Leading terms of anticyclotomic Stickelberger elements and 𝑝-adic periods

Abstract

Abstract. Let E be a quadratic extension of a totally real number field. We construct Stickelberger elements for Hilbert modular forms of parallel weight 2 in anticyclotomic extensions of E. Extending methods developed by Dasgupta and Spieß from the multiplicative group to an arbitrary one-dimensional torus we bound the order of vanishing of these Stickelberger elements from below and, in the analytic rank zero situation, we give a description of their leading terms via automorphic L-invariants. If the field E is totally imaginary, we use the p-adic uniformization of Shimura curves to show the equality between automorphic and arithmetic L-invariants. This generalizes a result of Bertolini and Darmon from the case that the ground field is the field of rationals to arbitrary totally real number fields.

Plectic Stark-Heegner points

“…When B is totally definite, the equality of automorphic and arithmetic Linvariants can be deduced from Manin and Drinfeld's uniformization of Jacobians of Mumford curves (cf. [BG18], Theorem 6.9).…”

“…Following ([Ber17], [BG18]) we attach an anticyclotomic p-adic L-functions to the elliptic curve A /F and the quadratic extension E/F . The main result of this section is a p-adic Gross-Zagier formula (Theorem 5.13) relating higher derivatives of these p-adic L-functions to plectic p-adic invariants.…”

Anticyclotomic exceptional zero phenomenon for Hilbert modular forms

“…Using their result, one may deduce a formula similar to (1.8) in the case of χ = 1 but less precise. Indeed, the automorphic periods q Sm,p in [3] are not showed to coincide with Tate's period; these periods q Sm,p are even not showed to be independent of S m . While in our formula (1.8) the leading term is precise.…”

Plectic p-adic invariants

“…When the algebraic rank r alg (A/E) is strictly smaller than r we cannot yet guess what arithmetic information is contained in the plectic p-adic invariant. Nevertheless, we expect that -assuming r alg (A/E) < r -the plectic point should be non-zero whenever the parity of r = |S| matches the parity of the order of vanishing of L S (A/E) (see [BG18], Corollary 5.7).…”