Integrality properties in the moduli space of elliptic curves: CM case

Moduli of Elliptic Curves

The purpose of these notes is to provide a quick introduction to the moduli of elliptic curves. There are many excellent and thorough references on the subject, ranging from the slightly archaic [Igu59] and [Shi94] to the more difficult [KM85] and [DR73]. Brian’s forthcoming book on the Ramanujan conjecture also covers some of this material and includes a careful comparison of the transcendenta...

Products of CM elliptic curves

Covers of Elliptic Curves and the Moduli Space of Stable Curves

Consider genus g curves that admit degree d covers of an elliptic curve. Varying a branch point, we get a one-parameter family W of simply branched covers. Varying the target elliptic curve, we get another one-parameter family Y of covers that have a unique branch point. We investigate the geometry of W and Y by using admissible covers to study their slopes, genera and components. The results c...

The Moduli Space of Curves

In this section, we will give a sketch of the construction of the moduli space Mg of curves of genus g and the closely related moduli space Mg,n of n-pointed curves of genus g using two different approaches. Throughout this section we always assume that 2g − 2 + n > 0. In the first approach, we will embed curves in a projective space P using a sufficiently high power n of their dualizing sheaf....

Realizations of the Elliptic Polylogarithm for CM elliptic curves

In these notes, we give an overview of our paper [BKT] which gives an explicit description of the de Rham and p-adic realizations of the elliptic polylogarithm, for a general elliptic curve defined over a subfield of C in the de Rham case and for a CM elliptic curve defined over its field of complex multiplication and with good reduction at the primes above p ≥ 5 in the p-adic case. As explaine...