43 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
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
Computational Arithmetic of Modular Forms
These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted student to implement it over any ring (such that a sufficient linear algebra theory is available in the chosen computer algebra system). The chosen approach is based on group cohomology and along the way the needed tools from homological algebra are provided
Endomorphism algebras of Abelian varieties with special reference to superelliptic Jacobians
This is (mostly) a survey article. We use an information about Galois properties of points of small order on an Abelian variety in order to describe its endomorphism algebra over an algebraic closure of the ground field. We discuss in detail applications to jacobians of cyclic covers of the projective line