Integrality of a ratio of Petersson norms and level-lowering congruences

Abstract

We prove integrality of the ratio f, f / g, g (outside an explicit finite set of primes), where g is an arithmetically normalized holomorphic newform on a Shimura curve, f is a normalized Hecke eigenform on GL(2) with the same Hecke eigenvalues as g and , denotes the Petersson inner product. The primes dividing this ratio are shown to be closely related to certain level-lowering congruences satisfied by f and to the central values of a family of Rankin-Selberg L-functions. Finally we give two applications, the first to proving the integrality of a certain triple product L-value and the second to the computation of the Faltings height of Jacobians of Shimura curves.

Arithmetic properties of the Shimura–Shintani–Waldspurger correspondence

“…We use the same conventions here as in [22]. In the interest of brevity, the reader is referred to §1 of that article for the measure normalizations used on the different local and adelic groups, the only difference being that the indefinite quaternion algebra is called D in [22] as opposed to B in the present article.…”

“…we need to compare the arithmetic properties of the form R χ q (g ) with those of g . We now apply the rationality and integrality criteria of [12] and [22], formulated more precisely in our context in Prop. 5.1 below.…”

“…SetF 0 = Q if 2k = 2 andF 0 = F 0 otherwise. Let F be any number field containingF 0 and all the Hecke eigenvalues of f , let p be a prime not dividing N and λ a prime in F lying over p. As shown in [22] and as will be recalled below, f and g may be normalized canonically up to λ-adic units in F . One has attached to f and g, canonical fundamental periods u ± (f, F, λ) and u ± (g, F, λ), well defined up to λ-adic units in F .…”

“…The main theorem of the appendix guarantees then that H has λ-integral Fourier coefficients as well.) Now one applies an argument similar to that of the authors' previous article [22] for a suitable CM period Ω. One needs to use here a refined study of congruences between θ κ and other forms as well as the main conjecture of Iwasawa theory for the imaginary quadratic field K, which is a deep theorem of Rubin [25].…”

Shimura Curves and Special Values of p-adic L-functions

“…To reduce to the G L 2 case, note that, by the argument on p. 919 of [31], there is a false isogeny (defined over a number field in which p is unramified)…”