Let F be a totally real field and χ an abelian totally odd character of F . In 1988, Gross stated a p-adic analogue of Stark's conjecture that relates the value of the derivative of the p-adic L-function associated to χ and the p-adic logarithm of a p-unit in the extension of F cut out by χ. In this paper we prove Gross's conjecture when F is a real quadratic field and χ is a narrow ring class character. The main result also applies to general totally real fields for which Leopoldt's conjecture holds, assuming that either there are at least two primes above p in F , or that a certain condition relating the Linvariants of χ and χ −1 holds. This condition on L -invariants is always satisfied when χ is quadratic.

Let F be a totally real number field and let p be a finite prime of F , such that p splits completely in the finite abelian extension H of F. Stark has proposed a conjecture stating the existence of a p-unit in H with absolute values at the places above p specified in terms of the values at zero of the partial zeta-functions associated to H/F. Gross proposed a refinement of Stark's conjecture which gives a conjectural formula for the image of Stark's unit in F × p / E, where F p denotes the completion of F at p and E denotes the topological closure of the group of totally positive units E of F. We propose a further refinement of Gross' conjecture by proposing a conjectural formula for the exact value of Stark's unit in F × p .

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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 ...