Let $d_1$ and $d_2$ be discriminants of distinct quadratic imaginary orders ${{\cal O}}_{d_1}$ and ${{\cal O}}_{d_2}$ and let $J(d_1,d_2)$ denote the product of differences of CM $j$ 
-invariants with discriminants $d_1$ and $d_2$ 
. In 1985, Gross and Zagier gave an elegant formula for the factorization of the integer $J(d_1,d_2)$ in the case that $d_1$ and $d_2$ 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 ${{\cal O}}_{d_1}$ and ${{\cal O}}_{d_2}$ 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 ${{\cal O}}_{d_1}$ and ${{\cal O}}_{d_2}$ 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 $v_{\ell }(J(d_1,d_2))$ for any pair of discriminants $d_1\neq d_2$ and any prime $\ell \gt 2$ 
. In the case that $d_1$ is squarefree and $d_2$ 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 $d_1$ and $d_2$ are relatively prime.

On Singular Moduli for Arbitrary Discriminants

In this thesis I investigate certain computational aspects of Modular Forms and Elliptic Curves. The main computational problem in the theory of Modular Forms is the computation of their Fourier coefficients. The first part of the thesis is concerned with the computational complexity of this problem. I show that if we do not fix the modular form, then the problem of computing Fourier coefficients is s P -hard. Next, in joint work with Eric Bach, we show that if we fix a modular form that is a Hecke eigenform, then computing the Fourier coefficients is at least as hard as factoring RSA moduli. I also provide a new algorithm to compute the Fourier coefficients of any fixed space of cusp forms. This algorithm has a faster asymptotic running time than any previous algorithm for computing Fourier coefficients of these forms. 
The arithmetic theory of Elliptic Curves has many interesting computational problems. I study two such problems in this thesis. One is the problem of distinguishing elliptic curves with Complex Multiplication from ordinary elliptic curves. I give two algorithms for this problem: one is a randomized polynomial time algorithm and the other is a deterministic polynomial time algorithm. I also show how the randomized polynomial time algorithm can be made to have one-sided error. The other problem that I study is the problem of computing the modular polynomial. The modular polynomial is an object that governs isogenies between elliptic curves. In joint work with Kristin Lauter, we provide a new algorithm to compute the modular polynomial over finite fields. This algorithm also yields an efficient method of generating random isogenies from an elliptic curve over a finite field.

Computational aspects of modular forms and elliptic curves

DENIS XAVIER CHARLESAbstract We consider the problem of checking whether an elliptic curve deﬁned over a givennumber ﬁeld has complex multiplication We study two polynomial time algorithms for this prob-lem, one randomized and the other deterministic The randomized algorithm can be adapted toyield the discriminant of the endomorphism ring of the curveKeywords Algorithms, Elliptic Curves, Complex Multiplication, Endomorphism Ring, l-adicrepresentations, Chebotarev Density Theorem

Complex multiplication tests for elliptic curves

For an odd prime p, define $H_{p}(z)=\sum_{u,v\ (\mathop{\mathrm{mod}}p)}(\frac{uv(1-u)(1-v)(1-uvz)}{p})$ , where z is an integer (mod p) and the summands are Legendre symbols. The function Hp(z) was explicitly evaluated for z=1 by Evans (1981) and for z=−1 by Greene and Stanton (1986). Koike (1992) determined Hp(1/4) (mod p), and Ono (1998) evaluated Hp(z) for z=1/4,−1/8, and 1/64. This paper evaluates Hp(z) for infinitely many new classes of arguments z.

Tim Hutchinson

Papers

A Conjectural Extension of the Gross-Zagier Formula on Singular Moduli