\(\mathcal{L}\)-invariants and logarithm derivatives of eigenvalues of Frobenius. (English) Zbl 1323.11096
Let \(g\) be a level \(\Gamma_0(N)\)(\(p||N\)) newform of even weight \(k\geq 2\), \(L(g, s)\) the classical \(L\)-function of g and \(L_p(g, s)\) the \(p\)-adic \(L\)-function associated to \(g\). Let \(\rho_g\) be the Galois representation of \(\mathrm{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p)\). There is a conjecture due to B. Mazur et al. [Invent. Math. 84, 1–48 (1986; Zbl 0699.14028)] stating that the quotient \(\mathcal L\): \(=L^\prime_p(g, k/2)/L(g, k/2)\) depends only on \(\rho_g\) (called the exceptional zero conjecture). In this paper, let \(K\) be a \(p\)-adic local field. The author studies a special kind of \(p\)-adic Galois representation of \(it\). These representations are similar to the Galois representations occurring in the exceptional zero conjecture for modular forms. In particular, the author generalized a formula of P. Colmez to general \(p\)-adic local fields [in: Représentations \(p\)-adiques de groupes \(p\)-adiques III: Méthodes globales et géométriques. Paris: Société Mathématique de France. 13–28 (2010; Zbl 1251.11080)].
Reviewer: Chun-Gang Ji (Nanjing)
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