Teitelbaum's exceptional zero conjecture in the anticyclotomic setting

Abstract

Teitelbaum formulated a conjecture relating first derivatives of the Mazur-Swinnerton-Dyer p -adic L -functions attached to modular forms of even weight k ≥ 2 to certain [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]-invariants arising from Shimura curve parametrizations. This article formulates an analogue of Teitelbaum's conjecture in which the cyclotomic [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] extension of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] is replaced by the anticyclotomic [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /]-extension of an imaginary quadratic field. This analogue is then proved by using the Cerednik-Drinfeld theory of p -adic uniformisation of Shimura curves.