On the real points of an arithmetic quotient of a bounded symmetric domain

Real algebraic curves

“…(2g, T). Conversely, using Weil's criterion [17], it is easy to show that any point in the cone ^ represents the period matrix of a principally polarized abelian variety over R. Fixing lifts H^ of the inequivalent symmetric bilinear forms on a vector space of dimension g over Z/2, we obtain the following (compare Shimura [14]). At the other extreme, if n(A)=l and H=I^ the group r^nis finite of order 2^(g!…”

Topology, Moduli and Automorphisms of Real Algebraic Surfaces

“…(For the similar result in the case of principally polarized Abelian surfaces, see, e.g., [20] and [19]). ‡ For K3and Enriques surfaces the result follows from the global Torelli theorem, see [17] and [5], respectively, or [4], where the proof is given for both the cases.…”

“…First counter examples appear already in the case of curves of genus ≥ 2, see [20]. As a consequence, the moduli space of real objects is extremely rarely the real part of the moduli space of complex ones.…”

Antiholomorphic Involutions of Analytic Families of Abelian Varieties

“…It is proved in [1] that for all q, Hq(U, K*) is canonically isomorphic toHq(T, W). 7. The mapping X: X X W^> E^ constructed in §5 maps X X L onto a subspace Ä of E^.…”

“…After we had finished our dissertations, we learned that Professor Shimura [7] and his student Professor Shih [8] had been led by entirely different considerations to examine the problems of antiholomorphic involutions of abelian varieties and of bounded symmetric domains. They were concerned with the existence of rational points on abelian varieties and on arithmetic varieties, and observed that if such a variety admitted an antiholomorphic involution which had no fixed points, the corresponding real algebraic variety would have no real points and a fortiori no rational points.…”