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28275 Implement isomorphism testing for modular abelian varieties klui "This ticket implements isomorphism testing for modular abelian varieties. This code was used for parts of my thesis.
= Algorithm =
The algorithm used here is described in Algorithm 2.11 here: https://wstein.org/papers/current/modabvar/modabvar.pdf . This is the same algorithm that William Stein implemented in Magma.
= Bonus Tests =
== Isomorphic to Dual? ==
This code determines which simple subvarieties of the new subvariety of `J0(N)` is isomorphic to its dual.
{{{
# about 2 minutes on i5-8350U
from sage.all import J0
not_isomorphic_to_dual = []
for N in range(1, 100):
Jnew = J0(N).new_subvariety()
for A in Jnew:
Ad = A.dual()[0]
buul, f, _ = A.is_isomorphic(Ad, both_maps=True)
if not buul:
not_isomorphic_to_dual.append((A.label(), A.dimension(),
A.hecke_operator(2).matrix().minpoly()))
else:
assert(f.matrix().determinant().abs() == 1)
print(not_isomorphic_to_dual)
}}}
which yields these 4 abelian varieties:
{{{
[('69b(1,69)', 2, x^2 - 5), ('77d(1,77)', 2, x^2 - 5), ('81a(1,81)', 2, x^2 - 3), ('85c(1,85)', 2, x^2 - 3)]
}}}
The corresponding Magma should be:
{{{
for N in [1..99] do
D := Decomposition(NewSubvariety(JZero(N)));
for A in D do
Ad := Dual(A);
try
buul := IsIsomorphic(A, Ad);
if not buul then
A, Dimension(A), MinimalPolynomial(HeckeOperator(A, 2));
end if;
catch e
print ""The following yields an error!!!"";
A, Dimension(A), MinimalPolynomial(HeckeOperator(A, 2));
end try;
end for;
end for;
}}}
which yields the same 4 abelian varieties:
{{{
Modular abelian variety image(69B) of dimension 2, level 3*23 and conductor 3^2*23^2 over Q
2 x^2 - 5
Modular abelian variety image(77D) of dimension 2, level 7*11 and conductor 7^2*11^2 over Q
2 x^2 - 5
The following yields an error!!!
Modular abelian variety image(81A) of dimension 2, level 3^4 and conductor 3^8 over Q
2 x^2 - 3
The following yields an error!!!
Modular abelian variety image(85C) of dimension 2, level 5*17 and conductor 5^2*17^2 over Q
2 x^2 - 3
}}}
When this code claims `A` is not isomorphic to `A.dual()[0]`, Magma either agrees or gives a error with a traceback that looks like
{{{
IsIsomorphic(
A: Modular abelian variety image(81A) of dimension 2, level 3^4...,
B: Modular abelian variety of dimension 2 and level 3^4 over Q
)
CanDetermineIsomorphism(
A: Modular abelian variety image(81A) of dimension 2, level 3^4...,
B: Modular abelian variety of dimension 2 and level 3^4 over Q
)
In file ""/home/user/magma-2.18/package/Geometry/ModAbVar/modabvar.m"", line 1072, column 23:
>> Append(~X, K!y);
^
Runtime error in 'Append': Bad argument types
Argument types given: *nothing ~, FldNumElt[FldRat]
}}}
== `A` is isomorphic to `(A/A.torsion_subgroup(3))[0]`
Here we check that `A` is isomorphic to `(A/A.torsion_subgroup(3))[0]`.
{{{
# about 18 seconds on i5-8350U
from sage.all import J0
for N in range(1, 50):
Jnew = J0(N).new_subvariety()
for A in Jnew:
B = (A / A.torsion_subgroup(3))[0]
buul, f, _ = A.is_isomorphic(B, both_maps=True)
assert(f.matrix().determinant().abs() == 1)
}}}" enhancement needs_review major sage-9.2 modular forms modular abelian variety isomorphism testing was Kevin Lui N/A u/klui/better_is_isomorphism be26efc2b894db924917423e9fab96df437adaef