On Higher regulators of Siegel varieties

Abstract

We construct classes in the middle degree motivic cohomology of the Siegel variety of almost any dimension. We compute their image by Beilinson's higher regulator in terms of Rankin-Selberg type automorphic integrals. In the case of GSp( 6), using results of Pollack and Shah, we relate the integral to a non-critical special value of a degree 8 spin L-function.

Norm-Compatible Systems of Galois Cohomology Classes for $\mathbf{GSp}_6$

“…For the group GSp 4 and the classes constructed in [LSZ19], recent progress on this has been announced by Loeffler and Zerbes. In [CLRJ19], we have shown that our motivic classes (in the case of trivial coefficients) are related to non-critical special values of the complex spin L-function, as predicted by the conjectures of Beilinson and Bloch-Kato, through Deligne's regulator taking values in Deligne cohomology. This gives a link between the special values of the arithmetic p-adic spin L-function at certain integers and non-critical special values of of the spin L-function.…”

“…We expect an explicit reciprocity law to hold, relating values of Bloch-Kato's dual exponential maps of our Iwasawa class to certain values of the complex spin L-function, which should also show the non-vanishing of the classes. Progress in this direction has been achieved in [CLRJ19], where the archimedean regulator of the motivic classes (for trivial coefficients) has been calculated in terms of the complex spin L-function at noncritical points, using techniques in [Kin98], [Lem17] and [PS18]. This establishes a connection between the arithmetic p-adic spin L-function constructed in this article with the complex spin L-function, in the spirit of Perrin-Riou conjectures.…”

“…Definition 5.5), interpolating the exponential and dual exponential maps of different twists of the class z π Iw,α . The p-adic L-function thus defined is not a priori related to special values of the complex spin L-function associated to the automorphic form π but, as we point out below, a relation between these objects is established in our recent work [CLRJ19].…”

“…This establishes a connection between the arithmetic p-adic spin L-function constructed in this article with the complex spin L-function, in the spirit of Perrin-Riou conjectures. Moreover, in [CLRJ19], possible generalisations to GSp 2n with n > 3, of the construction of motivic cohomology classes as well as the study of their archimedean regulator are considered. This work can hence be considered as our first attempt devoted to the study of the arithmetic of automorphic forms for symplectic groups.…”

Higher Hida theory and p-adic L-functions for GSp(4)

“…In particular, the previously-known techniques are not sufficient to prove an explicit reciprocity law for the Euler system constructed in [LSZ17] for automorphic representations of GSp 4 ; this is the major obstacle that must be solved in order to use this new Euler system to prove the Bloch-Kato conjecture for automorphic motives attached to this group. The same difficulty arises for several other recently-discovered Euler systems, such as those of [LLZ18,CLRJ19,LSZ19].…”