438 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
Quantum transport in weakly coupled superlattices at low temperature
We report on the study of the electrical current flowing in weakly coupled
superlattice (SL) structures under an applied electric field at very low
temperature, i.e. in the tunneling regime. This low temperature transport is
characterized by an extremely low tunneling probability between adjacent wells.
Experimentally, I(V) curves at low temperature display a striking feature, i.e
a plateau or null differential conductance. A theoretical model based on the
evaluation of scattering rates is developed in order to understand this
behaviour, exploring the different scattering mechanisms in AlGaAs alloys. The
dominant interaction in usual experimental conditions such as ours is found to
be the electron-ionized donors scattering. The existence of the plateau in the
I(V) characteristics is physically explained by a competition between the
electric field localization of the Wannier-Stark electron states in the weakly
coupled quantum wells and the electric field assisted tunneling between
adjacent wells. The influence of the doping concentration and profile as well
as the presence of impurities inside the barrier are discussed
Modular curves and N\'eron models of generalized Jacobians
Let be a smooth geometrically connected projective curve over the field
of fractions of a discrete valuation ring , and a modulus on
, given by a closed subscheme of which is geometrically reduced. The
generalized Jacobian of with respect to is
then an extension of the Jacobian of by a torus. We describe its N\'eron
model, together with the character and component groups of the special fibre,
in terms of a regular model of over . This generalizes Raynaud's
well-known description for the usual Jacobian. We also give some computations
for generalized Jacobians of modular curves with moduli supported on
the cusps.Comment: 36 pages, minor corrections and references added. Accepted version,
to appear in Compositio Mat
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