Executive Summary : | The Shimura-Taniyama conjecture over rational numbers (now called modularity theorem) proved by Wiles, Taylor-Wiles and Breuil-Conrad-Diamond-Taylor, shows that an elliptic curve over the rational numbers is modular i.e. it can be obtained via a rational map with integral coefficients from the classical modular curve. It then follows that the associated Hasse-Weil (complex) L-function has an analytic continuation to the whole complex plane and satisfies a functional equation with center at s =1. One of the most important conjectures in arithmetic geometry and number theory is Birch–-Swinnerton--Dyer (BSD) conjecture which states that the vanishing order of the Hasse-Weil L-function at s=1 is equal to the rank of the abelian group of rational points of the elliptic curve over rational numbers. Furthermore, there exists an explicit formula relating the leading coefficient of the Taylor expansion of the L-function at s=1 with the order of the Mordell-Weil group. For the rank part of BSD conjecture, by far the most general result for elliptic curves over rational numbers is obtained using the Gross-Zagier formula and Euler system of Heegner points by Kolyvagin. Deep results of Kato, Skinner-Urban, Zhang settled the p-part of the leading term formula in the rank 0 and rank 1. Iwasawa theory of elliptic curves has been at the core of all the above results. The main conjecture of Iwasawa theory is an intricate connection between algebraic invariants associated with the Tate–Shafarevich group (or in general Selmer group) and analytic invariants associated p-adic L-functions. However, the higher rank cases have turned out to be extremely difficult but it has fostered some of the most productive research of the recent decades. Over number fields there have been some recent breakthroughs by Bertolini, Darmon, Rotger, and collaborators. Fundamental work of Bertolini, Darmon, Rotger, Loeffler, Zerbes, and collaborators using Beilinson-Flach elements (in the higher chow group of the modular curve) construct a full cyclotomic Euler system of the tensor product of two motives attached to cusp forms and prove new cases of BSD conjectures. Guided by the analogy with Beilinson-Flach elements and Gross--Kudla--Schoen cycles, Darmon--Rotger initiated a study of the Euler system of diagonal cycles in chow groups to prove new cases of BSD conjectures. This concerns 2-dimensional Artin twists, the triple product of modular forms, and the Rankin product of two modular forms. These algebraic cycles have several important corollaries e.g. Dasgupta’s factorization formula for the tensor product of two modular forms, exceptional zero conjecture of symmetric square, construction of Stark-Heegner points, etc. In this proposal, we plan to continue to implement these cycles (e.g. Euler systems of Diagonal cycles, etc) in the (higher) Chow group to generalize Dasgupta's factorization formula to triple products of three modular cuspidal forms and other cases. |