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Teitelbaum’s exceptional zero conjecture in the anticyclotomic setting. (English) Zbl 1079.11036
Let $$\phi$$ be an eigenform of even weight $$k \geq 2$$ on $$\Gamma_0(N)$$. In ”On $$p$$-adic analogues of the conjectures of Birch and Swinnerton-Dyer“ [Invent. Math. 84, 1–48 (1986; Zbl 0699.14028)] B. Mazur, J. Tate and J. Teitelbaum formulated a $$p$$-adic variant of the conjecture of Birch and Swinnerton-Dyer – the case $$k=2$$ – for the value of the $$p$$-adic $$L$$-function $$L_p(\phi,s)$$ attached to the cyclotomic $$\mathbb Z_p$$-extension of $$\mathbb Q$$ at $$s=1$$. In the special case that $$p$$ divides $$N$$ exactly and that the $$p$$-th coefficient of $$\phi$$ equals $$1$$, the $$p$$-adic $$L$$-function $$L_p(\phi,s)$$ vanishes at $$s=1$$. In this situation the conjectures of Mazur, Tate and Teitelbaum imply the following relationship between the derivative $$L'_p(\phi,1)$$ and the special value $$L(\phi,1)$$ of the classical $$L$$-function at $$1$$: $L'_p(\phi,1)= \mathcal L(\phi)\cdot L(\phi,1)/\Omega,$ where $$\Omega$$ is a real period and $$\mathcal L(\phi)$$, the so-called $$\mathcal L$$-invariant, is defined using $$p$$-adic uniformization. This relation was proved by R. Greenberg and G. Stevens [Invent. Math. 111, 407–447 (1993; Zbl 0778.11034)].
For arbitrary modular forms of even weight $$k$$ J. Teitelbaum [Invent. Math. 101, 395–410 (1990; Zbl 0731.11065)] suggested a definition of the $$\mathcal L$$-invariant, which should give the analog of the Greenberg-Stevens result in the form $L'_p(\phi,k/2)= \mathcal L(\phi)\cdot L(\phi,k/2)/\Omega.$ In this paper the authors prove an analogue of Teitelbaum’s conjecture, in which the cyclotomic $$\mathbb Z_p$$-extension of $$\mathbb Q$$ is replaced by the anti-cyclotomic $$\mathbb Z_p$$-extension of an imaginary quadratic number field $$K$$, in which the prime $$p$$ splits. This generalizes results of M. Bertolini and H. Darmon [Duke Math. J. 98, 305–334 (1999; Zbl 1037.11045)] and emphasizes the role of $$p$$-adic integration. The authors also consider the case that the prime $$p$$ is inert in $$K$$, which leads to an interpretation of $$L'_p(\phi,k/2)$$ in terms of $$p$$-adic Coleman integrals and generalizes one of the main results of M. Bertolini and H. Darmon [Invent. Math. 131, 453–491 (1998; Zbl 0899.11029)] to even weights $$k \geq 2$$.

##### MSC:
 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11F33 Congruences for modular and $$p$$-adic modular forms 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11G18 Arithmetic aspects of modular and Shimura varieties
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