Arithmetic on a quintic threefold

This paper investigates some aspects of the arithmetic of a quintic threefo ld in Pr-4 with double points singularities. Particular emphasis is given t o the study of the L-function of the Galois action rho on the middle l-adic cohomology. The main result of the paper is the proof of the existence of a Hilbert modular form of weight (2,4) and conductor 30, on the real quadra tic field whose associated (continuous system of) Galois representation(s) appears to be the most likely candidate to induce the scalar extension rho circle times Q(l)(root5). The Hilbert modular form is interpreted as a comm on eigenvector of the Brandt matrices which describe the action of the Heck e operators on a space of theta series associated to the norm form of a qua ternion algebra over Q(root5) and a related Eichler order.