ETHZ-UZh-Sbg Arithmetic and Geometry Research Group

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**Workshop "Special cycles on Shimura curves"**

This workshop is organized by J. Ayoub, A. Kresch, G. Wüstholz, C. Fuchs together with M. Rapoport and A. Mihatsch. The latter are responsible for the scientific program. The workshop takes place from July 5-10, 2015 at the Böglerhof in Alpbach/Tyrol, Austria.

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.

**Participants:**

Joseph Ayoub

Réda Boumasmoud (Lausanne)

Ana Brecan (Mainz)

Ernest Brooks (Lausanne)

Peter Bruin (Leiden)

Andrea Ferraguti

Riccardo Ferrario

Clemens Fuchs (Salzburg)

Javier Fresan

Ziyang Gao (Leiden)

Wushi Goldring

Fritz Hörmann (Freiburg)

Ariyan Javanpeykar (Mainz)

Dimitar Jetchev (Lausanne)

Peter Jossen

Rafael von Känel (Princton)

Lars Kühne (MPIM Bonn)

Andrew Kresch

Andreas Mihatsch (Bonn)

Roland Paulin (Salzburg)

Simon Pepin Lehalleur

Maximilian Preisinger (Mainz)

Sonia Samol (Mainz)

Siddarth Sankaran (Toronto)

Michael Rapoport (Bonn)

Alberto Vezzani

Gisbert Wüstholz

The participants consist of PhD students and postdocs in arithmetic and geometry from Zurich as well as colleagues from Bonn, Freiburg, Lausanne, Leiden, Mainz, Salzburg, and Toronto.

**Program:**

The program will start on Sunday evening, the first talk is scheduled for 17:15-18:45. The first draft of the detailed program worked out by A. Mihatsch and M. Rapoport can be found here: Program-2015

(Tentative) Schedule:

09:00 - 10:30: First talk

10:30 - 11:00: Coffee break

11:00 - 12:30: Second talk

12:45: Lunch

13:45 - 15:15: Third talk

15:30 - 18:00: Time for informal discussion

19:00: Dinner

**Lecture Notes and Literature:**

Program written by A. Mihatsch and M. Rapoport (version from January 21, 2015): Program-2015 Suitable references can be found therein.

Notes of A. Mihatsch on Shimura curves from the pre-Alpbach minicourse: URL

Notes of F. Hörmann: PDF

[1] ARGOS seminar in Bonn, Intersections of modular correspondences, Astérisque, Vol. 312, 2007.

[2] R. Borcherds, The Gross-Kohnen-Zagier Theorem in higher dimensions, Duke Math. J., 97 (1999), 219-233.

[3] B. Howard, Moduli spaces of CM elliptic curves and derivatives of Eisenstein series, Lecture notes on his homepage.

[4] S. Kudla, M. Rapoport, Height pairings on Shimura curves and p-adic uniformization, Invent. math., 142 (2000), 153-223.

[5] S. Kudla, M. Rapoport, T. Yang, On the derivative of an Eisenstein series of weight one, Int. Math. Res. Notices (1999), 347-385.

[6] S. Kudla, M. Rapoport, T. Yang, Derivatives of Eisenstein series and Faltings heights, Compositio Math., 140 (2004), 887-951.

[7] S. Kudla, M. Rapoport, T. Yang, Modular forms and special cycles on Shimura curves, Annals of Math. Studies, 161, Princeton Univ. Press, 2006

Back to the main page.

This workshop is organized by J. Ayoub, A. Kresch, G. Wüstholz, C. Fuchs together with M. Rapoport and A. Mihatsch. The latter are responsible for the scientific program. The workshop takes place from July 5-10, 2015 at the Böglerhof in Alpbach/Tyrol, Austria.

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.

Joseph Ayoub

Réda Boumasmoud (Lausanne)

Ana Brecan (Mainz)

Ernest Brooks (Lausanne)

Peter Bruin (Leiden)

Andrea Ferraguti

Riccardo Ferrario

Clemens Fuchs (Salzburg)

Javier Fresan

Ziyang Gao (Leiden)

Wushi Goldring

Fritz Hörmann (Freiburg)

Ariyan Javanpeykar (Mainz)

Dimitar Jetchev (Lausanne)

Peter Jossen

Rafael von Känel (Princton)

Lars Kühne (MPIM Bonn)

Andrew Kresch

Andreas Mihatsch (Bonn)

Roland Paulin (Salzburg)

Simon Pepin Lehalleur

Maximilian Preisinger (Mainz)

Sonia Samol (Mainz)

Siddarth Sankaran (Toronto)

Michael Rapoport (Bonn)

Alberto Vezzani

Gisbert Wüstholz

The participants consist of PhD students and postdocs in arithmetic and geometry from Zurich as well as colleagues from Bonn, Freiburg, Lausanne, Leiden, Mainz, Salzburg, and Toronto.

The program will start on Sunday evening, the first talk is scheduled for 17:15-18:45. The first draft of the detailed program worked out by A. Mihatsch and M. Rapoport can be found here: Program-2015

Sunday, 05.07.2015 (Arrival day): Moduli of CM elliptic curves and special cycles (welcome talk given by M. Rapoport) |

Monday: Geometric points and orbital integrals (Ernest Brooks); formula of Gross (Roland Paulin); relation to Eisenstein series (Fritz Hörmann) |

Thuesday: Arithmetic models of Shimura curves (Ariyan Javanpeykar); Omega-hat and p-adic uniformization (Alberto Vezzani); special cycles on Omega-hat (Ziyang Gao) |

Wednesday: Intersection theory on regular surfaces (Peter Bruin); (Algebraic) special cycles on M and their components (Andreas Mihatsch); special cycles in arithmetic Chow Groups (Simon Pepin Lehalleur) |

Thursday: Faltings heights of CM elliptic curves (Javier Fresan) |

Friday, 10.07.2015: The Change under isogeny (Lars Kühne); modularity of the Mordell-Weil component and modularity of the analytic component (Siddarth Sankaran) |

(Tentative) Schedule:

09:00 - 10:30: First talk

10:30 - 11:00: Coffee break

11:00 - 12:30: Second talk

12:45: Lunch

13:45 - 15:15: Third talk

15:30 - 18:00: Time for informal discussion

19:00: Dinner

Program written by A. Mihatsch and M. Rapoport (version from January 21, 2015): Program-2015 Suitable references can be found therein.

Notes of A. Mihatsch on Shimura curves from the pre-Alpbach minicourse: URL

Notes of F. Hörmann: PDF

[1] ARGOS seminar in Bonn, Intersections of modular correspondences, Astérisque, Vol. 312, 2007.

[2] R. Borcherds, The Gross-Kohnen-Zagier Theorem in higher dimensions, Duke Math. J., 97 (1999), 219-233.

[3] B. Howard, Moduli spaces of CM elliptic curves and derivatives of Eisenstein series, Lecture notes on his homepage.

[4] S. Kudla, M. Rapoport, Height pairings on Shimura curves and p-adic uniformization, Invent. math., 142 (2000), 153-223.

[5] S. Kudla, M. Rapoport, T. Yang, On the derivative of an Eisenstein series of weight one, Int. Math. Res. Notices (1999), 347-385.

[6] S. Kudla, M. Rapoport, T. Yang, Derivatives of Eisenstein series and Faltings heights, Compositio Math., 140 (2004), 887-951.

[7] S. Kudla, M. Rapoport, T. Yang, Modular forms and special cycles on Shimura curves, Annals of Math. Studies, 161, Princeton Univ. Press, 2006

Impressum 21.03.2016