# Program

The fundamental conjecture of Birch and Swinnerton-Dyer relating the Mordell–Weil ranks of elliptic curves to their L-functions is one of the most important and motivating problems in number theory. It resides at the heart of a collection of important conjectures (due especially to Deligne, Beilinson, Bloch and Kato) that connect values of L-functions and their leading terms to cycles and Galois cohomology groups.
The study of special algebraic cycles on Shimura varieties has led to progress in our understanding of these conjectures. The arithmetic intersection numbers and the p-adic regulators of special cycles are directly related to the values and derivatives of L-functions, as shown in the pioneering theorem of Gross-Zagier and its p-adic avatars for Heegner points on modular curves. The cohomology classes of special cycles (and related constructions such as Eisenstein classes) form the foundation of the theory of Euler systems, providing one of the most powerful methods known to prove vanishing or finiteness results for Selmer groups of Galois representations.
The goal of this semester is to bring together researchers working on different aspects of this young but fast-developing subject, and to make progress on understanding the mysterious relations between L-functions, Euler systems, and algebraic cycles.
For details on the image, see http://www.elleneischen.com/gaussianperiods.html.
Bibliography

**Keywords and Mathematics Subject Classification (MSC)**

**Tags/Keywords**

algebraic cycles

arithmetic ge- ometry

automorphic forms and L-functions

Euler systems

Galois repre- sentations

and Shimura varieties.

**Primary Mathematics Subject Classification**

11G18 - Arithmetic aspects of modular and Shimura varieties [See also 14G35]

14G40 - Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30]

**Secondary Mathematics Subject Classification**No Secondary AMS MSC

January 19, 2023 - January 20, 2023 | Connections Workshop: Algebraic Cycles, L-Values, and Euler Systems |

January 23, 2023 - January 27, 2023 | Introductory Workshop: Algebraic Cycles, L-Values, and Euler Systems |

March 13, 2023 - March 17, 2023 | Shimura Varieties and L-Functions |