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The Teitelbaum conjecture in the indefinite setting. (English) Zbl 1288.11051
Let $$f$$ 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)]

##### MSC:
 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11F75 Cohomology of arithmetic groups 11F80 Galois representations
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