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

Group law computations on Jacobians of hyperelliptic curves

We derive an explicit method of computing the composition step in Cantor’s algorithm for group operations on Jacobians of hyperelliptic curves. Our technique is inspired by the geometric description of the group law and applies to hyperelliptic curves of arbitrary genus. While Cantor’s general composition involves arithmetic in the polynomial ring F_q[x], the algorithm we propose solves a linear system over the base field which can be written down directly from the Mumford coordinates of the group elements. We apply this method to give more efficient formulas for group operations in both affine and projective coordinates for cryptographic systems based on Jacobians of genus 2 hyperelliptic curves in general form

Modular polynomials for genus 2

Modular polynomials are an important tool in many algorithms involving elliptic curves. In this article we investigate their generalization to the genus 2 case following pioneering work by Gaudry and Dupont. We prove various properties of these genus 2 modular polynomials and give an improved way to explicitly compute them