Constructing Class invariants

Shimura reciprocity law allows us to verify that a modular function is a class invariant. Here we present a new method based on Shimura reciprocity that allows us not only to verify but to find new class invariants from a modular function of level $N$.Comment: 12 page

Implementing the asymptotically fast version of the elliptic curve primality proving algorithm

The elliptic curve primality proving (ECPP) algorithm is one of the current fastest practical algorithms for proving the primality of large numbers. Its running time cannot be proven rigorously, but heuristic arguments show that it should run in time O ((log N)^5) to prove the primality of N. An asymptotically fast version of it, attributed to J. O. Shallit, runs in time O ((log N)^4). The aim of this article is to describe this version in more details, leading to actual implementations able to handle numbers with several thousands of decimal digits

Constructing elliptic curves of prime order

We present a very efficient algorithm to construct an elliptic curve E and a finite field F such that the order of the point group E(F) is a given prime number N. Heuristically, this algorithm only takes polynomial time Otilde((\log N)^3), and it is so fast that it may profitably be used to tackle the related problem of finding elliptic curves with point groups of prime order of prescribed size. We also discuss the impact of the use of high level modular functions to reduce the run time by large constant factors and show that recent gonality bounds for modular curves imply limits on the time reduction that can be obtained.Comment: 13 page

Galois action on special theta values

For a primitive Dirichlet character χ of conductor N set θχ(τ) = ∑n ∈ℤ n∈ χ(n) eπin2τ/N (where ∈ = 0 for even χ, ∈ = 1 for odd χ) the associated theta series. Its value at its point of symmetry under the modular transformation τ(image found)−1/τ is related by θχ(i) = W(χ)θ(image found) (i) to the root number of the L-series of χ and hence can be used to calculate the latter quickly if it does not vanish. Using Shimura’s reciprocity law, we calculate the Galois action on these special values of theta functions with odd N normalised by the Dedekind eta function. As a consequence, we prove some experimental results of Cohen and Zagier and we deduce a partial result on the non-vanishing of these special theta values with prime N

The complexity of class polynomial computation via floating point approximations

We analyse the complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots. The heart of the algorithm is the evaluation of modular functions in several arguments. The fastest one of the presented approaches uses a technique devised by Dupont to evaluate modular functions by Newton iterations on an expression involving the arithmetic-geometric mean. It runs in time $O (|D| \log^5 |D| \log \log |D|) = O (|D|^{1 + \epsilon}) = O (h^{2 + \epsilon})$ for any $\epsilon > 0$, where $D$ is the CM discriminant and $h$ is the degree of the class polynomial. Another fast algorithm uses multipoint evaluation techniques known from symbolic computation; its asymptotic complexity is worse by a factor of $\log |D|$. Up to logarithmic factors, this running time matches the size of the constructed polynomials. The estimate also relies on a new result concerning the complexity of enumerating the class group of an imaginary-quadratic order and on a rigorously proven upper bound for the height of class polynomials