Modularity, Siegel forms and abelian surfaces (Q2890360)

Let \(f\) be a cuspidal Siegel eigenform of genus 2, weight \(\kappa = (k_1,k_2)\), \(k_1 \geq k_2\) and level \(K\). Let \(\rho _f\) be the associated Galois representation with values in ring of integers \(\mathcal{O}\) of some finite extension of \(\mathbb{Q}_p\). The author introduces hypotheses on the triple \((f,K,p)\): ordinarity of \(f\) at \(p\) (with stronger assumptions on weights), an assumption on \(p\), and the bigness of the image of the reduction \(\bar{\rho}_f\) of \(\rho_f\). Finally, he also puts hypotheses on the level \(K\) and the ramification of \(\rho\). A triple \((f,K,p)\) satisfying those conditions is called admissible.NEWLINENEWLINEThe main theorem of the paper states, that if \((f,K,p)\) is an admissible triple and if \(\rho: \mathrm{Gal}(\bar{\mathbb{Q}} / \mathbb{Q}) \rightarrow \mathrm{GSp}_4(\mathcal{O})\) is a Galois representation and if we suppose: {\parindent=6mm \begin{itemize}\item[{\(\bullet\)}] \(\bar{\rho}_f \simeq \bar{\rho}\) \item[{\(\bullet\)}] For each place \(v\) different from \(p\), \(\rho _{f | I_v} \simeq \rho _{| I_v}\) (restriction to inertia groups) \item[{\(\bullet\)}] There exists a weight \(\kappa ' = (k_1 ', k_2 ')\) such that \(\rho _{|D_p}\) (restriction to the decomposition group) is ordinary at \(p\) of weight \(0, 2-k _2 ', 1-k_1 ', 3-k_1 ' -k_2'\). NEWLINENEWLINE\end{itemize}} Then there exists an ordinary \(p\)-adic modular eigenform \(g\) which is cuspidal, of level \(K\) and weight \(\kappa '\) such that \(\rho _g = \rho\).NEWLINENEWLINEThe proof consists on identifying the universal ring \(\tilde{R}\) of ordinary deformations with variable Hodge-Tate weights of \(\bar{\rho}_f\) with the Hecke algebra \(\tilde{T}\) acting on the space of \(p\)-adic ordinary cuspidal eigenforms of level \(K\) (as in the theory of Hida). This kind of result is proved using the classical method of Taylor-Wiles, though there are differences due to the fact that one works with p-adic ordinary forms rather than classical forms.NEWLINENEWLINEFrom this theorem, with additional hypotheses on Hodge-Tate weights of \(\rho\), the author deduces modularity of \(\rho\).