Teichm�ller curves in moduli space, Eisenstein series and an application to triangular billiards

Moduli Interpretation of Eisenstein Series

Let l ≥ 3. Using the moduli interpretation, we define certain elliptic modular forms of level Γ(l), which make sense over any field k in which 6l 6= 0 and that contains the lth roots of unity. Over the complex numbers, these forms include all holomorphic Eisenstein series on Γ(l) in all weights, in a natural way. The graded ring Rl that is generated by our special modular forms turns out to be ...

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....

Billiards, quadrilaterals and moduli spaces

Eisenstein Series*

group GC defined over Q whose connected component G 0 Q has no rational character. It is also necessary to suppose that the centralizer of a maximal Q split torus of G0C meets every component of GC. The reduction theory of Borel applies, with trivial modifications, to G; it will be convenient to assume that Γ has a fundamental set with only one cusp. Fix a minimal parabolic subgroup P 0 C defin...

The Second Term of an Eisenstein Series

The leading term of Eisenstein series and L-functions are well-known to be extremely important and to have very important arithmetic implications. For example, one may consider the analytic class number formula, The Siegel-Weil formula, the Birch and Swinnerton-Dyer conjecture, Stark’s conjectures, and Beilinson’s and Bloch-Kato conjectures. These are well-documented and are central in current ...