Central values of Rankin L-functions

Heegner points and $p$-adic $L$-functions for elliptic curves over certain totally real fields

“…As in Section 3.1, let K be an anti-cyclotomic character over a CM extension K of F . In this section, we state the formula of Xue [38], that generalizes the works of Gross [18], Hatcher [20], and Zhang [41]. We make some simplifying assumptions.…”

“…We remark that in [38], the unitary normalization of automorphic L-function is used, so that in loc. cit.…”

“…We also note that the normalization of the Petersson inner product of Hilbert modular forms in Definition 3.5 above is different from [38], namely that the factor . …”

“…Heegner points and p-adic L-functions 897 and K is trivial on the image of A F in A K . Given a cuspidal Hilbert newform g over F , and an anti-cyclotomic character K over K, one defines the Rankin L-functions L.s; g=K; K / as in [41], [38].…”

Central values of $L$-functions and half-integral weight forms

“…Now we make a few comments on the assumptions stated above. The oddness of the class number h(F ) of F is used to guarantee the passage from the original central value formula, which is stated on B × /F × in [13], to the current setting on B × . This assumption may be dropped if one can prove a central value formula directly on B × .…”