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"Selmer groups and Mordell-Weil groups of
elliptic curves over towers of function fields," submitted. (2005)
In two recent papers, Silverman discusses the problem of bounding the
Mordell-Weil ranks of elliptic curves over towers of function fields.
We first prove generalizations of the theorems of those two papers,
allowing non-abelian Galois groups and removing the dependence on
Tate's conjectures. We then prove some theorems about the growth of
Mordell-Weil ranks in towers of function fields whose Galois groups
are $p$-adic Lie groups; in particular, we give some Galois-theoretic
criteria which guarantee that certain curves E/Q(t) have bounded
Mordell-Weil rank over C(t^{1/p^n}) as n grows, and show that these
criteria are met for elliptic K3 surfaces whose associated Galois
representations have sufficiently large image. This contrasts with
recent results of Ulmer showing that the Mordell-Weil rank of
E/k(t^{1/p^n}) can be unbounded in n when k is a finite field.
"On uniform bounds for rational points on non-rational curves," with A. Venkatesh, to appear, Int. Math. Res. Not..
A theorem of Heath-Brown shows that a plane curve has at most C H^{2/d} rational points of height at most H, where C is a constant that does not depend on the curve. If the curve has positive genus, the global geometry of its Jacobian can be used to improve this bound; we show that in this case the exponent can be improved to 2/d - delta for some positive delta. If d = 3, we show delta = 1/450 is small enough. (Slightly edited version posted 26 Mar 2005.)
"The number of extensions of a number field with fixed degree and bounded discriminant", with A. Venkatesh, to appear, Ann. of Math.
An old conjecture holds that the number N_n(X) of degree n number fields with discriminant less than X is asymptotic to c_n X when X grows and n is fixed. This conjecture was proved by Davenport and Heilbronn for n = 3, and recently for n = 4,5 by Bhargava. For general n, however, the best known upper bound, due to Schmidt, was N_n(X) << X^{(n+2)/4}. We prove the much stronger bound N_n(X) << X^{exp(C sqrt(log n))}. We also show that the number of degree n fields with Galois group S_n and discriminant < X is bounded below by a constant multiple of X^{1/2}, and we give an upper bound on the number of Galois extensions of a number field with fixed degree and bounded discriminant. All our results are proved in the general setting of counting degree-n extensions of an arbitrary number field K.
While the theorem is purely number-theoretic, the proof is almost entirely algebro-geometric; the main idea is to relate the problem of counting number fields to the problem of counting integral points on certain carefully chosen varieties related to Hilbert schemes of points on affine space.
Let G be a subgroup of S_n. Malle has conjectured that the number of number fields which have degree n over Q, Galois group G, and discriminant between -X and X is asymptotic to C X^a (log X)^b where a and b are constants depending on G. We study the problem of counting extensions of F_q(T) with the above properties; this is essentially a problem of counting F_q-rational points on Hurwitz varieties. We show that the heuristic "an irreducible d-fold over F_q has q^d rational points" yields the analogue of Malle's conjecture, with identical values of a and b. Moreover, the function field setting suggests more general heuristics about the distribution of discriminants of number fields.
(Remark:
"Serre's conjecture over
F9," to appear, Ann. of Math..
We prove that, under certain local conditions at 3,5, and
infinity, a representation of the absolute Galois group of Q in
GL2(F9) is modular. As a corollary, abelian surfaces over Q with real multiplication and ordinary reduction at 3 and 5 are modular.
September 2002: I have replaced this paper with a substantially revised version, containing more details and nicer arguments in several places.
"K3 surfaces over number fields with geometric Picard number one,", in Arithmetic of Higher Dimensional Algebraic Varieties, B. Poonen and Y. Tschinkel, eds. (2004)
The generic complex K3 surface of degree d has Picard number one. We prove the (somewhat surprisingly, subtler) fact that there exist K3 surfaces of arbitrary degree over Qbar with Picard number one. Note: After this paper was in press, I learned about the paper of T. Terasoma ("Complete intersections with middle Picard number 1 defined over Q", Math. Z. 189 (1985), no.2, 289--296) which treats a related problem and uses a similar method.
"Q-curves and Galois representations," in Modular Curves and Abelian Varieties, J. Cremona, J.C.Lario, J.Quer and K.Ribet, eds. (2004)
A survey paper about the arithmetic of Q-curves, including problems about rational points on modular curves, surjectivity of Galois representations, modularity, and applications to classical Diophantine problems. Accessible open questions are emphasized throughout.
"A sharp diameter bound for unipotent groups of classical type over Z/pZ", with J. Tymoczko, in preparation.
We give a sharp bound for the diameter of unipotent groups over Z/pZ, with respect to generators arising from simple roots. This paper generalizes previous unpublished work of the first author pertaining to the unipotent subgroup of GL_n(Z/pZ), as cited, e.g. in P. Diaconis and L. Saloff-Coste, "Moderate growth and random walk on finite groups."
"On the average number of octahedral modular forms," Math. Res. Lett. 10, 269--273 (2003)
In a recent paper, P. Michel and A. Venkatesh, sharpening a result of Duke, show that the number of modular forms of conductor N which are octahedral (associated to Galois representations with projective image S4) is at most N4/5 + e. Any octahedral form gives rise to a Galois representation Phi: Gal(Q) -> S3 by composition with the projection S4 -> S3. Using an amplifier constructed by S. Wong, we show that there are at most N2/3 + e octahedral forms of conductor N associated to a given such Phi. We use this fact, combined with the Davenport-Heilbronn theorem, to show that the average number of octahedral forms of conductor N, as N varies over square-free integers, is at most N2/3 + e.
"On the error term in Duke's estimate for the average special value of L-functions," to appear, Canad. Math. Bull.
Duke showed, in his 1995 Invent. Math. paper, that the average value of L(f,1), as f ranges over an orthogonal basis for the cusp forms of weight 2 and level N, is 4Pi + O(N^(-1/2) log N). We improve the bound on the error term to O(N^(-1 + e)) for any e > 0. (NOTE: Nathan Ng found an error in an earlier version of this paper that changed the power of log N in the main result. The version dated April 2005 is correct. This change does not affect the results of "Galois representations attached to Q-curves..." below.)
"Galois representations attached to Q-curves and the generalized
Fermat equation A^4 + B^2 = C^p," Amer. J. Math. 126(4), 763--787 (2004)
We bound the degree of rational isogenies of Q-curves over
quadratic number fields, by combining Mazur's formal immersion
method with an analytic argument showing that Jacobians of certain
twisted modular curves admit quotients with Mordell-Weil rank 0.
As a consequence, we show that the generalized Fermat equation
A4 + B2 = Cp has no nontrivial
primitive solutions for p sufficiently large. In order to make "sufficiently large" not too large, we use a refinement of a result of Duke (see "On the error term..." above.)
Minor revisions, August 2002.
"Galois invariants of dessins d'enfants," in Arithmetic Fundamental Groups and Noncommutative Algebra, M. Fried and Y.Ihara, eds. (2002)
This paper has two goals. First, we discuss some known Galois invariants of dessins d'enfants, such as the cartographic group and the lifting invariants of Fried; we show that these invariants are preserved by the Grothendieck-Teichmuller group. We then define a new invariant which associates to every genus 0 dessin d'enfant a triple of elements in a profinite spherical braid group. We give an example showing two dessins with the same monodromy group and local ramification data which are separated by the braid group invariant.
"Endomorphism algebras of Jacobians," Adv. Math. 162, 243--271 (2001)
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." "On the modularity of Q-curves,",with C. Skinner, Duke Math. J. 109, no. 1, 97--122 (2001)
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.
"Finite flatness of torsion subschemes of
Hilbert-Blumenthal abelian varieties," J. Reine Angew. Math. 532, 1--32 (2001)
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.
"Congruence ABC implies ABC," Indag. Math., N.S., 11 (2), 197--200 (2000)
A note proving the following fact: if the ABC conjecture holds
for all A,B,C satisfying a divisibility condition N | ABC, then the full
ABC conjecture holds.
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 * ellenber@math.princeton.edu * revised 22 Dec 2004