On {$q$}-clan geometry, {$q=2\sp e$}}

The Fundamental Theorem of q-Clan Geometry

Isomorphisms Between Subiaco q–Clan Geometries

For q = 2e, e ≥ 4, the Subiaco construction introduced in [2] provides one q–clan, one flock, and for e 6≡ 2 (mod 4), one oval in PG(2, q). When e ≡ 2 (mod 4), there are two inequivalent ovals. The associated generalised quadrangle of order (q, q) has a complete automorphism group G of order 2e(q − 1)q. For each Subiaco oval O there is a group of collineations of PG(2, q) induced by a subgroup ...

Integral calculus on E q ( 2 ) ?

The complexes of integral forms on the quantum Euclidean group Eq(2) and the quantum plane are defined and their isomorphisms with the corresponding de Rham complexes are established.

On the Joint Distribution Of Selφ(E/Q) and Selφ(E ′/Q) in Quadratic Twist Families

Strikingly, this is not true when E has a single rational point of order two. In this case E has a degree two isogeny φ : E → E ′ and an associated Selmer group Selφ(E/Q). Work of Xiong shows that if E does not have a cyclic 4-isogeny de ned over Q(E[2]), then the distribution of the ranks of Selφ(E /Q) as d varies among the squarefree integers less than X tends to the distribution Max{0,N (0, ...

On the geometry of the exceptional group G 2( q ), q even

We study some geometry of the exceptional group G2(q), q even, in terms of symplectic geometric configurations in the projective space PG(5, q). Using the spin representation of Sp6(q), we obtain an alternative description of the Split Cayley hexagon H(q) related to G2(q). We also give another geometric proof of the maximality of G2(q), q even, in PSp6(q).