Higher regulators, periods and special values of the degree 4 \(L\)-function of \(\text{GSp}(4)\) (Q952113)

In the context of mixed motives and motivic sheaves, one of A. A. Beilinson's famous conjectures relates the value \(L(0,M)\) of the \(L\)-function of a pure motive \(M\) over \(\mathbb Q\) of weight less than \(-2\) to the space of 1-extensions \(\text{Ext}^1(\mathbb Q(0),M)\) between the trivial motive \(\mathbb Q(0)\) and \(M\) in the category of mixed motives over \(\mathbb Q\) [cf. \textit{A. A. Beilinson}, Contemp. Math. 55, 1--34 (1986; Zbl 0609.14006)] Beilinson's construction of suitable higher regulators lead him to a proof of this conjecture for elliptic modular forms, which was later on extended to Hilbert modular forms over a real quadratic number field by \textit{G. Kings} [Duke Math. J. 92, No. 1, 61--127 (1998; Zbl 0962.11024)]. In the short note under review, the author announces and describes some of his recent extensions of Beilinson's ideas to the case of automorphic representations of the symplectic group \(G=\text{GSp}(V_4,\psi)\), where \((V_4,\psi)\) is a four-dimensional symplectic space over \(\mathbb Q\). In contrast to the approaches by Beilinson and Kings, this particular case requires a modified explicit construction of 1-extensions of motives, which is achieved here by using the so-called Eisenstein symbols in motivic cohomology (à la Beilinson) and a more recent cohomological vanishing theorem due to \textit{L. Saper} [Astérisque 298, 319--334 (2005; Zbl 1083.11033)]. As for the precise link between this 1-extension and the special value of the according motivic \(L\)-function, the author applies to his approach some earlier results about \(L\)-functions for the symplectic group \(\text{GSp}_4\) obtained by \textit{I. I. Piatetski-Shapiro} [Pac. J. Math., Spec. Issue, 259--275 (1998; Zbl 1001.11020)] and by \textit{M. Harris} [in: Hida, Haruzo (ed.) et al., Contributions to automorphic forms, geometry, and number theory. Papers from the conference in honor of Joseph Shalika on the occasion of his 60th birthday, Johns Hopkins University, Baltimore, MD, USA, May 14--17, 2002. Baltimore, MD: Johns Hopkins University Press, 331--354 (2004; Zbl 1173.11329)].