Detail Elliptic curves and alternating group extensions of the rational numbers Bibliografi Author: Evans, Mark Fraser ; Rohrlich, David E. (Advisor) Topik: MATHEMATICS Bahasa: (EN )    ISBN: 0-599-71801-3     Penerbit: Boston University     Tahun Terbit: 2001     Jenis: Theses - Dissertation Fulltext: 9967149.pdf (0.0B; 0 download) Abstract The inverse Galois problem asks if each finite group G is the Galois group of some extension of the rational numbers $Q$. The modern approach to this problem involves exhibiting a regular G-Galois cover of curves $C\to P1$, defined over $Q$. The corresponding extension of function fields $QC/QP1$ is then Galois with group G. One uses Hilbert's irreducibility theorem to conclude that there exist infinitely many rational points in $P1Q$ which “specialize” to give a Galois extension of $Q$ with group G. In this thesis, we consider the case where $P1$ is replaced by an elliptic curve $E/Q$ with positive Mordell-Weil rank. A theorem of Néron and Serre says that if a group G is perfect then a regular G-Galois cover $C\to E$ defined over $Q$ can be specialized to almost any point in $EQ$ to obtain a Galois extension of $Q$ with group G. We use this theorem to realize the alternating groups $An$ (n $\not\equiv$ 3 mod 6) as Galois groups over $Q$. This is the first time an infinite family of groups has been realized using this variant of the classical Hilbert irreducibility theorem. Opini Anda Klik untuk menuliskan opini Anda tentang koleksi ini! Lihat Sejarah Pengadaan  Konversi Metadata   Kembali

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