3. "On the modularity of Q-curves" (2000), with C.Skinner, submitted.
2. "Finite flatness of torsion subschemes of Hilbert-Blumenthal abelian varieties," (1999), to appear, Jour. f. die Reine und Angew. Math.
1. "Congruence ABC implies ABC," (1999), to appear, Indag.Math.
4. "Endomorphism algebras of Jacobians," (2000), submitted.
Van der Geer and Oort have written:
"...one expects excess
intersection of the Torelli locus and the loci corresponding to
abelian varieties with very large endomorphism rings; that is, one
expects that they intersect much more than their dimensions suggest." 3. "On the modularity of Q-curves,"(2000),with C. Skinner, submitted.
A Q-curve is an elliptic curve over a number field K which
is isogenous to its Galois conjugates. Ribet proved that every
quotient of J1(N) is a Q-curve, and conjectured
conversely that every Q-curve is a quotient of some J_1(N). We
prove this conjecture subject to certain local conditions at 3. The
main tools are the deformation theorems of Conrad-Diamond-Taylor and
Skinner-Wiles.
2. "Finite flatness of torsion subschemes of
Hilbert-Blumenthal abelian varieties," (1999), to appear,
Jour. f. die Reine und Angew. Math.
Let E be a totally real number field of degree d over Q.
We give a method for constructing a set of Hilbert modular cuspforms
f1,...,fd with the following property. Let K be the
fraction field of a complete dvr A, and let X/K be a
Hilbert-Blumenthal abelian variety with multiplicative reduction and
real multiplication by the ring of integers of E. Suppose n is
an integer such that n divides the minimal valuation of fi(X) for
all i. Then X[n']/K extends to a finite flat group scheme over
A, where n' is a divisor of n with n'/n bounded by a constant
depending only on f1,..., fd. When E = Q, the theorem
reduces to a well-known property of f1 = D. When E is a quadratic field of discriminant 5
or 8, we produce the desired pairs of Hilbert modular forms explicitly
and show how they can be used to compute the group of Néron
components of a Hilbert-Blumenthal abelian variety with real
multiplication by E.
Previous works of Brumer, Mestre, Ekedahl-Serre, and others have
justified this expectation by providing examples of families of curves
whose Jacobians have large endomorphism rings. We give a general
procedure for producing families of branched covers of the line whose
Jacobians have extra endomorphisms. We show that many of the examples
produced by the above authors are "explained" in this way, and produce some
new examples. For instance, we obtain curves whose Jacobians have
real multiplication by the index-n subfield of Q(zp), where n is one of 2,4,6,8,10, and
p is any prime congruent to 1 mod n. At the end,
we discuss some questions about upper bounds for endomorphism algebras
of Jacobians.
Jordan Ellenberg * [email protected] * revised 16 Jun 2000