We construct vertex transitive lattices on products of trees of arbitrary dimension d ≥ 1 based on quaternion algebras over global fields with exactly two ramified places. Starting from arithmetic examples, we find non-residually finite groups generalizing earlier results of Wise, Burger and Mozes to higher dimension. We make effective use of the combinatorial language of cubical sets and the doubling construction generalized to arbitrary dimension.Congruence subgroups of these quaternion lattices yield explicit cubical Ramanujan complexes, a higher dimensional cubical version of Ramanujan graphs (optimal expanders).
We construct a torsion-free arithmetic lattice in PGL2(F2((t))) × PGL2(F2((t))) arising from a quaternion algebra over F2(z). It is the fundamental group of a square complex with universal covering T3 × T3, a product of trees with constant valency 3, which has minimal Euler characteristic. Furthermore, our lattice gives rise to a fake quadric over F2((t)) by means of non-archimedean uniformization.
Abstract. We show that for any given field k and natural number r ≥ 2, every continuous extension of the absolute Galois group Gal k by a finite group is the arithmetic fundamental group of a geometrically connected smooth projective variety over k of dimension r.
By considering the norm of elements in the ring of integers in Q( √ −a), we give an algebraic approach to count the number of integral solutions of diophantine equations of the form x 2 + ay 2 = n where a is a Heegner number or a = 27.
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