14.6. Further generalisations🔗

The three constructions and the equalities between them are expected to hold in great generality — for a Galois representation V arising from a motive M and corresponding under Langlands to an automorphic representation \pi — but very few cases are complete. We record the shape of the general expectation.

State of the art. The known cases are fragmentary. On the analytic side, \GL(1) and \GL(2) are as above, \GL(3) was handled only recently (with the symmetric-square case known much earlier), and no general construction is known for \GL(n) with n \ge 4; there are scattered results for unitary groups, Siegel modular forms and \GL_{n+1}\times\GL_n. On the arithmetic side, Euler systems remain rare — Kato's system, cyclotomic units and elliptic units were long almost the only examples — though the last decade has seen new constructions (Rankin–Selberg products, diagonal cycles for triple products, and \mathrm{GSp}_4), each demanding its own explicit reciprocity law. On the algebraic side, Main Conjectures are known in wide ordinary generality, and an Euler system together with L_p^{\mathrm{an}} = L_p^{\mathrm{arith}} always yields torsionness of the Selmer group and the divisibility L_p^{\mathrm{alg}} \mid (L_p^{\mathrm{an}}).