1,483 research outputs found
Abelian varieties over Q and modular forms
This paper gives a conjectural characterization of those elliptic curves over
the field of complex numbers which "should" be covered by standard modular
curves. The elliptic curves in question all have algebraic j-invariant, so they
can be viewed as curves over Q-bar, the field of algebraic numbers. The
condition that they satisfy is that they must be isogenous to all their Galois
conjugates. Borrowing a term from B.H. Gross, "Arithmetic on elliptic curves
with complex multiplication," we say that the elliptic curves in question are
"Q-curves." Since all complex multiplication elliptic curves are Q-curves (with
this definition), and since they are all uniformized by modular forms
(Shimura), we consider only non-CM curves for the remainder of this abstract.
We prove:
1. Let C be an elliptic curve over Q-bar. Then C is a Q-curve if and only if
C is a Q-bar simple factor of an abelian variety A over Q whose algebra of
Q-endomorphisms is a number field of degree dim(A). (We say that abelian
varieties A/Q with this property are of "GL(2) type.")
2. Suppose that Serre's conjecture on mod p modular forms are correct (Ref:
Duke Journal, 1987). Then an abelian variety A over Q is of GL(2)-type if and
only if it is a simple factor (over Q) of the Jacobian J_1(N) for some integer
N\ge1. (The abelian variety J_1(N) is the Jacobian of the standard modularComment: 19 pages, AMS-TeX 2.
Galois theory and torsion points on curves
In this paper, we survey some Galois-theoretic techniques for studying
torsion points on curves. In particular, we give new proofs of some results of
A. Tamagawa and the present authors for studying torsion points on curves with
"ordinary good" or "ordinary semistable" reduction at a given prime. We also
give new proofs of: (1) The Manin-Mumford conjecture: There are only finitely
many torsion points lying on a curve of genus at least 2 embedded in its
Jacobian by an Albanese map; and (2) The Coleman-Kaskel-Ribet conjecture: If p
is a prime number which is at least 23, then the only torsion points lying on
the curve X_0(p), embedded in its Jacobian by a cuspidal embedding, are the
cusps (together with the hyperelliptic branch points when X_0(p) is
hyperelliptic and p is not 37). In an effort to make the exposition as useful
as possible, we provide references for all of the facts about modular curves
which are needed for our discussion.Comment: 18 page
On the modularity level of modular abelian varieties over number fields
Let f be a weight two newform for Gamma_1(N) without complex multiplication.
In this article we study the conductor of the absolutely simple factors B of
the variety A_f over certain number fields L. The strategy we follow is to
compute the restriction of scalars Res_{L/\Q}(B), and then to apply Milne's
formula for the conductor of the restriction of scalars. In this way we obtain
an expression for the local exponents of the conductor N_L(B). Under some
hypothesis it is possible to give global formulas relating this conductor with
N. For instance, if N is squarefree we find that N_L(B) belongs to Z and
N_L(B)*f_L^{dim B}=N^{dim B}, where f_L is the conductor of L
Documento elaborado en el marco del Seminario Virtual 2004
Este documento es producto de un seminario virtual realizado durante el primer semestre de 2004, con el objetivo de consolidar federalmente una mirada sobre los problemas que afectan a la educación en establecimientos penitenciarios
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Emergent Disability and the Limits of Equality: A Critical Reading of the UN Convention on the Rights of Persons with Disabilities
The UN Convention on the Rights of Persons with Disabilities marks a
shift in international legal relationships to, and conceptions of,
disability. The Convention is the first binding international instrument
of its kind related to disability. Its premises differ from the earlier World
Programme on Disability, and more closely integrate the frameworks of
U.S. domestic equal protection and disability civil rights law. Drawing
on critical race and feminist theory, this Article critically examines the
implications of internationalizing a U.S. disability law framework, with
particular attention to the problem of emergent disability, or disability
which is specifically produced as a consequence of social inequity or state
violence
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