In this paper we derived a nice general formula for the local integrals of triple product formula whenever one of the representations has sufficiently higher level than the other two. As an application we generalized Venkatesh and Woodbury’s work on the subconvexity bound of triple product L-function in level aspect, allowing joint ramifications, higher ramifications, general unitary central characters and general special values of local epsilon factors.
We define the Heegner--Drinfeld cycle on the moduli stack of Drinfeld Shtukas of rank two with r-modifications for an even integer r. We prove an identity between (1) The r-th central derivative of the quadratic base change L-function associated to an everywhere unramified cuspidal automorphic representation π of PGL2; (2) The self-intersection number of the π-isotypic component of the Heegner--Drinfeld cycle. This identity can be viewed as a function-field analog of the Waldspurger and Gross--Zagier formula for higher derivatives of L-functions.
In the relative trace formula approach to the arithmetic Gan–Gross–Prasad conjecture, we formulate a local conjecture (arithmetic transfer) in the case of an exotic smooth formal moduli space of p-divisible groups, associated to a unitary group relative to a ramified quadratic extension of a p-adic field. We prove our conjecture in the case of a unitary group in three variables.