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Uniform construction of non-singular ternary and quaternary homogeneous forms violating the local-global principle (本文)

平川, 義之輔 慶應義塾大学

2020.09.21

概要

In number theory, rational points on algebraic varieties have been studied by many people from the time of Diophantus of Alexandria. In particular, it is an important problem to determine whether a given algebraic variety has a rational point or not. It is obvious that an algebraic variety has no rational points if it has no local points. However, it is a very deep problem to determine whether an algebraic variety with local points has a rational point or not. In fact, many SPORADIC examples of algebraic varieties are known to have local points but no rational points, i.e., violate the local-global principle. In this thesis, we give two kinds of conjectural but UNIFORM constructions of algebraic varieties which violate the local-global principle. More precisely, our UNIFORM construction of algebraic varieties of specified dimension means an algorithm to obtain a non-singular projective hypersurface of the projective space PN of every sufficiently large degree n 1 which violates the local-global principle (N = 3 or 4 in this thesis).

In the first part of this thesis, we construct a family of non-singular curves of odd degree n > 3 which violate the local-global principle under a certain mild hypothesis on the degree n. In fact, we conjecture that EVERY odd integer n > 3 satisfies our hypothesis, and we prove that our construction actually produces non-singular curves which violate the local-global principle for at least 90% (if ordered by heights) of the odd degrees n > 3. Moreover, for each fixed n, our construction gives infinitely many algebraic curves of degree n which are not geometrically isomorphic to each other. Our construction gives a vast generalization of Fujiwara’s quintic curve (1972).

The contents of the first part are based on the joint work [28] with Yosuke Shimizu, and the author contributed to the major part including the formulations of the theorems and the details of their proofs.

In the second part of this thesis, for every odd prime number p > 3 such that p ≡ 3 (mod 4), we construct a family of non-singular projective surfaces of degrees n = (p −1)/2 which violate the local-global principle. In fact, we construct non-singular projective surfaces of both GENERAL odd and even degrees n ≥ 3 which violate the local-global principle under a certain mild hypothesis that the arithmetic progression {1 + nr}r∈N contains a sufficiently small prime number. Our construction gives a vast generalization of (modified) Swinnerton-Dyer’s cubic surface (1962). The contents of the second part is based on [26] and its generalization.

Although there is a vast literature on the violation of the local-global principle, there is no known such a uniform construction of non-singular homogeneous forms violating the local-global principle before the results in this thesis.

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