Euler systems and arithmetic applications Euler systems can be thought of as a collection of points or cohomology classes attached to an algebraic object, which satisfy some very precise compatibility relations. These systems are very hard to construct, but they have very powerful arithmetic applications, such as proving new cases of the Bloch-Kato conjecture as well as the Birch and Swinnerton-Dyer conjecture, as Professor Sarah Zerbes explains. A collection of cohomology classes for a given algebraic object, the first example of an Euler system was constructed by the Russian mathematician Victor Kolyvagin. While these systems are named after the highly influential 18th century Swiss mathematician Leonhard Euler, they were only introduced relatively recently. “In the case of Kolyvagin’s work, an Euler system is a collection of points on modular elliptic curves,” explains Sarah Zerbes, Professor of Maths at University College London. Euler systems, once constructed, have very powerful applications to some of the central open conjectures in number theory. “Kolyvagin recognised that his Euler system could be used to prove new cases of the conjecture of Birch and Swinnerton-Dyer.” says Professor Zerbes. These Euler systems are extremely difficult to construct however, and until relatively recently only four were known to exist. Professor Zerbes and her collaborators have since developed a new approach to constructing them. “It started with the Euler system of Beilinson-Flach elements. We found that the method with which we had constructed this system could be applied in great generality, so now we have greatly increased the number of known Euler systems,” she says. As the Principal Investigator of an ERC-funded research project based at UCL, Professor Zerbes is now investigating the arithmetic applications of these systems. “Euler systems are fundamentally geometric constructions. The underlying objects are socalled Shimura varieties, which are algebraic varieties arising in the representation theory of matrix groups,” she outlines.
Euler systems The construction of these systems is based on the use of geometric methods, as well as methods from representation theory and number theory. While researchers were able to construct new examples of Euler systems, it was not initially possible to use them for arithmetic applications. “We constructed these geometric objects, but it was difficult to establish that they were non-zero. We were in the frustrating situation where we
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Graphs of the L-functions attached to elliptic curves of ranks 0,1 and 2, respectively.
had many new Euler systems, but we didn’t know that they were non-zero,” explains Professor Zerbes. New results have since emerged which formed the missing piece of the puzzle, enabling Professor Zerbes and her colleagues to give a criterion for the nonvanishing of the Euler systems, and to study the various arithmetical applications. “The way to prove the non-vanishing of an Euler system is by relating it to the value of an L-function. Such a relation is called an explicit reciprocity law.” This opens up many new exciting avenues of research, with Professor Zerbes particularly interested in proving new cases of the BlochKato conjecture, which predicts that certain values of the L-function govern the size of a Selmer group, attached to the underlying
Shimura variety. An Euler system is extremely useful in terms of proving new cases of the conjecture. “Two things can be attached to an Euler system. Firstly, we can attach to it a Selmer group, which is a certain cohomology group. We can also attach to it an L-function, which is a complex analytic function,” outlines Professor Zerbes. This provides the basis for researchers to investigate possible new cases of the Bloch-Kato conjecture. “If the L-function at a certain point does not vanish, then this Selmer group should be zero. This is the case that an Euler system will prove,” continues Professor Zerbes. “We are also interested in the Birch and SwinnertonDyer conjecture. One can think of the BlochKato conjecture as a vast generalisation of the Birch and Swinnerton-Dyer conjecture.”
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