5 research outputs found
A Gross-Zagier formula for quaternion algebras over totally real fields
We prove a higher dimensional generalization of Gross and Zagier's theorem on
the factorization of differences of singular moduli. Their result is proved by
giving a counting formula for the number of isomorphisms between elliptic
curves with complex multiplication by two different imaginary quadratic fields
and , when the curves are reduced modulo a supersingular prime
and its powers. Equivalently, the Gross-Zagier formula counts optimal
embeddings of the ring of integers of an imaginary quadratic field into
particular maximal orders in , the definite quaternion algebra
over \QQ ramified only at and infinity. Our work gives an analogous
counting formula for the number of simultaneous embeddings of the rings of
integers of primitive CM fields into superspecial orders in definite quaternion
algebras over totally real fields of strict class number 1. Our results can
also be viewed as a counting formula for the number of isomorphisms modulo
between abelian varieties with CM by different fields. Our
counting formula can also be used to determine which superspecial primes appear
in the factorizations of differences of values of Siegel modular functions at
CM points associated to two different CM fields, and to give a bound on those
supersingular primes which can appear. In the special case of Jacobians of
genus 2 curves, this provides information about the factorizations of
numerators of Igusa invariants, and so is also relevant to the problem of
constructing genus 2 curves for use in cryptography.Comment: 32 page
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
Moduli of CM False Elliptic Curves
Thesis advisor: Benjamin HowardWe study two moduli problems involving false elliptic curves with complex multiplication (CM), generalizing theorems about the arithmetic degree of certain moduli spaces of CM elliptic curves. The first moduli problem generalizes a space considered by Howard and Yang, and the formula for its arithmetic degree can be seen as a calculation of the intersection multiplicity of two CM divisors on a Shimura curve. This formula is an extension of the Gross-Zagier theorem on singular moduli to certain Shimura curves. The second moduli problem we consider deals with special endomorphisms of false elliptic curves. The formula for its arithmetic degree generalizes a theorem of Kudla, Rapoport, and Yang.Thesis (PhD) — Boston College, 2015.Submitted to: Boston College. Graduate School of Arts and Sciences.Discipline: Mathematics