Lectures on moduli spaces of elliptic curves (Q2837135)

The article under review is an expanded version of lectures provided by the author at Zhejiang University (2008). It concerns the moduli spaces of genus one Riemann surfaces and its Deligne-Mumford's compactification.NEWLINENEWLINEThe author states that the notes are intended for students, but I find them quite interesting inclusive for anyone trying to understand the concepts of moduli spaces of Riemann surfaces and algebraic curves.NEWLINENEWLINEThe reader should take care on some misprints on these notes (for instance, in page 97, when defining the periods of a holomorphic \(1\)-form, it does not says that the form should be different from zero, and in the proof of Lemma 1.3., it should say \(\lambda_{2}=a \lambda_{1}\). In page 115, in line 5 it says ``unramified'', but it seems that it should say ``ramified''). Each section consists of a set of results and exercises directed to obtain the main facts.NEWLINENEWLINESection 1 is dedicated to show that each compact Riemann surface of genus \(1\) (with a distinguished point) is equivalent to some quotient \({\mathbb C}/\Lambda\), where \(\Lambda \subset {\mathbb C}\) is a suitable lattice. The lattice is constructed by choosing a non-zero holomorphic \(1\)-form and integrate it along a basis of the homology. The author then introduces the moduli space of elliptic curves. For it, it is introduced the moduli space of elliptic curves together a frame (a basis \(\{a,b\}\) of the homology with \(a \cdot b=1\)) and shown it to be isomorphic to the hyperbolic plane \({\mathfrak h}\). The moduli space of elliptic curves is the quotient orbifold \(\text{SL}_{2}({\mathbb Z}) \setminus {\mathfrak h}\); which carries a Riemann surface structure (the complex plane) and also a structure of an orbifold.NEWLINENEWLINESection 2 introduces families of elliptic curves and the universal elliptic curve.NEWLINENEWLINEIn Section 3 it is introduced the concept of basic orbifolds, which are triples \((X,\Gamma,\rho)\), where \(X\) is a connected and simply-connected space, \(\Gamma\) is a group and \(\rho:\Gamma \to \text{Aut}(X)\) is a homomorphism with \(\rho(\Gamma)\) acting discontinuously on \(X\). Important is to note that \(\rho\) may not be injective (the situation under study \(X={\mathfrak h}\), \(\Gamma=\text{SL}_{2}({\mathbb Z})\) where the element \(-I\) acts as the identity). The fundamental groups, homology groups, cohomology groups and the Euler characteristic of orbifolds are defined. In this section the moduli space \({\mathcal M}_{1,1}=\text{SL}_{2}({\mathbb Z}) \setminus {\mathfrak h}\) is introduced as an orbifold and it is computed its first fundamental group and its homology and cohomology groups. Then, the universal elliptic curve \({\mathcal E}=(\text{SL}_{2}({\mathbb Z}) \ltimes {\mathbb Z}^{2}) \setminus ({\mathbb C} \times {\mathfrak h})\) is introduced (again as an orbifold).NEWLINENEWLINESection 4 introduces the Deligne-Mumford compactification \(\overline{\mathcal M}_{1,1} \) and modular forms. In page 121 it is sated that the Euler characteristic of \(\overline{\mathcal M}_{1,1}\) is \(5/12\), but it appears to be \(1/6\).NEWLINENEWLINESection 5 relates elliptic curves to plane cubics by using the Weierstrass \(\wp\)-function. With this equivalent description, it is shown how to extend in an explicit manner the universal elliptic curve over \({\mathcal M}_{1,1}\) to \(\overline{\mathcal M}_{1,1}\).NEWLINENEWLINESection 6 computes the Picard groups of \({\mathcal M}_{1,1}\) and \(\overline{\mathcal M}_{1,1}\).NEWLINENEWLINESection 7 describes the topology of \(\overline{\mathcal M}_{1,1}\) (this an orbifold whose underlying Riemann surface structure is the Riemann sphere).NEWLINENEWLINEFor the entire collection see [Zbl 1245.00047].