48 research outputs found
Transcendental Brauer elements via descent on elliptic surfaces
Transcendental Brauer elements are notoriously difficult to compute. Work of
Wittenberg, and later, Ieronymou, gives a method for computing 2-torsion
transcendental classes on surfaces that have a genus 1 fibration with rational
2-torsion in the Jacobian fibration. We use ideas from a descent paper of
Poonen and Schaefer to remove this assumption on the rational 2-torsion.Comment: 10 pages, small edits made to the introduction, references added in
the introductio
A family of varieties with exactly one pointless rational fiber
We construct a concrete example of a 1-parameter family of smooth projective
geometrically integral varieties over an open subscheme of P^1_Q such that
there is exactly one rational fiber with no rational points. This makes
explicit a construction of Poonen.Comment: 4 pages. Some stylistic changes, replaced an argument in Lemma 3.1
with a simpler argument as suggested by the referee. To appear in J. Th\'eor.
Nombres Bordeau
On singular moduli for arbitrary discriminants
Let d1 and d2 be discriminants of distinct quadratic imaginary orders O_d1
and O_d2 and let J(d1,d2) denote the product of differences of CM j-invariants
with discriminants d1 and d2. In 1985, Gross and Zagier gave an elegant formula
for the factorization of the integer J(d1,d2) in the case that d1 and d2 are
relatively prime and discriminants of maximal orders. To compute this formula,
they first reduce the problem to counting the number of simultaneous embeddings
of O_d1 and O_d2 into endomorphism rings of supersingular curves, and then
solve this counting problem.
Interestingly, this counting problem also appears when computing class
polynomials for invariants of genus 2 curves. However, in this application, one
must consider orders O_d1 and O_d2 that are non-maximal. Motivated by the
application to genus 2 curves, we generalize the methods of Gross and Zagier
and give a computable formula for v_p(J(d1,d2)) for any distinct pair of
discriminants d1,d2 and any prime p>2. In the case that d1 is squarefree and d2
is the discriminant of any quadratic imaginary order, our formula can be stated
in a simple closed form. We also give a conjectural closed formula when the
conductors of d1 and d2 are relatively prime.Comment: 33 pages. Changed the abstract and made small changes to the
introduction. Reorganized section 3.2, 4, and proof of Proposition 8.1. Some
remarks added to section
An arithmetic intersection formula for denominators of Igusa class polynomials
In this paper we prove an explicit formula for the arithmetic intersection
number (CM(K).G1)_{\ell} on the Siegel moduli space of abelian surfaces,
generalizing the work of Bruinier-Yang and Yang. These intersection numbers
allow one to compute the denominators of Igusa class polynomials, which has
important applications to the construction of genus 2 curves for use in
cryptography.
Bruinier and Yang conjectured a formula for intersection numbers on an
arithmetic Hilbert modular surface, and as a consequence obtained a conjectural
formula for the intersection number (CM(K).G1)_{\ell} under strong assumptions
on the ramification of the primitive quartic CM field K. Yang later proved this
conjecture assuming that O_K is freely generated by one element over the ring
of integers of the real quadratic subfield. In this paper, we prove a formula
for (CM(K).G1)_{\ell} for more general primitive quartic CM fields, and we use
a different method of proof than Yang. We prove a tight bound on this
intersection number which holds for all primitive quartic CM fields. As a
consequence, we obtain a formula for a multiple of the denominators of the
Igusa class polynomials for an arbitrary primitive quartic CM field. Our proof
entails studying the Embedding Problem posed by Goren and Lauter and counting
solutions using our previous article that generalized work of Gross-Zagier and
Dorman to arbitrary discriminants.Comment: 30 pages. Minor edit
Rational points on varieties and the Brauer-Manin obstruction
These lecture notes give an introduction to the Brauer-Manin obstruction to
the existence of rational points, focusing on the interplay between theory and
computation.Comment: 19 page
Failure of the Hasse principle for Chatelet surfaces in characteristic 2
Given any global field k of characteristic 2, we construct a Chatelet surface
over k which fails to satisfy the Hasse principle. This failure is due to a
Brauer-Manin obstruction. This construction extends a result of Poonen to
characteristic 2, thereby showing that the etale-Brauer obstruction is
insufficient to explain all failures of the Hasse principle over a global field
of any characteristic.Comment: 5 pages. Changed the title, added Lemma 3.2, made small changes to
the introductio