A study on theoretical and practical aspects of Weil-restrictions of varieties (Q2763624)

From the foreword: Let \(K|k\) be a finite Galois extension of fields, \(X'\) a quasi-projective \(K\)-variety. Then there exists a quasi-projective \(k\)-variety \(W\) which has in particular the properties that \(X' (K)\simeq W(k)\) and \(W_K=W\otimes_kK\) is a product of Galois-conjugates of \(X'\). \(W\) is called the Weil restriction of \(X'\) with respect to \(K|k)\). After Weil-restrictions where successfully studied to solve problems of ``pure mathematics'' for decades, a new direction of research was shown by Frey in a talk in 1998. He suggested to use Weil-restrictions of elliptic curves both as a tool to construct as well as to break discrete-logarithm problems (D-L problems).NEWLINENEWLINENEWLINEIn this work we study Weil-restrictions of varieties both from a pure as well as from an applied point of view. In particular, we show how questions on Weil-restrictions of abelian varieties motivated by cryptoanalytical applications can often be proven directly from the defining functorial properties.NEWLINENEWLINENEWLINEIn chapter one, we first give basic definitions related to Weil-restrictions of varieties and schemes. We study the Weil-restriction of a projective variety \(X'/K\) with a rational point with respect to a Galois field extension \(K|k\), we analyze the Picard functor of the Weil-restriction \(W\) and we derive the structure of the endomorphism ring of Weil-restrictions of an abelian variety over finite fields.NEWLINENEWLINENEWLINEFor the second chapter, if \(A\) is an elliptic curve \(E\), then \(W\) is isogenous to the product of \(E\) and an abelian variety \(N\) called its trace-zero-hypersurface. We study the Néron-Severi group of \(N\) and in particular the polarizations of \(N\).NEWLINENEWLINENEWLINEThe third chapter is entirely devoted to cryptoanalytical applications.