Families of automorphic forms on definite quaternion algebras and Teitelbaum’s conjecture. (Familles de formes automorphes sur les algèbres quaternioniques et conjecture de Teitelbaum.) (English. French summary) Zbl 1251.11033
Berger, Laurent (ed.) et al., Représentations \(p\)-adiques de groupes \(p\)-adiques III: Méthodes globales et géométriques. Paris: Société Mathématique de France (ISBN 978-2-85629-282-2/pbk). Astérisque 331, 29-64 (2010).
Let \(f\) be a classical newform of even weight \(k_0+ 2\geq 2\) for \(\Gamma_0(Np)\), with a prime \(p> 2\) and \(N\) a positive integer not divisible by \(p\). Assume that \(f\) is split multiplicative at \(p\), i.e. the eigenvalue \(a_p(f)\) of the operator \(U_p\) acting on \(f\) satisfies \(a_p(f)= p^{k_0/2}\). Let \(L(f, s)\) denote the complex \(L\)-function attached to \(f\).
Let \(L_p(f,-)\) be the corresponding \(p\)-adic \(L\)-function associated to \(f\) (depending on the choice of a complex period \(\Omega_f\)). Let \(L^*(f, 1+ k_0/2):= L(f,1+ k_0/2)/\Omega_f\) denote the algebraic part of \(L(f, s)\) at \(s= 1+ k_0/2\).
B. Mazur, J. Tate and J. Teitelbaum [Invent. Math. 84, 1–48 (1986; Zbl 0699.14028)] made the following conjecture: There exists a constant \({\mathcal L}(f)\in\mathbb{C}_p\), such that \(L_p'(f, 1+ k_0/2)={\mathcal L}(f)L^*(f, 1+ k_0/2)\).
The \({\mathcal L}\)-invariant \({\mathcal L}(f)\) was defined by Mazur, Tate and Teitelbaum (loc. cit.) only in the weight two case. In the higher weight case, several (a priori inequivalent) definitions of \({\mathcal L}(f)\) were proposed: \({\mathcal L}_T(f)\) by J. T. Teitelbaum [Invent. Math. 101, No. 2, 395–410 (1990; Zbl 0731.11065)]; \({\mathcal L}_C(f)\) by R. F. Coleman [Contemp. Math. 165, 21–51 (1994; Zbl 0838.11033)]; \({\mathcal L}_{FM}(f)\) by J.-M. Fontaine and B. Mazur [Cambridge, MA: International Press. Ser. Number Theory. 1, 41–78 (1995; Zbl 0839.14011); \({\mathcal L}_0(f)\) by L. Orton [Can. J. Math. 56, No. 2, 373–405 (2004; Zbl 1060.11030)]; \({\mathcal L}_{Br}(f)\) by C. Breuil [Astérisque 331, 65–115 (2010; Zbl 1246.11106)].
We now know that all the above \({\mathcal L}\)-invariants are equal (when they are defined) as result of work of many people: \({\mathcal L}_C(f)={\mathcal L}_{FM}(f)\) (Coleman-Iovita, Colmez); \({\mathcal L}_T(f)={\mathcal L}_C(f)\) (Iovita-Spiess); \({\mathcal L}_{Br}(f)={\mathcal L}_0(f)\) (Breuil).
The conjecture of Mazur, Tate and Teitelbaum was first proved by R. Greenberg and G. Stevens [Invent. Math. 111, No. 2, 407–447 (1993; Zbl 0778.11034)] for the weight two case. In the higher weight case, several proofs have been announced: by Kato-Kurihara-Tsuji, working with \({\mathcal L}_{FM}(f)\) (unpublished); by Stevens, working with \({\mathcal L}_C(f)\) (unpublished); by Orton, working with \({\mathcal L}_C(f)\) [loc. cit.], and by M. Emerton, working with \({\mathcal L}_{Br}(f)\) [Duke Math. J. 130, No. 2, 353–392 (2005; Zbl 1092.11024)].
The authors describe a new proof of the “exceptional zero conjecture” of Mazur, Tate and Teitelbaum. Their proof relies on Teitelbaum’s approach to the \({\mathcal L}\)-invariant, based on the Cherednik-Drinfeld theory of \(p\)-adic uniformisation of Shimura curves.
For the entire collection see [Zbl 1192.11002].
Let \(L_p(f,-)\) be the corresponding \(p\)-adic \(L\)-function associated to \(f\) (depending on the choice of a complex period \(\Omega_f\)). Let \(L^*(f, 1+ k_0/2):= L(f,1+ k_0/2)/\Omega_f\) denote the algebraic part of \(L(f, s)\) at \(s= 1+ k_0/2\).
B. Mazur, J. Tate and J. Teitelbaum [Invent. Math. 84, 1–48 (1986; Zbl 0699.14028)] made the following conjecture: There exists a constant \({\mathcal L}(f)\in\mathbb{C}_p\), such that \(L_p'(f, 1+ k_0/2)={\mathcal L}(f)L^*(f, 1+ k_0/2)\).
The \({\mathcal L}\)-invariant \({\mathcal L}(f)\) was defined by Mazur, Tate and Teitelbaum (loc. cit.) only in the weight two case. In the higher weight case, several (a priori inequivalent) definitions of \({\mathcal L}(f)\) were proposed: \({\mathcal L}_T(f)\) by J. T. Teitelbaum [Invent. Math. 101, No. 2, 395–410 (1990; Zbl 0731.11065)]; \({\mathcal L}_C(f)\) by R. F. Coleman [Contemp. Math. 165, 21–51 (1994; Zbl 0838.11033)]; \({\mathcal L}_{FM}(f)\) by J.-M. Fontaine and B. Mazur [Cambridge, MA: International Press. Ser. Number Theory. 1, 41–78 (1995; Zbl 0839.14011); \({\mathcal L}_0(f)\) by L. Orton [Can. J. Math. 56, No. 2, 373–405 (2004; Zbl 1060.11030)]; \({\mathcal L}_{Br}(f)\) by C. Breuil [Astérisque 331, 65–115 (2010; Zbl 1246.11106)].
We now know that all the above \({\mathcal L}\)-invariants are equal (when they are defined) as result of work of many people: \({\mathcal L}_C(f)={\mathcal L}_{FM}(f)\) (Coleman-Iovita, Colmez); \({\mathcal L}_T(f)={\mathcal L}_C(f)\) (Iovita-Spiess); \({\mathcal L}_{Br}(f)={\mathcal L}_0(f)\) (Breuil).
The conjecture of Mazur, Tate and Teitelbaum was first proved by R. Greenberg and G. Stevens [Invent. Math. 111, No. 2, 407–447 (1993; Zbl 0778.11034)] for the weight two case. In the higher weight case, several proofs have been announced: by Kato-Kurihara-Tsuji, working with \({\mathcal L}_{FM}(f)\) (unpublished); by Stevens, working with \({\mathcal L}_C(f)\) (unpublished); by Orton, working with \({\mathcal L}_C(f)\) [loc. cit.], and by M. Emerton, working with \({\mathcal L}_{Br}(f)\) [Duke Math. J. 130, No. 2, 353–392 (2005; Zbl 1092.11024)].
The authors describe a new proof of the “exceptional zero conjecture” of Mazur, Tate and Teitelbaum. Their proof relies on Teitelbaum’s approach to the \({\mathcal L}\)-invariant, based on the Cherednik-Drinfeld theory of \(p\)-adic uniformisation of Shimura curves.
For the entire collection see [Zbl 1192.11002].
Reviewer: Andrzej Dąbrowski (Szczecin)
MSC:
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 11G05 | Elliptic curves over global fields |
| 11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |
