Factorization of p-adic Rankin L-series

Abstract

We prove that the p-adic L-series of the tensor square of a p-ordinary modular form factors as the product of the symmetric square p-adic L-series of the form and a KubotaLeopoldt p-adic L-series. This establishes a generalization of a conjecture of Citro. Greenberg's exceptional zero conjecture for the adjoint follows as a corollary of our theorem.Our method of proof follows that of Gross, who proved a factorization result for the Katz p-adic L-series associated to the restriction of a Dirichlet character. Whereas Gross's method is based on comparing circular units with elliptic units, our method is based on comparing these same circular units with a new family of units (called Beilinson-Flach units) that we construct. The Beilinson-Flach units are constructed using Bloch's intersection theory of higher Chow groups applied to products of modular curves. We relate these units to special values of classical and p-adic L-functions using work of Beilinson (as generalized by Lei-Loeffler-Zerbes) in the archimedean case and Bertolini-Darmon-Rotger (as generalized by Kings-Loeffler-Zerbes) in the p-adic case. Central to our method are two compatibility theorems regarding Bloch's intersection pairing and the classical and p-adic Beilinson regulators defined on higher Chow groups. We thank Henri Darmon and Victor Rotger for several inspiring and helpful discussions at the outset of this project, and particularly for their seminal papers [BDR1] and [BDR2] that have opened the door to a wide range of arithmetic applications of p-adic Beilinson formulae. We are also greatly indebted to David Loeffler and Sarah Zerbes, with whom we have engaged in several discussions regarding this project, in particular regarding the extraction of the results needed in this paper from their wonderful articles [LLZ] and [KLZ]. We thank Denis Auroux, Akshay Venkatesh, Brian Conrad, and Guido Kings for helpful discussions regarding the compatiblity results stated in §5. We thank Joël Bellaïche, Robert Harron, and Luis Garcia for suggestions and stimulating discussions. Finally, we thank the anonymous referee for making many detailed suggestions that greatly improved the quality of the exposition, especially regarding the compatibility results in §5. Factorization of p-adic Rankin L-series IntroductionThe main result of this paper is a factorization formula for the p-adic L-function associated to the tensor square of a p-ordinary cuspidal eigenform. We introduce some notation to state our result. Letb n q n ∈ S (Γ 1 (N g ), χ g ) be two normalized cuspidal eigenforms of weights k, 2 and nebentype characters χ f , χ g , respectively. Let ψ be an auxiliary Dirichlet character of conductor N ψ , and letThe Rankin L-series of f and g twisted by ψ is defined bywhere L N denotes a Dirichlet L-function with the Euler factors at primes dividing N removed. The Rankin series D N (f, g, ψ, s) has an Euler product equal to that of the primitive L-series L(f ⊗ g ⊗ ψ, s) outside of primes dividing N ; see §2.3.Shimura proved that the values of D N ...

Iwasawa theory for the symmetric square of a modular form

“…If εψ is not the trivial character, but maps p to 1, then we conclude that (for a suitable choice of c) we have exp * Gr 1 V (−k) c BF f ψ = 0 if and only ifL p (f , f , ψ)(k, k) = 0. It seems likely thatL p (f , f , ψ)(k, k) is never zero, and it may be possible to prove this using an adaptation of the methods of [Das16]. We hope to return to this issue in a future paper.…”

“…Complex L-functions. We introduce the various L-functions which we shall consider, following [Sch88,Das16]. For a prime ℓ ∤ N , we write…”

Iwasawa theory for Rankin-Selberg products of p-nonordinary eigenforms

“…Note that the motive associated to f ⊗ f ⊗ χ does not possess any critical values and as a result, one may not appeal to non-vanishing statements for complex L-values to deduce the required non-triviality. However, Arlandini's work in progress (extending Dasgupta's result [Das16, Theorem 1] in the p-ordinary case) shows that the p-adic L-functions in question factors as a product of the symmetric square p-adic L-function and a Kubota-Leopoldt p-adic L-function. The required non-triviality easily follows from generic non-vanishing statements for symmetric square L-values; see Section 2.2 for details.…”

Beilinson–Flach elements, Stark units and 𝑝-adic iterated integrals

“…Proof. According to [Das1,Section 3.2], we know that Besides, the assumptions we have fixed imply that dim L (V gh ) G Q = 0, and since the functional equation introduces no extra zero or pole at s = 1 (see [Das2]), we conclude that the order of vanishing at s = 1 is also 2.…”