Óscar Rivero defended his PhD thesis Arithmetic applications of the Euler systems of Beilinson-Flach elements and diagonal cycles, supervised by Víctor Rotger, on 12 February 2021 within the UPC doctoral program in Applied Mathematics. Currently, he is in the group of David Loeffler at the University of Warwick and holding a Newton International Fellowship grant of the Royal Society.
Thesis summary
This thesis studies some applications to arithmetic problems of the so-called Euler systems, which are Galois cohomology classes that vary in a compatible way over towers of fields.
Following a fairly general philosophy introduced by Perrin– Riou, the image of these Euler systems under certain regulators allows us to recover the p-adic L-function associated with a Galois representation. In this thesis we focus mainly on the systems of Beilinson-Flach and diagonal cycles, although we also study
others that share properties with the previous ones and that help us to better understand them. Let us mention that different works in the last decade have already succeeded, by means of these Euler systems, in proving new cases of the equivariant Birch and Swinnerton-Dyer conjecture, one of the great mathematical challenges of our times.
The arithmetic applications we discuss in this monograph are diverse: exceptional zeros, special value formulas, non-vanishing results, connections with Iwasawa theory … The first chapters of the thesis study a phenomenon of exceptional zeros. Recall that the p-adic L-functions interpolate, over a certain region, values of the complex L-function multiplied by appropriate Euler factors. The vanishing of these factors often leads to interesting arithmetic phenomena. This, far from being fortuitous, supports an algebraic interpretation in terms of Selmer groups. For example, the cancellation at s = 0 of the Kubota-Leopoldt function is related to the fact that there is an extra p-unit in the corresponding component of the group of units, and its logarithm is related to the derivative of the p-adic L-function. This is one of Gross’ best known conjectures.
Here we begin by studying the case of the adjoint representation of a weight 1 modular form. In this setting, we prove a conjecture of Darmon, Lauder, and Rotger that expresses the value of the derivative of the associated p-adic L-function in terms of a combination of logarithms of units and p-units in the field cut-out by the representation. The proof uses the theory of p-adic L-functions and improved p-adic L-functions, as well as Galois deformations.
In addition, we observe a phenomenon that complements this study. The p-adic L-functions that come out of it are the image of the Euler system of Beilinson-Flach elements by the PerrinRiou morphism. These exceptional zeros are also observed at the level of Euler systems, and one can introduce the concept of derived class, that allows us to recover the L-invariant that controls the arithmetic of the Galois representation. Not only that: with this notion of derivative, we can give an alternative proof of the previous result by exploiting the geometry of these systems.
This first part of the thesis is complemented by two chapters where we work on the phenomenon of exceptional zeros for both elliptic units and diagonal cycles.
The final chapters delve into the study of other issues around Euler systems, and begin with the development of an Artin formalism at the level of cohomology classes. The most basic case is to consider a cuspidal eigenform f of weight 2 which is congruent to an Eisenstein series. The cohomology class associated with f gives rise, under appropriate conditions, to two different components modulo p. We suggest congruences relating each of them to expressions involving circular units. This uses, on the one hand, factorizations of p-adic L-functions and reciprocity laws; on the other hand, we recover some results of Fukaya–Kato developed during the study of Sharifi’s conjectures.
Highlighted publication: O. Rivero, V. Rotger. Derived Beilinson–Flach elements and the arithmetic of the adjoint of a modular form. Journal of the European Mathematical Society
JEMS (published online 2021-03-15). arXiv preprint.