A note on trilinear forms for reducible representations and Beilinson’s conjectures

Let F be a non-Archimedean local field, and πi (i = 1, 2, 3) irreducible admissible representations of G = GL2(F ), such that the product of their central characters is trivial. In [8], Prasad shows that there exists, up to a scalar factor, at most one G-invariant linear form on π1 ⊗ π2 ⊗ π3, and determines exactly when such a form exists. These results have been used by Harris and Kudla [6] in the study of the triple product L-function attached to three cuspidal automorphic representations of GL2 of a global field. In this note we consider the case when πi is permitted to be a reducible principal series representation, whose unique irreducible subspace is infinitedimensional. It is relatively trivial to extend Prasad’s results to cover these cases. The interest in so doing is global. In [1] Beilinson constructs certain subspaces of the motivic cohomology of the product of two modular curves using modular units. His construction can be interpreted as a certain invariant trilinear form on π⊗π′⊗π′′ taking values in motivic cohomology: here π, π are weight 2 cuspidal (irreducible) representations of GL2 of the finite adeles of Q, and π is the space of weight 2 holomorphic Eisenstein series (which is highly reducible). The regulators of these elements of motivic cohomology can be computed as special values of Rankin double product L-functions attached to π and π, and Beilinson’s calculation of the regulator, together with his general conjectures, predict that these subspaces are one-dimensional. The main aim of the present note is to verify this prediction unconditionally (Theorem 3.1 below). Acknowledgements. The authors grateful acknowledge the support of the European Commision through the TMR Network Arithmetic Algebraic Geometry, which enabled this collaboration to take place. The second author also wishes to thank the EPSRC for support during his stay at the Isaac Newton Institute in 1998, when some of the work was done.