Research on the descent problem of base fields of open affine algebraic plane curves

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Let k be a field and let K be an algebraic closure of k. A commutative k-algebra A is called a k-form of an affine line if the K-algebra obtained by an extension of the base field k of A to K is K-isomorphic to an affine line over K. The main purpose of this project is to study k-algebraic structures of an arbitrary k-form A of an affine line. Only a sporadic examples of non trivial (i.e. non polynomial) k-forms of the affine line have been known before the start of the project. During a period of two years for the project the head investigator obtained the following results. First, we found a new series of non trivial k-forms of the affine line, and next proved any k-form A of the affine line is k-isomorphic to one of those examples. In particular, such a k-form A is given as a residue of a polynomial ring over k in three variables modulo a prime ideal P generated by three elements which can be explicitly written (Structure theorem of k-forms of affine line). As a corollary of this theorem, we have the following: A k-form A of the affine line is generated by two elements over k if and only if the prime ideal P corresponding A defined above is an ideal theoretic complete intersection. Using these results we can also find all groups (up to isomorphisms) obtained as the k-automorphism group of some k-form of an affine line. For the proof of these results, we use a Galois theory for extensions of rings, which the head investigator has been proved. A survey of these results can be found in [T. Asanuma, On A^1 -forms, Memoirs of the faculty of education Toyama University No.56 (2002)43-51].