Infinitesimal deformations and the \(\ell\)-invariant
The main results of this paper are two formulas for Greenberg's generalized \(\ell\)-invariant in terms of derivatives of eigenvalues of Frobenius. Rather heavy preliminaries are required, which we shall try to summarize in a minimal way. Let \(M\) be a pure motive over \({\mathbb Q}\) with coefficients in a number field \(E\), and assume that the \(L\)-function \(L(M, s)\) is well-defined. Suppose moreover that \(M\) is critical and that \(L(M,0) \not= 0.\) Fix a \(p\)-place \(\lambda\) of \(E\) and assume that the \(\lambda\)-adic realization \(M_\lambda\) of \(M\) is semistable in the sense of Fontaine. The \((\varphi, N)\)-module \({\mathbb D}_{st} \;(M_\lambda)\) associated to \(M_\lambda\) is a finite dimensional \(E_\lambda\)-vector space equipped with an exhaustive decreasing filtration, an \(E_\lambda\)-linear bijective Frobenius map \(\varphi\) and a nilpotent monodromy operator \(N\) such that \(N \varphi = p \varphi N\). A \((\varphi, N)\)-submodule \(D\) of \({\mathbb D}_{st}\;(M_\lambda)\) is called regular if \( {\mathbb D}_{st}\;(M_\lambda) = D \oplus \;Fil^o\;{\mathbb D}_{st}\;(M_\lambda)\) as \(E_\lambda\)-vector spaces. The theory of [\textit{B. Perrin-Riou}, \(p\)-adic \(L\)-functions and \(p\)-adic representations. Astérisque. 229. Paris: Société Mathématique de France (1995; Zbl 0845.11040)] suggests that any regular \(D\) should give rise to a \(p\)-adic \(L\)-function \(L_p (M, D, s)\) interpolating rational parts of special values of \(L(M, s).\) In particular, the interpolation formula at \(s = 0\) should read: \(L_p (M, D, 0) = \text{Eul}(M, D)\;L(M, 0)/\Omega_\infty (M),\) where \(\Omega_\infty (M)\) is the Deligne period and \(\text{Eul}(M, D)\) is a certain product of Euler-like factors. One can naturally expect that \(L_p (M, D, 0) = 0\) if and only if \(\text{Eul}(M, D) = 0,\) in which case one says that \(L_p(M, D, s) = 0\) has a trivial zero at \(s = 0\). In his preprint [``A generalization of Greenberg's \(\mathcal{L}\)-invariant,'' Am. J. Math. 133, No. 6, 1573--1632 (2011; Zbl 1307.11079), \url{arXiv : 0906-2857 (2009)}], the author formulated the following ``trivial zero conjecture'': \(L_p (M, D, s)\) has a zero of order \(e \geq 0\) at \(s = 0\) and \(\displaystyle \lim_{s \to 0}\;L_p (M, 0, s)/s^e = \text{Eul}^+ (M, D)\;{\mathfrak L} (M_\lambda^\ast (1), D^\ast)\;L(M, O)_{\displaystyle/\Omega_\infty (M)},\) where \(\text{Eul}^+ (M, D)\) is the subproduct of \(\text{Eul}(M, D)\) obtained by ``excluding zero factors'' and \(D^\ast = \Hom({\mathbb D}_{st}\;(M_\lambda)/D\;{\mathbb D}_{st} (E_\lambda (1)))\) is the dual regular module. The new ingredient is the invariant \({\mathfrak L} (V, D)\) which the author defines for any regular \(D \subset {\mathbb D}_{st} (V)\) of a ``reasonably behaved'' (in a precise sense, the prototype being \(M_\lambda)\) pseudo-geometric semi-stable representation \(V\) over a \(p\)-adic field \(L\). If \(M_\lambda\) is ordinary at \(p,\) the above conjecture coincides with a conjecture of \textit{R. Greenberg} [Contemp. Math. 165, 149--174 (1994; Zbl 0838.11070)]. Another particular case is that of the motive \(M_f\) attached to a normalized newform \(f = \displaystyle\sum^\infty_0\;a_n q^n\) of weight \(2 k\) on \(\Gamma_0 (N p)\) with \(p \nmid N\). The twisted motive \(M_f (k)\) is critical. The eigenvalues of \(\varphi\) acting on \({\mathbb D}_{st}\;(M_{f, \lambda} (k))\) are \(\alpha = p^{-k} a_p\) and \(\beta = p^{1-k} a_p,\) with \(v_p (a_p) = k-1\). The unique regular submodule is \(D = E_\lambda d,\) where \(\varphi (d) = \alpha\;d,\) and \(L_p (M_f (k), D, s) = L_p (f, s + k),\) where \(L_p (f, s)\) is the classical modular \(p\)-adic \(L\)-function. If \(a_p = p^{k-1},\) the function \(L_p (f, s)\) vanishes at \( s = k\). In this case, several constructions of the \({\mathfrak L}\)-invariant have been proposed by many authors (Breuil, Colmez, Fontaine, Mazur, Orton, Teitelbaum \(\ldots\)) but it is known that they all coincide, and the above conjecture takes the form of the Mazur-Tate-Teitelbaum conjecture: \[ L'_p (f, k) = {\mathfrak L} (f)\;L(f, k)/\Omega_\infty (f) \] \(\Bigl( = -2 (d\;\log) a_p (2 k)\;L(f, k)/\Omega_\infty(f),\) where \((d\;\log) a_p(x) = a_p(x)^{-1} {\displaystyle d a_p(x)\over dx} \Bigr)\). Thus the M-T-T conjecture is equivalent to the formula : \({\mathfrak L} (f) = -2 (d\;\log) a_p (2 k)\). In this paper, working with his generalized \({\mathfrak L}\)-invariant, the author extends the formula of M-T-T to some infinitesimal deformations of a ``reasonably behaved'' pseudo-geometric representation. Assuming that \(e = 1,\) he gives two formulas (too technical to be reproduced here) expressing \({\mathfrak L} (V^\ast (1), D^\ast)\) in terms of derivatives of eigenvalues of Frobenius, one in the crystalline case, the other in the stable case. For classical modular forms, the existence of deformations having the required properties follows from the theory of Coleman-Mazur together with deep results of Kisin and Saito; in this case, the author's second result yields a new proof of the M-T-T conjecture. The proof of both results is purely algebraic, a direct generalization of \textit{R. Greenberg} and \textit{G. Stevens}' proof of the M-T-T conjecture in weight 2 [Contemp. Math. 165, 183--211 (1994; Zbl 0846.11030)], using the cohomology of \((\varphi, \Gamma)\)-modules instead of Galois cohomology.
scientific article; zbMATH DE number 5827437