The Teitelbaum conjecture in the indefinite setting.

*(English)*Zbl 1288.11051Let \(f\) be a new form of level \(N\) and even weight \(k+2\geq 2\). Assume \(N=pN^+N^-\) where the factors are prime to each other, and \(N^-\) is square free. There is an associated invariant \({\mathcal L}^{N^-}(f)\) attached to this factorization. In the setting of \(p\)-adic \(L\)-functions associated to \(f\) by Mazur-Tate-Teitelbaum, there are \({\mathcal L}\)-invariants associated to \(f\): \({\mathcal L}_C(f)\) by R. F. Coleman [Contemp. Math. 165, 21–51 (1994; Zbl 0838.11033)], \({\mathcal L}_{FM}(f)\) by Fontaine and B. Mazur [Contemp. Math. 165, 1–20 (1994; Zbl 0846.11039)], \({\mathcal L}_B(f)\) by Ch. Breuil [Astérisque 331, 65–115 (2010; Zbl 1246.11106)]. This paper proves that when \(N^-\) has an even number of prime factors (thus the quaternion algebra ramified over all primes dividing \(N^-\) is indefinite),
\[
{\mathcal L}^{N^-}(f)=-2(\log a_p)'(k).
\]
Here one associates a Hida family to \(f\) such that for \(n\geq k\) integer, there is \(f^n:=\sum_{i\geq 1} a_i(n)q^i\) modular form of weight \(n+2\) and level \(N\) in a neighborhood of \(k\) (under \(p\)-adic topology). The derivative is \(\frac{d}{dn}(\log a_p(n))|_{n=k}\).

The author shows that \({\mathcal L}^{N^-}(f)\) equals the other invariants \({\mathcal L}_C(f)\), \({\mathcal L}_{FM}(f)\) and \({\mathcal L}_{B}(f)\), and also is independent of the choice of \(N^-\) in the factorization of \(N\). When \(N^-\) has odd number of prime factors (thus the corresponding quaternion algebra is definite), the results are known (at least when \(k=0\)) thanks to the work of M. Bertolini, H. Darmon and A. Iovita [Astérisque 331, 65–115 (2010; Zbl 1251.11033)]

The author shows that \({\mathcal L}^{N^-}(f)\) equals the other invariants \({\mathcal L}_C(f)\), \({\mathcal L}_{FM}(f)\) and \({\mathcal L}_{B}(f)\), and also is independent of the choice of \(N^-\) in the factorization of \(N\). When \(N^-\) has odd number of prime factors (thus the corresponding quaternion algebra is definite), the results are known (at least when \(k=0\)) thanks to the work of M. Bertolini, H. Darmon and A. Iovita [Astérisque 331, 65–115 (2010; Zbl 1251.11033)]

Reviewer: Zhengyu Mao (Newark)