P-adic L-functions and explicit reciprocity laws

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An important strand of research in number theory concerns the relation between algebraic properties of arithmetical objects, such as elliptic curves, and the values of analytic functions associated to them (L-functions). The best-known example of this is the Birch and Swinnerton-Dyer conjecture (one of the Clay Millennium Prize problems), which predicts that the size of the set of rational points on an elliptic curve is determined by the behaviour of its L-function at a specific point -- in particular, rational points exist if and only if the value of the L-function at this point is zero. However, this is only the first instance of a much more general theme.In recent years there has been some very exciting progress in understanding the links between L-functions and arithmetic, using an algebraic tool called an "Euler system". These are powerful tools, but difficult to construct. In my previous work with Lei and Zerbes in 2014, I discovered a new Euler system arising from products of modular forms, and this new construction has played a central role in many recent works on the BSD conjecture and related problems. The focus of my research program at present is to try to find a systematic approach to constructing new Euler systems, using methods from a variety of mathematical fields including representation theory and algebraic geometry; this research is funded by a grant from the Royal Society. My team have already found several new examples of Euler systems, including one related to Siegel modular forms which could potentially have very interesting consequences for the arithmetic of genus 2 algebraic curves.However, there is a significant gap in our understanding of these objects, which is that in many cases we cannot prove that the new objects are not zero. In the earlier constructions of Euler systems, this input was provided by theorems called "explicit reciprocity laws", which relate the Euler system to the values of an L-function. The goal of the proposed research is to prove explicit reciprocity laws for some of the newly-discovered Euler systems. Until recently this problem seemed to be entirely inaccessible; but recent breakthroughs in the theory of p-adic automorphic forms, arising from work of Vincent Pilloni, suggest a strategy for attacking the problem.