Number Theory and Geometry related to Algebraic Groups

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(1) Yukie investigated zeta functions associated with prehomogeneous vector spaces and obtained an estimate of the number of quintic fields with discriminant less than or equal to X.(2) Hanamura investigated mixed motifs and obtained a result on the Kunneth formula for modular varieties.(3) Ishida established a theory relating ideal theory (of rings) to rational and real fans for toric varieties. Also he found a real fan analogue of blow-ups of algebraic varieties. Especially for a finite number of blow-ups, he confirmed similarities with the case of algebraic varieties. In algebraic geometry, Zariski-Riemann topology can be defined on the set of valuation rings of the function fields. He defined this notion for rational and real fans using the set of all additive orders on free modules and real vector spaces. Using this Zariski-Riemann topology, he proved the existence of the compactification of toric varieties.(4) Hara generalized the notion of tight closure of ideals I of rings R of … More positive characteristic to "I-tight closure" and proved various properties for the generalized determinantal ideal τ(I), thus made a foundation of the theory. Using this method, he applied to the proof of a special case of the Fujita conjecture on the global generation of adjoint bundles and to a new proof of the Ein-Lazarsfeld-Smith comparison theorem on the symbolic power of ideals of regular local rings.(5) Nakamura : If a CM elliptic curve is isogenous to all its Galois conjugate, it is called a Q-curve and has important properties. He classified all Q-curves over the absolute class field of a given imaginary quadratic field. It is well-known that the torsion group of an elliptic curve over a number field is finite. He investigated how the torsion changes among isogenous elliptic curves.(6) Ogata investigated projective normality and the degrees of the generators of the defining ideals of toric varieties by very ample line bundles, and opbtained some criteria of projective normality and some estimates of the degrees of the defining ideals. Less