A norm compatible system of Galois cohomology classes for \(\text{GSp}(4)\)
Given any de Rham \(p\)-adic Galois representation \(M,\) a conjecture of Perrin-Riou associates a \(p\)-adic \(L\)-function for \(M\) to a compatible system of Galois cohomology classes in the inverse limit \(\lim_\leftarrow H^1 ({\mathbb Q} (\xi_{p^n}), M),\) via \(p\)-adic Hodge theory and a \(p\)-adic interpolation of Bloch-Kato exponential maps (\textit{B. Perrin-Riou} [Fonctions \(L\) \(p\)-adiques des représentations \(p\)-adiques. Astérisque 229. Paris: Société Mathématique de France (1995; Zbl 0845.11040)], chapitre 4). Only a few examples of such Euler systems are known: the system of cyclotomic units, leading to the Kubota-Leopoldt \(L_p\)-function; the system of Beilinson-Kato's elements defined as \(K\)-theoretical cup-products of modular units, leading to the \(L_p\)-function of elliptic modular forms. Actually, these two systems can be seen as the \(p\)-adic realization of the torsion sections of the classical and elliptic polylogarithm respectively. Building on this idea, the author provides here another system for the symplectic group in four variables \(G Sp_4,\) the norm compatible cohomology classes being the \(p\)-adic realization of cup-products of Beilinson's Eisenstein symbols, which are torsion sections of the elliptic polylogarithm pro-sheaf [\textit{G. Kings}, J. Reine Angew. Math. 517, 103--116 (1999; Zbl 1007.11043]. The main ingredients are the norm compatibility of the above pro-sheaf, due to Wildeshaus, and the computations of weights in the cohomology of Siegel threefolds in the author's previous work. In fact, once the relevant \(L_p\)-function is constructed from the norm compatible system, the difficult part is to show that it interpolates the special values of the usual \(L\)-function. The author intends to carry this over in a forthcoming paper by computing explicitly the image of his system under the dual exponential map.
scientific article; zbMATH DE number 5797238
scientific article