The (4,p)-arithmetic hyperbolic lattices, p ≥ 2, in three dimensions. G.J. Martin, K. Salehi and Y. Yamashita * Abstract We identify the finitely many arithmetic lattices Γ in the orienta- tion preserving isometry group of hyperbolic 3-space H 3 generated by an element of order 4 and and element of order p ≥ 2. Thus Γ has a presentation of the form Γ ∼ = hf,g : f 4 = g p = w(f,g)= ··· =1i We find that necessarily p ∈{2, 3, 4, 5, 6, ∞}, where p = ∞ denotes that g is a parabolic element, the total degree of the invariant trace field kΓ= Q({tr 2 (h): h ∈ Γ}) is at most 4, and each orbifold is either a two bridge link of slope r/s surgered with (4, 0), (p, 0) Dehn surgery (possibly a two bridge knot if p = 4) or a Heckoid group with slope r/s and w(f,g)=(w r/s ) r with r ∈{1, 2, 3, 4}. We give a discrete and faithful representation in PSL(2, C) for each group and identify the associated number theoretic data. 1 Introduction In this paper we continue our long running programme to identify (up to con- jugacy) all the finitely many arithmetic lattices Γ in the group of orientation preserving isometries of hyperbolic 3-space Isom + (H 3 ) ∼ = PSL(2, C) gener- ated by two elements of finite order p and q, we also allow p = ∞ or q = ∞ to mean that a generator is parabolic. In [26] it is proved that there are only finitely many such lattices. In fact it is widely expected that there are only * Research of all authors supported in part by the NZ Marsden Fund. 1 arXiv:2206.14174v1 [math.GT] 28 Jun 2022