Euler factors and Tate's algorithm for genus 2 hyperelliptic curve
Nick Alexander
I am at the L-series conference at PCMI in Utah. We're interested in
understanding Euler factors at primes of bad reductions of genus 2
hyperelliptic curves. Any pointers?
Nick
David R. Kohel
You probably know that Qing Liu has a package for genus 2 reduction, now
maintained in Sage (see his home page http://www.math.u-bordeaux.fr/~liu).
In terms of his publications documenting this algorithm, this article:
Modeles minimaux des courbes de genre deux
is perhaps the most relevant (available from his web page).
Cheers,
David
William Stein
>
> Hi Nick,
>
> You probably know that Qing Liu has a package for genus 2 reduction, now
> maintained in Sage (see his home page http://www.math.u-bordeaux.fr/~liu).
>
> In terms of his publications documenting this algorithm, this article:
>
> Modeles minimaux des courbes de genre deux
>
> is perhaps the most relevant (available from his web page).
>
> Cheers,
>
> David
+1 to David's remark. ALSO, see Tim Dokchiter's paper(s) on computing
L-series. Maybe part of the point of them is to "reverse engineer"
information about bad factors from knowledge of good factors, when
possible. I'll let Tim comment further.
William
Michael Stoll
> Hello sage-nt,
>
> I am at the L-series conference at PCMI in Utah. We're interested in
> understanding Euler factors at primes of bad reductions of genus 2
> hyperelliptic curves. Any pointers?
>
> Nick
Basically, you need to know enough information on the special fiber of a
regular proper model of the curve over Z_p. Liu's program provides most of
that when p is odd. What it does not tell you is how Frobenius acts on the
associated graph (whose vertices are the components of and multiple points or
points of intersection on the special fiber, connected by an edge for every
branch of the component passing through the point) and what the isomorphism
class over F_p is of the genus 1 curves that show up as components. Both are
needed in general to determine the Euler factor.
The Euler factor L_p(T) is the product of L_{p,toric}(T) and L_{p,abelian}(T).
(The actual factor in the L-series Euler product is L_p(p^{-s})^{-1}.)
L_{p,abelian}(T) is simply the product of the Euler factors associated to
(the smooth projective models of) the components of positive genus.
L_{p,toric}(T) is the reciprocal characteristic polynomial of Frob_p acting
on the first homology of the graph mentioned above.
Examples:
(1) You have two elliptic curves E_1 and E_2 in the special fiber (which then
have to be joined by a chain of P^1's). Then
L_p(T) = L_{p,E_1}(T) L_{p,E_2}(T)
(2) One elliptic curve E plus a nodal cubic, meeting in one point. Then
L_p(T) = L_{p,E}(T) (1 +/- T)
The sign is negative iff the tangent directions at the node are defined over
F_p (this is where the action of Frobenius comes in).
(3) Two P^1's meeting in three points (which may be replaced by chains of
P^1's). In this case, L_p = L_{p,toric}, and L_{p,toric} is a polynomial of
degree 2 that depends on how Frob_p permutes the intersection points -
trivial action --> (1 - T)^2
swap two points --> 1 - T^2
cyclic permutation --> 1 + T + T^2
(4) No higher genus components and no loops in the graph ==> L_p(T) = 1
So in order to determine L_p(T), one needs to know how Frobenius acts on the
graph, and one needs to know exactly which elliptic curves show up (not just
their j-invariants). I would assume it's not that hard to upgrade Liu's
program to also provide this information. This still leaves p=2 completely
out of the picture, however.
Michael
--
Michael Stoll * http://www.mathe2.uni-bayreuth.de/stoll/
Mathematisches Institut * Universität Bayreuth * 95440 Bayreuth, Germany
Michae...@uni-bayreuth.de
Nick Alexander
>>> reduction, now
>>> maintained in Sage
We did, but many thanks for the reference.
>> +1 to David's remark. ALSO, see Tim Dokchiter's paper(s) on
>> computing
>> L-series. Maybe part of the point of them is to "reverse engineer"
>> information about bad factors from knowledge of good factors, when
>> possible. I'll let Tim comment further.
Tim, anything you'd care to add would be appreciated.
> So in order to determine L_p(T), one needs to know how Frobenius
> acts on the
> graph, and one needs to know exactly which elliptic curves show up
> (not just
> their j-invariants). I would assume it's not that hard to upgrade
> Liu's
> program to also provide this information. This still leaves p=2
> completely
> out of the picture, however.
Michael, thanks for such a detailed reply. I will think about what
you've said and get back to the list.
Thanks!
Nick Alexander
Tim Dokchitser
> >> computing
> >> L-series. Maybe part of the point of them is to "reverse engineer"
> >> information about bad factors from knowledge of good factors, when
> >> possible. I'll let Tim comment further.
>
> Tim, anything you'd care to add would be appreciated.
factors from the functional equation of the L-function. Say you have a
curve C for which you know all bad factors except at p=2, where C has
some horrible reduction type. Then you can try to go through possible
exponents of the conductor at 2 (it is bounded by that of the
discriminant) and possible local factors at 2 (again, there are
finitely many choices) and check the functional equation of the L-
series of C - there is only one choice where it is satisfied, and that
is the correct one.
There is an examples in Magma,
http://magma.maths.usyd.edu.au/magma/htmlhelp/text1412.htm#14432
that works with an L-series of a genus 2 curve (except that here I
honestly work out the exponent of the conductor and the local factor
at the bad prime, they are harmless here so there is no guessing);
maybe this one may be modified to suit your example. Actually, if you
have a specific hyperelliptic curve in mind, you can send it to me and
I can try to figure out the bad local factors using my old scripts, if
I can remember what they did...
(As William said, stuff like this is mentioned in http://arxiv.org/abs/math/0207280,
in section 7)
Hope this helps!
Tim
Maxim P.
P.S. I've asked a question at mathoverflow: http://mathoverflow.net/questions/171708/calculate-reduction-of-jacobian-of-hyperelliptic-curve
суббота, 4 июля 2009 г., 2:12:52 UTC+4 пользователь Tim Dokchitser написал: