The Ruth Moufang Lectures is a yearly distinguished lecture series addressing a broad mathematical audience in honour of Ruth Moufang, one of the major figures at Goethe University Frankfurt and a pioneer among women in mathematics. Each year, we will invite a major, established or up-and-coming international figure working in the area of Arithmetic Algebraic Geometry.

This year’s speaker of 2024 will be Ana Caraiani.

Ana Caraiani is a Professor at Imperial College London, as well as a Royal Society University Research Fellow. She has previously held positions in Bonn, Princeton and Chicago after completing her PhD at Harvard University.

Her research lies in algebraic number theory and arithmetic geometry. She has made important contributions to the study of the geometry and cohomology of Shimura varieties, as well as to the Langlands program, in particular to local global compatibility, torsion vanishing and modularity.

Ana Caraiani is a Fellow of the AMS, and has been awarded with the Whitehead Prize (2018), EMS Prize (2020), Philip Leverhulme Prize (2020) and a New Horizons in Mathematics Prize (2023) among other recognitions.

**Ana Caraiani – Colloquium ****“Elliptic curves and modularity”**

*Abstract: The goal of this talk is to give you a glimpse of the Langlands program, a central topic at the intersection of algebraic number theory, algebraic geometry and representation theory. I will focus on a celebrated instance of the Langlands correspondence, namely the modularity of elliptic curves. In the first part of the talk, I will give an explicit example, discuss the different meanings of modularity for rational elliptic curves, and mention applications. In the second part of the talk, I will discuss what is known about the modularity of elliptic curves over more general number fields.*

**Ana Caraiani – Seminar talks **

**“On the cohomology of Shimura varieties with torsion coefficients”**

*Abstract: I will survey results concerning the cohomology of Shimura varieties with torsion coefficients from the past few years. I will discuss the geometry of the Hodge-Tate period morphism, including a recent generalisation of Igusa varieties to Igusa stacks due to Mingjia Zhang. Then I will contrast the original approach of computing cohomology with torsion coefficients due to myself and Peter Scholze, which relies on the trace formula, with more recent approaches due to Teruhisa Koshikawa, Linus Hamann and Si Ying Lee, who rely on deep local results. Finally, I will explain how, by combining the two approaches, one can obtain a new instance of local-global compatibility.*

**“On the modularity of elliptic curves over imaginary quadratic fields”**

*Abstract: I will survey how to prove the modularity of elliptic curves defined over the rational numbers, as pioneered by Wiles and Taylor-Wiles and completed by Breuil, Conrad, Diamond and Taylor. I will also mention the case of elliptic curves defined over real quadratic fields, more recently completed by Freitas, Le Hung and Siksek. I will then explain why the case of imaginary quadratic fields is qualitatively different from the previous ones. Finally, I will discuss joint work with James Newton, where we prove a local-global compatibility result in the crystalline case for Galois representations attached to torsion classes occurring in the cohomology of locally symmetric spaces. This has an application to the modularity of elliptic curves over imaginary quadratic fields, which also builds on recent work of Allen, Khare and Thorne.*

CEST | Room | Thursday, March 21, 2024 | Friday, March 22, 2024 | |

10.30 – 11:30 | Rob.Mayer-Str. 6-8, Hilbertraum |
Ana Caraiani – Seminar talks On the cohomology of Shimura varieties with torsion coefficients |
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12.00 – 13:00 | Rob.Mayer-Str. 6-8, Hilbertraum |
Ana Caraiani – Seminar talks On the modularity of elliptic curves over imaginary quadratic fields |
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17.00 |
Rob.Mayer-Str. 10, |
Ana Caraiani – ColloquiumElliptic curves and modularity |