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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 \(k \geq 2\).

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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