Construction of anticyclotomic Euler systems using diagonal cycles

Abstract

In this thesis, we construct a new anticyclotomic Euler system for the four-dimensional Galois representation attached to two modular forms and a Hecke character of an imaginary quadratic field. To state the results more precisely, let g and h be newforms of weights l≥m of the same parity and let ψ be a Hecke character of an imaginary quadratic field K of infinity-type (1−k,0) for some even integer k≥2. Assume that the product of the characters of g, h and the CM-form attached to ψ is trivial. Let p be a prime which splits in K. We then study the p-adic GK-representation V:=Vg⊗Vh(ψ−1)(1−c), where c=(k+l+m−2)/2. Combining a geometric construction using modified diagonal cycles in the product of three modular curves with a result of Lei--Loeffler--Zerbes, we obtain cohomology classes over ring class field extensions of K, and we prove that they form a split anticyclotomic Euler system in the sense of Jetchev--Nekov'a\v{r}--Skinner. The bottom Λ-adic class of our Euler system is related to a one-variable specialization of the triple product p-adic L-function constructed by Darmon--Rotger via a reciprocity law proved by Bertolini--Seveso--Venerucci and Darmon--Rotger. This reciprocity law, together with the Euler-system machinery developed by Jetchev--Nekov'a\v{r}--Skinner, allows us to deduce, under some additional hypotheses, different cases of the Bloch--Kato conjecture for the representation V in analytic rank zero and one. As a different application, we also give two equivalent formulations of an Iwasawa--Greenberg Main Conjecture in this setting and prove one divisibility. When h=g∗, i.e., the modular form obtained by conjugating the Fourier coefficients of g, we obtain an Euler system for the three-dimensional GK-representation V′:=ad0(Vg)(ψ−1)(1−k/2)⊂V and use it to derive similar applications towards the Bloch--Kato conjecture in analytic rank zero and one and towards a divisibility of an Iwasawa--Greenberg Main Conjecture.