17 research outputs found
Elements of prime order in Tate-Shafarevich groups of abelian varieties over
No full textFor each prime , we show that there exist geometrically simple abelian
varieties with non-trivial -torsion in their Tate-Shafarevich
groups. Specifically, for any prime , let be an
optimal quotient of with a rational point of order , and let . Then the number of positive integers , such
that the Tate-Shafarevich group of has non-trivial -torsion, is
, where is the dual of the -th quadratic twist of
. We prove this more generally for abelian varieties of -type
with a -isogeny satisfying a mild technical condition. In the special case
of elliptic curves, we give stronger results, including many examples where
for an explicit positive proportion of integers
.Comment: 10 pages. Final version, with improved exposition and a new section
with explicit calculations for elliptic curves. To appear in Forum of
Mathematics, Sigm
The geometry and arithmetic of bielliptic Picard curves
No full textWe study the geometry and arithmetic of the curves and their associated Prym abelian surfaces . We prove a Torelli theorem in this context and give a geometric proof of the fact that has quaternionic multiplication (QM) by the quaternion order of discriminant . This allows us to describe the Galois action on the geometric endomorphism algebra of . As an application, we classify the torsion subgroups of the Mordell-Weil groups , as both abelian groups and -modules.37 pages, minor change
Ranks of abelian varieties in cyclotomic twist families
No full textLet be an abelian variety over a number field , and suppose that embeds in , for some root of unity of order . Assuming that the Galois action on the finite group is sufficiently reducible, we bound the average rank of the Mordell--Weil groups , as varies through the family of -twists of . Combining this with the recently proved uniform Mordell--Lang conjecture, we prove near-uniform bounds for the number of rational points in twist families of bicyclic trigonal curves , as well as in twist families of theta divisors of cyclic trigonal curves . Our main technical result is the determination of the average size of a -isogeny Selmer group in a family of -twists.32 pages. Final version, to appear in Algebra & Number Theor
Rank growth of elliptic curves over -th root extensions
No full textFix an elliptic curve over a number field and an integer which is
a power of . We study the growth of the Mordell--Weil rank of after base
change to the fields . If admits a -isogeny, then
we show that the average ``new rank'' of over , appropriately defined,
is bounded as the height of goes to infinity. When , we moreover
show that for many elliptic curves , there are no new points on
over , for a positive proportion of integers .
This is a horizontal analogue of a well-known result of Cornut and Vatsal. As a
corollary, we show that Hilbert's tenth problem has a negative solution over a
positive proportion of pure sextic fields .
The proofs combine our recent work on ranks of abelian varieties in
cyclotomic twist families with a technique we call the ``correlation trick'',
which applies in a more general context where one is trying to show
simultaneous vanishing of multiple Selmer groups. We also apply this technique
to families of twists of Prym surfaces, which leads to bounds on the number of
rational points in sextic twist families of bielliptic genus 3 curves.Comment: 24 pages. Revised following referee comments. Section added with
application to Hilbert's 10th problem. To appear in Transactions of the AMS.
Comments welcome
Integers expressible as the sum of two rational cubes
No full textWe prove that a positive proportion of integers are expressible as the sum of two rational cubes, and a positive proportion are not so expressible, thus proving a conjecture of Davenport. More generally, we prove that a positive proportion (in fact, at least one sixth) of elliptic curves in any cubic twist family have rank 0, and a positive proportion (in fact, at least one sixth) of elliptic curves with good reduction at 2 in any cubic twist family have rank 1.
Our method involves proving that the average size of the 2-Selmer group of elliptic curves in any cubic twist family, having any given root number, is 3. We accomplish this by generalizing a parametrization, due to the second author and Ho, of elliptic curves with extra structure by pairs of binary cubic forms. We then use a novel combination of geometry-of-numbers methods and the circle method that builds on earlier work of Ruth and the first author. In particular, we make use of a new interpretation of the singular integral and series arising in the circle method in terms of real and -adic Haar measures on the relevant group. We prove a uniformity estimate for integral points on the relevant quadric, which along with a sieve allows us to prove that the average size of the 2-Selmer group over the cubic twist family is 3. By suitably partitioning the subset of curves in the family with given root number, we effect a further sieve to show that the root number is equidistributed and that the same average, now taken over only those curves of given root number, is again 3. Finally, we apply the -parity theorem of Dokchitser-Dokchitser and a -converse theorem of Burungale-Skinner to conclude.
We also prove the analogue of the above results for the sequence of square numbers: namely, we prove that a positive proportion of square integers are expressible as the sum of two rational cubes, and a positive proportion are not.54 pages; a number of details and additional results have been added in this versio
Vanishing criteria for Ceresa cycles
No full textLet be a smooth projective curve, and let be its Jacobian. We prove
vanishing criteria for the Ceresa cycle in the Chow group of 1-cycles on . Namely,
If , then
vanishes;
If and the Hodge
conjecture holds, then vanishes modulo algebraic equivalence.
We then study the first interesting case where holds but does
not, namely the case of Picard curves .
Using work of Schoen on the Hodge conjecture, we show that the Ceresa cycle of
a Picard curve is torsion in the Griffiths group. Moreover, we determine
exactly when it is torsion in the Chow group. As a byproduct, we show that
there are infinitely many plane quartic curves over with torsion
Ceresa cycle (in fact, there is a one parameter family of such curves).
Finally, we determine which automorphism group strata are contained in the
vanishing locus of the universal Ceresa cycle over .Comment: 21 pages, comments welcom
Arbitrarily large -torsion in Tate-Shafarevich groups
No full textWe show that, for any prime , there exist absolutely simple abelian varieties over with arbitrarily large -torsion in their Tate-Shafarevich group. To prove this, we construct explicit -covers of Jacobians of the form which violate the Hasse principle. In the appendix, Tom Fisher explains how to interpret our proof in terms of a Cassels-Tate pairing.Added an appendix by Tom Fisher and made a few minor changes. To appear in J. Inst. Math. Jussie
