Cusp eigenforms and the Hall algebra of an elliptic curve (Q2845014)

Using the theory of Hall algebras and the Langlands correspondence for function fields and \(\mathrm{GL}_n\), the author gives an explicit construction of the cusp eigenforms on an elliptic curve defined over a finite field. As a consequence the Hall algebra of an elliptic curve is presented as an infinite tensor product of simpler algebras. It is proved that all these algebras are specializations of a universal spherical Hall algebra (as defined and studied by \textit{I. Burban} and \textit{O. Schiffmann} [Duke Math. J. 161, No. 7, 1171--1231 (2012; Zbl 1286.16029)] and \textit{O. Schiffmann} and \textit{E. Vasserot} [Compos. Math. 147, No. 1, 188--234 (2011; Zbl 1234.20005)]).NEWLINENEWLINEThe paper under review consists of an introduction, eight sections and an appendix. In the introduction, the author explains the setting and the connections to different fields of mathematics. The main results of the paper are formulated here in a simplified form. In section~1, the notations are introduced, a reminder on the classification of vector bundles on elliptic curves is given. In section~2, the Hall algebra \(\mathbf{H}_X\) and its Drinfeld's double \(\mathbf{DH}_X\) are defined for an elliptic curve \(X\). The action of \(\mathrm{SL}_2(\mathbb Z)\) on \(\mathbf{DH}_X\) is briefly explained. The notion of cuspidality for elements in the Hall algebra is defined here, Hecke operators are introduced. The twisted spherical Hall algebra, which is the main object of study in the paper, is defined in section~3. Section~4 is based on the results from [\textit{M. M. Kapranov}, J. Math. Sci., New York 84, No. 5, 1311--1360 (1997; Zbl 0929.11015)]. A reminder on automorphic forms and Rankin-Selberg \(L\)-functions is presented here. The main results of the paper, Theorem~5.1 and Theorem~5.2, are stated in section~5. In section~6, the author computes the Frobenius eigenvalues for an irreducible \(l\)-adic representation of the fundamental group of \(X\) and the action of the Hecke operators on the corresponding cusp eigenforms. Some technical lemmas are presented in section~7. The main results of the paper are proved in Section~8 using the results from the previous two sections, which are in fact the technical heart of the paper. Some lemmas used in the paper are presented and proven in the appendix.