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The Teitelbaum conjecture in the indefinite setting
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 135, Number 6, December 2013
- pp. 1525-1557
- 10.1353/ajm.2013.0055
- Article
- Additional Information
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Let $f$ be a new cusp form on $\Gamma_0(N)$ of even weight $k+2\geq 2$.
Suppose that there is a prime $p\parallel N$ and that we may write
$N=pN^{+}N^{-}$, where $N^{-}$ is the squarefree product of an even number
of primes. There is a Darmon style ${\cal L}$-invariant ${\cal
L}^{N^{-}}(f)$ attached to this factorization, which is the Orton ${\cal
L}$-invariant when $N^{-}=1$. We prove that ${\cal L}^{N^{-}}(f)$ does not
depend on the chosen factorization of $N$ and it is equal to the other
known ${\cal L}$-invariants. We also give a formula for the computation of
the logarithmic $p$-adic Abel-Jacobi image of the Darmon cycles. This
formula is crucial for the computations of the derivatives of the $p$-adic
$L$-functions of the weight variable attached to a real quadratic field
$K/\Bbb{Q}$ such that the primes dividing $N^{+}$ are split and the primes
dividing $pN^{-}$ are inert.