Numerical verification of a conjecture of Harris and Venkatesh

Abstract

In [Ven16], Venkatesh conjectures a relation between the action of derived Hecke operators and an action by motivic cohomology groups. In [HV18], Harris and Venkatesh reformulate this conjecture for the case of cohomology groups of coherent sheaves associated to modular forms of weight one. We refer to the latter as the HV conjecture. Recently, the work of Darmon, Harris, Rotger and Venkatesh in [DHRV] proves that this conjecture holds for dihedral weight one forms. The following article focuses therefore on the case of exotic forms, describes methods to compute explicitly all key ingredients appearing in the HV conjecture and provides further numerical evidence for it.