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This summer school will continue a tradition of summer schools in Alpbach (see here) in arithmetic and geometry (e.g. 2013, p-adic modular forms; 2014, periods and heights of CM abelian varieties). The topic builds on themes from the previous two summers. PhD students and postdocs (mainly from Zurich/Freiburg/Mainz) present the topics.
Shimura curves arise in geometric and arithmetic settings, generalizing modular curves. Special cycles on Shimura curves play an important role in the Kudla programme, relating height pairings and L-functions. The goal of the summer school is to understand, first, the consequence for modular curves (Gross-Zagier formula) and its wide reaching generalizations.
In more details, the aim is to understand Theorem 3 from [5] and Theorem A from [7]. First, we will treat the one-dimensional case of special cycles on the moduli of CM elliptic curves, see [3, 5]. This serves both as motivation and as an entry to the subject. The proof of Theorem 3 in [5] relies on the computation of lengths of special cycles on the moduli space of CM elliptic curves. These lengths are given by the formula of Gross (and Keating) which also plays an important role in other parts of the Kudla program. We will follow the original proof of Gross of this formula, as presented in [1]. Then we will turn to the case of (arithmetic models of) Shimura curves. First, we develop the formalism of arithmetic Chow groups on surfaces in order to formulate Theorem A. In particular, we will discuss the decomposition of the first arithmetic Chow group into four direct summands. Modularity of the generating series from Theorem A is proved for each component individually. We will focus on the vertical component and the Faltings height component, whereas the Borcherds component will only be surveyed and the spectral component will be neglected. For the vertical component, we will recall the p-adic uniformization of the fibers at places of bad reduction and work through parts of [3]. For the Faltings height component, we will start with the Chowla-Selberg formula giving the height of a product of a CM elliptic curve with itself (which was considered in the last summer school), and then determine the deviation from this height caused by an isogeny, as explained in [6]. Prerequisites: Knowledge of the following topics would be useful: Dieudonné theory of p-divisible groups over an algebraically closed field, Lubin-Tate formal groups e.g. from [1, Chapter 6]. Also, it wouldn't hurt to have looked at basic definitions concerning Deligne-Mumford stacks.
A pre-Alpbach minicourse (covering the prerequisites) will take place at FIM (further information can be found here):
Monday, 30.03.2015: 4-6pm,
Tuesday, 31.03.2015: 2-4pm and 5-7pm,
Wednesday, 01.04.2015: 1-3pm.
The notes of A. Mihatsch on Shimura curves can be found here.