Computation of Hilbert class polynomials and modular polynomials from supersingular elliptic curves

We present several new heuristic algorithms to compute class polynomials and modular polynomials modulo a prime $P$. For that, we revisit the idea of working with supersingular elliptic curves. The best known algorithms to this date are based on ordinary curves, due to the supposed inefficiency of the supersingular case. While this was true a decade ago, it is not anymore due to the recent advances in the study of supersingular curves. Our main ingredients are two new heuristic algorithms to compute the $j$-invariants of supersingular curves having an endomorphism ring contained in some set of isomorphism class of maximal orders

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

Archimedean local height differences on elliptic curves

To compute generators for the Mordell-Weil group of an elliptic curve over a number field, one needs to bound the difference between the naive and the canonical height from above. We give an elementary and fast method to compute an upper bound for the local contribution to this difference at an archimedean place, which sometimes gives better results than previous algorithms.Comment: 10 pages, comments welcom

Slopes of overconvergent Hilbert modular forms

We give an explicit description of the matrix associated to the $U_p$ operator acting on spaces of overconvergent Hilbert modular forms over totally real fields. Using this, we compute slopes for weights in the centre and near the boundary of weight space for certain real quadratic fields. \added[id=h]{Near the boundary of weight space we see that the slopes do not appear to be given by finite unions of arithmetic progressions but instead can be produced by a simple recipe from which we make a conjecture on the structure of slopes. We also prove a lower bound on the Newton polygon of the $U_p$.Comment: 22 pages, 7 figure, 7 tables. Final version, to appear in Experiment. Math. arXiv admin note: text overlap with arXiv:1610.0971