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, up-and-coming international figure working in the area of Arithmetic Algebraic Geometry.

This year’s speaker will be Sarah Zerbes with three lectures. The lectures will circle around the BSD conjecture and Euler systems.

Sarah Livia Zerbes is a German algebraic number theorist, working as a full professor at ETH Zürich. She obtained a bachelor’s degree with first class honors from the University of Cambridge, and she then wrote her PhD thesis on non-commutative Iwasawa theory under the supervision of John Coates. Focusing on Euler systems and special values of L-functions, her research is concerned with problems such as the Birch and Swinnerton-Dyer conjecture and the Bloch-Kato conjecture. More broadly, her interests include L-functions, automorphic forms, p-adic Hodge theory and Iwasawa theory. Together with her husband David Loeffler, she discovered a new approach to constructing Euler systems, which gave new insight towards the Birch and Swinnerton-Dyer conjecture. They jointly received the Philip Leverhulme Prize in 2014 and the Whitehead Prize, awarded by the London Mathematical Society, in 2015. They were invited speakers at the ICM 2022. From 2015 – 20, Sarah’s research was supported by the European Research Council. In her spare time, Sarah enjoys climbing and mountaineering and learning to speak Latin.

The 1st lecture, December 21, will be given as a colloquium talk at the usual time of the Darmstadt colloquium and should be accessible for a more general audience.

Title: *Euler systems and the Birch—Swinnerton-Dyer conjecture*Abstract: L-functions are one of the central objects of study in number theory. There are many beautiful theorems and many more open conjectures linking their values to arithmetic problems. The most famous example is the conjecture of Birch and Swinnerton-Dyer, which is one of the Clay Millenium Prize Problems. I will discuss this conjecture and some related open problems, and I will describe some recent progress on these conjectures, using tools called `Euler systems’.

You can follow the talk online on zoom following the link below.

Join Zoom Meeting

https://tu-darmstadt.zoom.us/j/62512397856?pwd=RE9Ea2VpZG5uWWJvd2tuT2hzbzZpUT09

Meeting ID: 625 1239 7856

Passcode: Moufang

Lectures 2 and 3, tba, are aimed at mathematicians with a background in number theory.

– – – The second two talks will probably take place in summer 2023. – – –

Title 1: *Euler systems: what they are and where to find them*Title 2:

*Euler systems and the BSD conjecture for abelian surfaces*

Abstract: Euler systems are one of the powerful tools for attacking the Bloch–Kato and Birch–Swinnerton-Dyer conjectures. In the first talk, I will sketch the construction of the Euler system attached to Rankin–Selberg convolutions of modular forms (joint work with Kings, Lei and Loeffler), which was motivated by the Rankin–Selberg integral formula.

In the second talk, I will recall a series of recent works (variously joint with Loeffler, Pilloni, Skinner) giving rise to an Euler system in the cohomology of Shimura varieties for GSp(4), and an explicit reciprocity law relating the Euler system to values of L-functions. I will then explain recent work with Loeffler, where we use this Euler system to prove new cases of the BSD conjecture for modular abelian surfaces over Q, and for modular elliptic curves over imaginary quadratic fields.

CEST | Room | Wednesday, December 21, 2022 | tba (The second two talks will probably take place in summer 2023.) | |

16:45 – 17:15 | S2|15 244 | tea | ||

17:15 – 18:15 | S2|08, 171 | Sarah Zerbes – Euler systems andthe Birch—Swinnerton-Dyer conjecture |
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tba | Sarah Zerbes – Euler systems: what they are and where to find them |
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tba | Sarah Zerbes –Euler systems and the BSD conjecture for abelian surfaces |