Units from 5-Torsion on the Jacobian of y=x+1 4 and the Conjectures of Stark and Rubin

Let K k be a normal extension of number fields, and / a character on its Galois group G. Stark's conjectures relate the lead term of the Taylor expansion of the Artin L-series L(s, /) at s=0 to the determinant of a matrix whose entries are linear combinations of logs of absolute values of units in K [StI IV, Tat1]. In the case that G is abelian and the L-series L(s, /) has a zero of order one at s=0, Stark gave a refined conjecture for the precise value of L$(0, /). Stark proved this refined conjecture in the case that k=Q or k is an imaginary quadratic field, making use of cyclotomic and elliptic units [StIV]. Only scant progress has been made in generalizing elliptic units to ``abelian units'' attached to abelian varieties of dimension greater than 1 [BaBo, BoBa, dSG, Gra2, A]. Recently Rubin gave a generalized refined Stark's conjecture for the value of the lead term of L(s, /) at s=0 whenever G is abelian [R]. In this paper we describe a set of abelian units attached to the 5-torsion of the Jacobian of the curve y=x+1 4 which can be used to verify Rubin's conjecture in a case where the L-series has a second order zero at s=0. Stark also questioned what the lead term should be when there is a second order zero [St2]. These units can also be used to affirm his question in this case. In the first section of the paper we recall the various conjectures. In the second section we describe the fields k and K of our example. In Section 3 we discuss the appropriate units, and in Section 4 we describe how to numerically evaluate the L-series. In the fifth section we establish a key equality, and in the last section we explain for this example how to derive the conjectures from the key equality. Finally, the appendix contains a description of the geometry behind the construction of the units. Article ID jnth.1998.2346, available online at http: www.idealibrary.com on