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We give bounds on the primes of geometric bad reduction for curves of genus 3 of primitive complex multiplication (CM) type in terms of the CM orders. In the case of elliptic curves, there are no primes of geometric bad reduction because CM elliptic curves are CM abelian varieties, which have potential good reduction everywhere. However, for genus at least 2, the curve can have bad reduction at a prime although the Jacobian has good reduction. Goren and Lauter gave the first bound in the case of genus 2. In the cases of hyperelliptic and Picard curves, our results imply bounds on primes appearing in the denominators of invariants and class polynomials, which are important for algorithmic construction of curves with given characteristic polynomials over finite fields. [ABSTRACT FROM AUTHOR]

The moduli space of principally polarized abelian surfaces is parametrized by three Igusa functions. In this article we investigate a new way to evaluate these functions by using the Siegel Eisenstein series. We explain how to compute the Fourier coefficients of certain Siegel modular forms using classical modular forms of half-integral weight. One of the results in this paper is an explicit algorithm to evaluate the Igusa functions to a prescribed precision.The moduli space of principally polarized abelian surfaces is parametrized by three Igusa functions. In this article we investigate a new way to evaluate these functions by using the Siegel Eisenstein series. We explain how to compute the Fourier coefficients of certain Siegel modular forms using classical modular forms of half-integral weight. One of the results in this paper is an explicit algorithm to evaluate the Igusa functions to a prescribed precision. [ABSTRACT FROM AUTHOR]

We present a new algorithm to compute the classical modular polynomial Φl in the rings Z[X, Y ] and (Z/mZ)[X, Y ], for a prime l and any positive integer m. Our approach uses the graph of l-isogenies to efficiently compute Φl mod p for many primes p of a suitable form, and then applies the Chinese Remainder Theorem (CRT). Under the Generalized Riemann Hypothesis (GRH), we achieve an expected running time of O(l3(log l)3 log log l), and compute Φl mod m using O(l2(log l)2+l2 logm) space. We have used the new algorithm to compute Φl with l over 5000, and Φl mod m with l over 20000. We also consider several modular functions g for which Φg l is smaller than Φl, allowing us to handle l over 60000. [ABSTRACT FROM AUTHOR]

We present a new algorithm to compute the classical modular polynomial ϕl in the rings Z[X, Y ] and (Z/mZ)[X, Y ], for a prime l and any positive integer m. Our approach uses the graph of l-isogenies to efficiently compute ϕl mod p for many primes p of a suitable form, and then applies the Chinese Remainder Theorem (CRT). Under the Generalized Riemann Hypothesis (GRH), we achieve an expected running time of O(I³(log l)³ log log l), and compute ϕl mod m using O(l²(log l)²+l² logm) space. We have used the new algorithm to compute ϕl with l over 5000, and ϕl mod m with l over 20000. We also consider several modular functions g for which ϕ9 l is smaller than ϕl, allowing us to handle l over 60000. [ABSTRACT FROM AUTHOR]