Anticyclotomic p-adic L-functions and Ichino’s formula

Abstract

We give a new construction of a p-adic L-function L(f, Ξ), for f a holomorphic newform and Ξ an anticyclotomic family of Hecke characters of Q( √ −d). The construction uses Ichino's triple product formula to express the central values of L(f, ξ, s) in terms of Petersson inner products, and then uses results of Hida to interpolate them. The resulting construction is well-suited for studying what happens when f is replaced by a modular form congruent to it modulo p, and has future applications in the case where f is residually reducible.1991 Mathematics Subject Classification. Primary 11F67.We will work heavily with the theory of classical modular forms and newforms, as developed in e.g.[Miy06] or [Shi94]. If χ is a Dirichlet character modulo N we let M k (N, χ) = M k (Γ 0 (N ), χ) be the C-vector space of modular forms that transform under Γ 0 (N ) with weight k and character χ. We let S k (N, χ) denote the subspace of cusp forms; on this space we have the Petersson inner product, which we always take to be normalized by the volume of the corresponding modular curve:The anticyclotomic p-adic L-function. Given this setup, we will fix a newform f ∈ S k (N ) with trivial central character. We will also want to fix an "anticyclotomic family" of Hecke characters ξ m for our imaginary quadratic field K. The precise meaning of this is defined in Section 5.2, but it amounts to starting with a fixed character ξ a (normalized to have infinity-type (a + 1, −a + k − 1)) and then constructing closely-which we move to Q ⊆ Q p via our embeddings i ∞ and i p , and then modify to a "p-adic part" L p (f, ξ −1 m , 0). The basic algebraicity result is due to Shimura, and the exact choices we make to define these values are specified in Section 2.3 Also, rather than literally take L p (f, Ξ −1 ) an analytic function on Z p , we instead use an algebraic analogue of this: we construct L p (f, Ξ −1 ) as an element of a power series ring I ∼ = Z ur p X . Certain continuous functions P m : I → Z ur p serve as "evaluation at m"; this is defined in Section 2.2. With all of these definitions made, we can precisely specify what L p (f, Ξ −1 ) should be: Definition 1.0.1. The anticyclotomic p-adic L-function L p (f, Ξ −1 ) is the unique element of I such that, for integers m > k satisfying m ≡ a (mod p − 1), we have P m (L p (f, Ξ −1 )) = L p (f, ξ −1 m , 0).Ichino's formula, classically. The definition of L-values does not lend itself to p-adic interpolation. Instead, p-adic L-functions are constructed by relating L-values to something else that is more readily interpolated. This often comes from the theory of automorphic representations, where there are many formulas relating Lvalues to integrals of automorphic forms. Our approach is to use Ichino's triple product formula [Ich08], which relates a certain global integral (for three automorphic representations π 1 , π 2 , π 3 on GL 2 ) to a product of local integrals. The constant relating them is the central value of a triple-product L-function L(π 1 × π 2 × π 3 , s). We will apply this by tak...

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