Examples
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Computing L-functions
Computing L-functions without the underlying
object
Stefan Lemurell
joint with David Farmer and Sally Koutsoliotas
Department of Mathematics
University of Gothenburg
and
Chalmers University of Technology
Curves and Automorphic Forms,
Arizona State University, March 2014
Stefan Lemurell joint with David Farmer and Sally Koutsoliotas
Computing L-functions
Examples
Axioms
Computing L-functions
Riemann zeta function
ζ(s) =
∞
X
1
, <(s) > 1.
ns
n=1
Two main features are the functional equation
ξ(s) = ΓR (s)ζ(s) = ξ(1 − s),
and the Euler product
ζ(s) =
Y
1 − p−s
−1
.
p
Notation: ΓR (s) = π −s/2 Γ(s/2) and ΓC (s) = 2(2π)−s Γ(s).
Stefan Lemurell joint with David Farmer and Sally Koutsoliotas
Computing L-functions
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Computing L-functions
Dirichlet L-functions
L(s, χ) =
∞
X
χ(n)
n=1
ns
, <(s) > 1,
where χ is a Dirichlet character. In this case there is a
functional equation
Λ(s, χ) = q s/2 ΓR (s + a)L(s, χ) = εΛ(1 − s, χ)
where q is the conductor of χ, a is 0 or 1 depending on if χ is
even or odd and |ε| = 1, and an Euler product
Y
−1
L(s, χ) =
1 − χ(p)p−s
.
p
Stefan Lemurell joint with David Farmer and Sally Koutsoliotas
Computing L-functions
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Computing L-functions
L-function of modular form on Γ0 (N)
Let
f (z) =
∞
X
bn q n
n=1
be a modular form of weight k on Γ0 (N). Then the L-function of
f (in analytic normalization) is
L(s, f ) =
∞
X
an
n=1
ns
, <(s) > 1,
where an = bn /n(k −1)/2 .
Stefan Lemurell joint with David Farmer and Sally Koutsoliotas
Computing L-functions
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Computing L-functions
L-function of modular form on Γ0 (N)
The functional equation in this case looks like
k −1
s/2
Λ(s, f ) = N ΓC s +
· L(s, f ) = εΛ(1 − s, f ),
2
and the Euler product
L(s, f ) =
Y
1 − ap p−s
−1 Y p|N
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1 − ap p−s + p−2s
p-N
Computing L-functions
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Computing L-functions
L-function of elliptic curve over Q
Let E be an elliptic curve defined over Q and and
√
bp = p + 1 − |E(Fp )| and ap = bp / p.
Then the L-function of E is
L(s, E) =
Y
1 − ap p−s
−1 Y p|N
1 − ap p−s + p−2s
−1
p-N
Thanks to modularity we know that it satisfies a functional
equation
Λ(s, E) = N s/2 ΓC (s + 1/2) · L(s, E) = εΛ(1 − s, E).
Stefan Lemurell joint with David Farmer and Sally Koutsoliotas
Computing L-functions
.
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Computing L-functions
L-function of Maass form on SL(2, Z)
Let
f (z) =
√ X
a(n)KiR (2π|n|y )e2πinx
y
n6=0
be a Maass form on Γ0 (N) with eigenvalue λ = 1/4 + R 2 . Then
the L-function of f is
L(s, f ) =
∞
X
an
n=1
Stefan Lemurell joint with David Farmer and Sally Koutsoliotas
ns
, <(s) > 1.
Computing L-functions
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Computing L-functions
L-function of Maass form on Γ0 (N)
The functional equation in this case looks like
Λ(s, f ) = N s/2 ΓR (s + a + iR) ΓR (s + a − iR)·L(s, f ) = εΛ(1−s, f ),
with (conjecturally) R ≥ 0 and a = 0 or 1 depending on whether
the form is even or odd. The Euler product looks like
L(s, f ) =
Y
1 − ap p−s
−1 Y p|N
Stefan Lemurell joint with David Farmer and Sally Koutsoliotas
1 − ap p−s + p−2s
p-N
Computing L-functions
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Computing L-functions
L-function of Maass forms on SL(n, Z), n > 2
The functional equation in this case looks like
Λ(s, f ) =
n
Y
ΓR (s + iµi ) · L(s, f ) = Λ(1 − s, f ),
i=1
with µi ∈ R and µ1 + µ2 + . . . + µn = 0 and the general form of
the Euler product is
L(s, f ) =
n
Y Y
p
(1 − αj,p p−s )−1 .
j=1
Stefan Lemurell joint with David Farmer and Sally Koutsoliotas
Computing L-functions
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Computing L-functions
Selberg axioms
It should be fairly obvious from the examples that there is a
pattern. This inspired Selberg to formulate a set of axioms that
an L-function should satisfy and to conjecture that every object
satisfying these axioms actually is the L-function of an
automorphic representation.
Stefan Lemurell joint with David Farmer and Sally Koutsoliotas
Computing L-functions
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Axioms
Computing L-functions
Dirichlet series
Axiom 1
L(s) is given by a Dirichlet series
L(s) =
∞
X
an
n=1
ns
where a1 = 1 and the series converges for <(s) > 1.
Meromorphic continuation with at most finitely many poles.
Stefan Lemurell joint with David Farmer and Sally Koutsoliotas
Computing L-functions
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Computing L-functions
Functional equation
Axiom 2
It satifies a functional equation
Λ(s) = εΛ(1 − s),
with |ε| = 1 where
Λ(s) := N
s/2
d1
Y
ΓR (s + µi )
i=1
d2
Y
ΓC (s + νj ) · L(s)
j=1
with N ∈ Z+ and µi , νj ∈ C. The integer d = d1 + 2d2 is called
the degree.
Stefan Lemurell joint with David Farmer and Sally Koutsoliotas
Computing L-functions
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Computing L-functions
Euler product and Ramanujan bound
Axiom 3
L(s) can can be written as
L(s) =
Y
Lp (p−s )−1 ,
p
where the product is over the primes, and Lp is a polynomial
with Lp (0) = 1.
Axiom 4
The coefficients satisfy the bound
an = O(nε )
for any ε > 0.
Stefan Lemurell joint with David Farmer and Sally Koutsoliotas
Computing L-functions
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Computing L-functions
Converse theorems
A converse theorem is a result stating that something that looks
like an L-function of a certain object, actually is an L-function of
such an object.
Hecke converse theorem
A Dirichlet series satisfying a functional equation like the
L-function of a modular form on SL(2, Z) is the L-function of a
modular form on SL(2, Z)
Weil converse theorem
If a Dirichlet series and all twists of it have functional equations
like the L-function of a modular form on Γ0 (N) (and its twists),
then it is the L-function of a modular form on Γ0 (N).
Stefan Lemurell joint with David Farmer and Sally Koutsoliotas
Computing L-functions
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Computing L-functions
Converse theorems
Conrey-Farmer converse theorem
A Dirichlet-series with a functional equation like the L-function
of a modular form on Γ0 (N) for N ≤ 17 and having at least a
partial Euler product is the L-function of a modular form on
Γ0 (N).
Stefan Lemurell joint with David Farmer and Sally Koutsoliotas
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Philosophy
Inspired by the Conrey-Farmer converse theorem you would
think that it should apply also in other cases.
In other words, it might be possible to compute an L-function
assuming only a functional equation and Euler product of
prescribed forms.
Stefan Lemurell joint with David Farmer and Sally Koutsoliotas
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Knowing an L-function
If you know
the functional equation and
the Dirichlet coefficients
then you “know” the L-function. At least in the sense that you
can compute everything you want to know (within reasonable
limits).
Stefan Lemurell joint with David Farmer and Sally Koutsoliotas
Computing L-functions
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Finding an L-function
The basic idea of our method to find L-functions is the
following: Given that we know
the (shape of the) functional equation and
the shape of the Euler product,
we can determine good approximations to both the functional
equation and the Dirichlet coefficients.
(Typically we know N and (at least) the real parts of the spectral
parameters µi and νj in the functional equation.)
Stefan Lemurell joint with David Farmer and Sally Koutsoliotas
Computing L-functions
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Approximate functional equation
The functional equation is modelled by the “approximate”
functional equation
Λ(s)g(s) = N
s/2
∞
X
an
n=1
ns
f1 (s, n) + εN
(1−s)/2
∞
X
an
f2 (1 − s, n)
n1−s
n=1
where g(s) is a test function with lots of freedom and
f1 (s, n) =
1
2πi
f2 (1−s, n) =
Z
ν+i∞
√
Gammafactors · z −1 g(s + z)( N/n)z dz,
ν−i∞
1
2πi
Z
ν+i∞
√
Gammafactors·z −1 g(s −z)( N/n)z dz
ν−i∞
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Linear equation
Note that once you fix the parameters in the functional
equation, the function g and a point s then the right hand side
is a linear combination of the Dirichlet coefficients. Hence, if
you know the coefficients then you may compute Λ(s), and if
you don’t then at least you have the value of Λ(s) expressed as
a linear combination in a2 , a3 , . . ..
If you do this with two different test functions, truncate and
subtract you get a linear equation in a2 , a3 , . . . , an .
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Linear system of equations
A linear system of equations with the coefficients a2 , a3 , . . . , an
as unknowns is created by
Fixing the parameters in the gamma functions.
Truncating the sums.
Choosing a number of points s (on the critical line).
For each point the approximate functional equation is
evaluted for two different test functions g. Subtracting
these gives one linear equation for each point.
Stefan Lemurell joint with David Farmer and Sally Koutsoliotas
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Euler product
In order to be successful it’s necessary to include (at least
some of) the information from the Euler product.
This creates a non-linear system of equations.
This is solved repeatedly and each of the solutions that are
found are plugged into a few extra equations generated by
other points on the critical line.
The goal is to find parameters for which the error in the
extra equations is as small as possible.
Stefan Lemurell joint with David Farmer and Sally Koutsoliotas
Computing L-functions
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Ramanujan bound
The Ramanujan bound is used implicity, since we start with
small initial values when searching for solutions.
Can be used to prove that there is no L-function with given
spectral parameters.
Stefan Lemurell joint with David Farmer and Sally Koutsoliotas
Computing L-functions
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Maass forms on SL(3, Z)
There is a fourier expansion also in this case, but at present it
seems hard to use it for finding the Maass forms.
Stefan Lemurell joint with David Farmer and Sally Koutsoliotas
Computing L-functions
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L-function data
Remember that the functional equation in this case looks like
Λ(s, f ) =
3
Y
ΓR (s + iµi ) · L(s, f ) = Λ(1 − s, f ),
i=1
with µi ∈ R and µ1 + µ2 + µ3 = 0 so there are two unknown real
spectral parameters µ1 , µ2 which may be assumed to be ≥ 0.
(Changing the sign of the parameters give the dual L-function.)
Stefan Lemurell joint with David Farmer and Sally Koutsoliotas
Computing L-functions
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Euler product
The Euler product in this case looks like
∞
X
an
n=1
ns
=
Y
1 − ap p−s + ap p−2s − p−3s
p
which gives
an am = anm
ap2 =
ap3 =
ap2
ap3
if (n, m) = 1
+ ap
− 2|ap |2 + 1.
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Data to determine
In order to determine an L-function corresponding to a SL(3, Z)
Maass form we need to find
The two real spectral parameters µ1 and µ2 .
The prime coefficients a2 , a3 , a5 , a7 . . . .
Stefan Lemurell joint with David Farmer and Sally Koutsoliotas
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Results for SL(3, Z)
Booker and Bian found the first 4 L-functions associated to
Maass forms on SL(3, Z) in 2008. They used twists of
L-functions and had huge systems of linear equations.
We have found over 2000 L-functions for SL(3, Z) of which
a selection is browsable on lmfdb.org.
For the first L-functions we only need 10-15 coefficients
and yet not more than around 100.
Also found some for non-trivial level up to 9.
Stefan Lemurell joint with David Farmer and Sally Koutsoliotas
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L-function for Maass forms on Sp(4, Z)
L-functions of Maass forms on Sp(4, Z) also have two
parameters and the functional equation in this case looks like
Λ(s, f ) = Λ(1 − s, f ),
where
Λ(s, f ) =
2
Y
ΓR s + iRj
j=1
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2
Y
ΓR s − iRj · L(s, f )
j=1
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Results for Sp(4, Z)
They have real coefficients and are just slightly harder than
SL(3, Z).
Have found about 100 L-functions of Maass forms on
Sp(4, Z).
The first of them beat the former world record (held by
ζ(s)) of longest zero free stretch on the critical line. First
zeros at ≈ ±14.49606.
Stefan Lemurell joint with David Farmer and Sally Koutsoliotas
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L-function for Maass forms on SL(4, Z)
L-functions of Maass forms on SL(4, Z) have three parameters
and the functional equation in this case looks like
Λ(s, f ) = Λ(1 − s, f ),
where
Λ(s, f ) :=
4
Y
ΓR s + iRj · L(s)
j=1
with Ri ∈ R and R1 + R2 + R3 + R4 = 0.
Have found over 100 L-functions of Maass forms on SL(4, Z) .
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Summary
In the original application that started the LMFDB-project it was
promised that we would:
Find thousands of GL(3) and GL(4) Maass forms:
These would be the first examples of such Maass
forms ever computed and this will require important
new algorithms.
Stefan Lemurell joint with David Farmer and Sally Koutsoliotas
Computing L-functions