Jacobians with complex multiplication

Cyclic Coverings, Calabi-Yau Manifolds and Complex Multiplication

“…All new examples C → P n of the preceding remark have a variation V of Hodge structures similar to the examples of J. de Jong and R. Noot [25], and of E. Viehweg and K. Zuo [47], which we call pure (1, n) − V HS. Let Hg(V) denote the generic Hodge group of V and let K denote an arbitrary maximal compact subgroup of Hg ad (V)(R).…”

“…In [25] J. de Jong and R. Noot gave counterexamples for g = 4 and g = 6 to the conjecture above. In [47] E. Viehweg and K. Zuo gave an additional counterexample for g = 6.…”

“…J. de Jong and R. Noot [25] resp., E. Viehweg and K. Zuo [47] have given counterexamples of families with infinitely many CM fibers for g = 4, 6. In our lists here we have counterexamples for g = 5, 7.…”

“…The case with m = 3 is one of the examples of a family of covers onto P 1 with a dense set of CM fibers by J. de Jong and R. Noot [25]. We must a bit work to give a suitable modified construction of a Viehweg-Zuo tower for this example.…”

Rational Conformal Field Theories and Complex Multiplication

“…Even though there is a dense set of point in the moduli of complex structure of the tori admitting complex multiplication this may not hold true for the measure zero subspace of it corresponding to those coming from Riemann surfaces. Unlikely as this sounds, indeed there is evidence and a standing mathematical conjecture by Coleman [41] (see also [42] (4) respectively. On the other hand, for the case of the one parameter family of quintic threefolds, complex multiplication is conjectured to occur at most at finite number of points.…”