Some computations with Hecke rings and deformation rings. With an appendix by Amod Agashe and William Stein.

Lario, Joan-C.; Schoof, René

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Let $k$ be a positive integer divisible by 4, $p > k$ a prime, $f$ an elliptic cuspidal eigenform (ordinary at $p$ ) of weight $k - 1$ , level 4 and non-trivial character. In this paper we provide evidence for the Bloch-Kato conjecture for the motives ${ad}^{0} M (- 1)$ and ${ad}^{0} M (2)$ , where $M$ is the motif attached to $f$ . More precisely, we prove that under certain conditions the $p$ -adic valuation of the algebraic part of the symmetric square $L$ -function of $f$ evaluated at $k$ provides a lower bound for the $p$ -adic valuation of the order...