We prove the two-variable anticyclotomic Iwasawa main conjecture for Hida families and discuss its arithmetic application to a definite version of horizontal non-vanishing conjecture, which is formulated in [LV11]. Our approach based on control theorem Selmer groups relation between L-function built out p-adic Gross points Shimura curves studied [CL16] [CKL17] self-dual twist specialisation lin...

The Eichler-Selberg trace formulas express the traces of Hecke operators on a spaces cusp forms in terms weighted sums Hurwitz-Kronecker class numbers. For $\text {\rm SL}_2(\mathbb{Z}),$ Zagier proved these by cleverly making use weight 3/2 nonholomorphic Eisenstein series he discovered 1970s. holomorphic part this form, its so-called {\it mock modular form}, is generating function for In expo...

We study the non-vanishing on the line Re(s) = 1 of the convolution series associated to two Dirichlet series in a certain class of Dirichlet series. The non-vanishing of various L-functions on the line Re(s) = 1 will be simple corollaries of our general theorems. Let f(z) = ∑∞ n=1 âf (n)e 2πinz and g(z) = ∑∞ n=1 âg(n)e 2πinz be cusp forms of weight k and level N with trivial character. Let Lf ...

Relevant objects and the strategy 9 3.2. Weak to strong lifting, and the cuspidality criterion 13 3.3. Triple product L-functions: local factors and holomorphy 15 3.4. Boundedness in vertical strips 18 3.5. Modularity in the good case 30 3.6. A descent criterion 32 3.7. Modularity in the general case 35 4. Applications 37 4.1. A multiplicity one theorem for SL(2) 37 4.2. Some new functional equ...

1.1. The refined Gan–Gross–Prasad conjecture. In this paper, as a sequel of [Xue14], we formulate a refinement to the global Gan–Gross–Prasad conjecture for the Fourier–Jacobi periods on U(n) × U(n) and prove it under some local conditions, assuming some expected properties of the L-packets and some parts of the local Gan–Gross–Prasad conjecture. This refinement is modeled on some recent work o...

Abstract Let F be a totally real field, and let $E/F$ CM quadratic extension. We construct p -adic L -function attached to Hida families for the group $\mathrm {GL}_{2/F}\times \mathrm {Res}_{E/F}\mathrm {GL}_{1}$ . It is characterized by an exact interpolation property critical Rankin–Selberg -values, at classical points corresponding representations $\pi \boxtimes \chi $ with weights of $\chi...