Unimodular lattices in dimensions 14 and 15 over the Eisenstein integers

All indecomposable unimodular hermitian lattices in dimensions 14 and 15 over the ring of integers in Q( √ −3) are determined. Precisely one lattice in dimension 14 and two lattices in dimension 15 have minimal norm 3. In 1978 W. Feit [10] classified the unimodular hermitian lattices of dimensions up to 12 over the ring Z[ω] of Eisenstein integers, where ω is a primitive third root of unity. These lattices all have roots, that is, vectors of norm 2. In dimension 13, for the first time a unimodular lattice without roots appears [1, 3]. In [2] the unimodular lattices in dimension 13 are completely classified. The root-free lattice turns out to be unique. It has minimal norm 3, and its automorphism group is isomorphic to the group Z6 × PSp6(3) of order 2.3.5.7.13. The remaining lattices all have roots; the rank of the root system is 12 in all cases. In this paper, we classify the unimodular lattices in dimensions 14 and 15. There are exactly 58, respectively 259 classes of indecomposable lattices in these dimensions. Below, we list their root systems and the orders of their automorphism groups. Gram matrices for all lattices are available electronically via www.mathematik.uni-dortmund.de/~scharlau There is only one root-free unimodular lattice of rank 14, and there are two root-free unimodular lattices of rank 15. The lattices without roots have minimal norm 3; they are extremal as introduced for unimodular Eisenstein lattices in [8], Chapter 10.7. They give rise to 3-modular extremal Z-lattices in twice the dimension, as defined by Quebbemann in [17]. See [8, 19, 20] for more information on extremal and modular lattices and their relation to modular forms. In this context, the lattices classified in this paper can be considered as complex structures on (extremal) 3-modular lattices. The question for existence, uniqueness, and possibly a full classification of extremal modular lattices has been an ongoing challenge, both computationally and theoretically, after the appearance of the influential paper [17]. Let V be a vector space over Q( √ −3) with a positive definite hermitian product (, ). A lattice L in V is a finitely generated Z[ω]-module contained in V such that L contains a basis of V and (x, y) ∈ Z[ω] for all x, y ∈ L. More precisely, one Received by the editor October 19, 2007 and, in revised form, January 2, 2008. 2000 Mathematics Subject Classification. Primary 11H06, 11H56; Secondary 11E39, 11H71, 11F11.