arXiv subjects
Publications and source records attributed to Ananyo Kazi.
Inspired by the pullback method in the recent work of Sangiovanni-Vincentelli--Skinner, we reconstruct the diagonal class of Darmon--Rotger. Moreover, we reinterpret the computation of the $p$-adic regulator formula.
We define a two-variable $p$-adic Asai $L$-function for a finite-slope family of Hilbert modular forms over a real quadratic field (with one component of the weight, and the cyclotomic twist variable, varying independently); and a two-variable ``twisted triple product'' $L$-function, interpolating the central $L$-value of the tensor product of such a family with a family of elliptic modular forms. The former construction generalizes a construction due to Grossi, Zerbes and the second author for ordinary families; the latter is a counterpart of the twisted triple product $L$-function of arXiv:2401.13230, but differs in that it interpolates classical $L$-values in a different range of weights, in which the dominant weight comes from the Hilbert modular form. Our construction relies on a ``nearly-overconvergent'' version of higher Coleman theory for Hilbert modular surfaces.
Let $L$ be a totally real field, and $p$ be a rational prime that is unramified in $L$. We construct overconvergent families of classes of relative de Rham cohomology of the universal abelian scheme over Hilbert modular varieties associated to $L$. We show that these classes come equipped with Gauss-Manin connection. We prove convergence for $p$-adic iteration of this connection, improving upon a technique due to Andreatta-Iovita. We use this to construct a $p$-adic twisted triple product $L$-function associated to finite slope families of Hilbert modular forms, extending work of Blanco-Chacon-Fornea for Hida families.
Our main objective in the present paper is to generalise the work of Blanco-Chacón and Fornea on the $p$-adic Gross-Zagier formula for twisted triple product $p$-aidc $L$-function. We extend their main result to the case of finite slope families of Hilbert modular forms and also allow the prime $p$ to be inert in the real quadratic field $L$.