p 6= 2.
For each bad p 6= 2, write f ≡ f̄ mod p
Ȳ : y2 = f̄ (x) := r̄(x)2 · s̄(x)
,
s̄ squarefree .
Ȳ is semistable ⇐⇒ gcd(r̄, s̄) = 1.
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
13 / 21
Computational Aspects of L-functions
Algorithm
Bad factor p 6= 2: Two curves
Case IIa: p 6= 2.
For each bad p 6= 2, write f ≡ f̄ mod p
Ȳ : y2 = f̄ (x) := r̄(x)2 · s̄(x)
,
s̄ squarefree .
Ȳ is semistable ⇐⇒ gcd(r̄, s̄) = 1.
Normalization
Ȳ0 :
Michel Börner, Inst. for Pure Mathematics
y2 = s̄(x)
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
13 / 21
Computational Aspects of L-functions
Algorithm
Bad factor p 6= 2: Loops
Ȳ0 :
y2 = s̄(x)
Ȳ : y2 = r̄(x)2 · s̄(x)
X̄
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
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Computational Aspects of L-functions
Algorithm
Bad factor p 6= 2: Loops
Ȳ0 :
r̄1
y2 = s̄(x)
r̄2
Ȳ : y2 = r̄(x)2 · s̄(x)
r̄ = ∏i r̄i ∈ Fp [x]
X̄
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
14 / 21
Computational Aspects of L-functions
Algorithm
Bad factor p 6= 2: L-factors
Results:
Exponent fp = # loops over F̄p = ∑i deg(r̄i )
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
15 / 21
Computational Aspects of L-functions
Algorithm
Bad factor p 6= 2: L-factors
Results:
Exponent fp = # loops over F̄p = ∑i deg(r̄i )
Nodes have split or non-split reduction and
Lp (Y, T ) = Lp (Ȳ0 , T ) · ∏(1 − ε i Tdi )−1
i
with ε i := ±1, di = deg(r̄i ).
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
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Computational Aspects of L-functions
Algorithm
Next steps
Next steps:
Case p = 2: very similar.
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
16 / 21
Computational Aspects of L-functions
Algorithm
Next steps
Next steps:
Case p = 2: very similar.
Adapt algorithm to large class of examples, e.g.
non-hyperelliptic curves of genus ≥ 3 using
I. Bouw, S. Wewers: Computing L-functions and
semistable reduction of superelliptic curves,
arXiv:1211.4459.
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
16 / 21
Computational Aspects of L-functions
Examples
Examples
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
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Computational Aspects of L-functions
Examples
Example for g = 3
Y hyperelliptic, g(Y) = 3,
Y : y2 + (3x3 + 3x2 + 2x + 1)y = x7 − 2x6 − 2x4 + x3 + 3x2 + x
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
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Computational Aspects of L-functions
Examples
Example for g = 3
Y hyperelliptic, g(Y) = 3,
Y : y2 + (3x3 + 3x2 + 2x + 1)y = x7 − 2x6 − 2x4 + x3 + 3x2 + x
disc(f ) = 212 · 113 · 29
Michel Börner, Inst. for Pure Mathematics
no sing mod 2
=⇒
Ulm University
p ∈ {11, 29}.
( Computational Aspects of L-functions
)
October 3, 2014
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Computational Aspects of L-functions
Examples
Example for g = 3
Y hyperelliptic, g(Y) = 3,
Y : y2 + (3x3 + 3x2 + 2x + 1)y = x7 − 2x6 − 2x4 + x3 + 3x2 + x
disc(f ) = 212 · 113 · 29
no sing mod 2
=⇒
p ∈ {11, 29}.
N = 113 · 29 ≈ 105 , M ≈ 4000,
point counting for 500 primes, time ≈ 1 min
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
18 / 21
Computational Aspects of L-functions
Examples
Example for g = 3
Y hyperelliptic, g(Y) = 3,
Y : y2 + (3x3 + 3x2 + 2x + 1)y = x7 − 2x6 − 2x4 + x3 + 3x2 + x
disc(f ) = 212 · 113 · 29
no sing mod 2
=⇒
p ∈ {11, 29}.
N = 113 · 29 ≈ 105 , M ≈ 4000,
point counting for 500 primes, time ≈ 1 min
L11 = (1 − T ) · (1 + T )2
L29 = (1 + T ) · (29T2 + 1) · (29T2 + 2T + 1)
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
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Computational Aspects of L-functions
Examples
Finding Examples
’Random’ polynomials g and h:
g = x7 + 2x6 + 3x5 + 4x4 + 5x3 + 6x2 + 7x − 8
h = x3 + 2x2 + 3x + 4
f = 4g + h2
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
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Computational Aspects of L-functions
Examples
Finding Examples
’Random’ polynomials g and h:
g = x7 + 2x6 + 3x5 + 4x4 + 5x3 + 6x2 + 7x − 8
h = x3 + 2x2 + 3x + 4
f = 4g + h2
disc(f ) = 222 · 5 · 17 · 31 · 5015746709
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
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Computational Aspects of L-functions
Examples
Finding Examples
’Random’ polynomials g and h:
g = x7 + 2x6 + 3x5 + 4x4 + 5x3 + 6x2 + 7x − 8
h = x3 + 2x2 + 3x + 4
f = 4g + h2
disc(f ) = 222 · 5 · 17 · 31 · 5015746709
12
=⇒ N ≥ 5 · 17 · 31 · 5015746709 ≈ 10
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
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M∼
√
N!
October 3, 2014
19 / 21
Computational Aspects of L-functions
Examples
Example for g = 5
Y hyperelliptic, g(Y) = 5,
Y : y2 + (−x5 − x4 − x − 1)y = x11 − x10 − x7 − x5 − x4 − x
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
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Computational Aspects of L-functions
Examples
Example for g = 5
Y hyperelliptic, g(Y) = 5,
Y : y2 + (−x5 − x4 − x − 1)y = x11 − x10 − x7 − x5 − x4 − x
disc(f ) = 240 · 32 · 13 · 19 · 97
sing mod 2
=⇒
p ∈ {2, 3, 13, 19, 97}.
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
20 / 21
Computational Aspects of L-functions
Examples
Example for g = 5
Y hyperelliptic, g(Y) = 5,
Y : y2 + (−x5 − x4 − x − 1)y = x11 − x10 − x7 − x5 − x4 − x
disc(f ) = 240 · 32 · 13 · 19 · 97
sing mod 2
=⇒
p ∈ {2, 3, 13, 19, 97}.
N = 216 · 32 · 13 · 19 · 97 ≈ 1010 ,
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
20 / 21
Computational Aspects of L-functions
Examples
Example for g = 5
Y hyperelliptic, g(Y) = 5,
Y : y2 + (−x5 − x4 − x − 1)y = x11 − x10 − x7 − x5 − x4 − x
disc(f ) = 240 · 32 · 13 · 19 · 97
sing mod 2
=⇒
p ∈ {2, 3, 13, 19, 97}.
N = 216 · 32 · 13 · 19 · 97 ≈ 1010 ,
M ≈ 1300000,
point counting for ≈ 100000 primes, time: 12 hours
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
)
October 3, 2014
20 / 21
Computational Aspects of L-functions
Examples
The last slide
Thank you!
Related papers:
I. Bouw, S. Wewers: Computing L-functions and semistable
reduction of superelliptic curves, arXiv:1211.4459.
T. Dokchitser: Computing special values of motivic
L-functions, arXiv:math/0207280.
M. B., I. Bouw, S. Wewers: in preparation.
Michel Börner, Inst. for Pure Mathematics
Ulm University
( Computational Aspects of L-functions
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October 3, 2014
21 / 21