On projective linear groups over finite fields as Galois groups over the rational numbers
نویسنده
چکیده
Ideas and techniques from Khare’s and Wintenberger’s article on the proof of Serre’s conjecture for odd conductors are used to establish that for a fixed prime l infinitely many of the groups PSL2(Flr ) (for r running) occur as Galois groups over the rationals such that the corresponding number fields are unramified outside a set consisting of l, the infinite place and only one other prime.
منابع مشابه
Conditional Results on the Birational Section Conjecture over Small Number Fields
In the present paper, we give necessary and sufficient conditions for a birational Galois section of a projective smooth curve over either the field of rational numbers or an imaginary quadratic field to be geometric. As a consequence, we prove that, over such a small number field, to prove the birational section conjecture for projective smooth curves, it suffices to verify that, roughly speak...
متن کاملRecent Advances in the Langlands Program
The Langlands Program has emerged in the late 60’s in the form of a series of far-reaching conjectures tying together seemingly unrelated objects in number theory, algebraic geometry, and the theory of automorphic forms [L1]. To motivate it, consider the old question from number theory: what is the structure of the Galois group Gal(Q/Q) of the field Q of rational numbers, i.e., the group of aut...
متن کاملProjective Pairs of Profinite Groups
We generalize the notion of a projective profinite group to a projective pair of a profinite group and a closed subgroup. We establish the connection with Pseudo Algebraically Closed (PAC) extensions of PAC fields: Let M be an algebraic extension of a PAC field K. Then M/K is PAC if and only if the corresponding pair of absolute Galois groups (Gal(M),Gal(K)) is projective. Moreover any projecti...
متن کاملMotifs, L-Functions, and the K-Cohomology of Rational Surfaces over Finite Fields
Let X be a smooth projective variety over a field. Let ~f~ denote the Zariski sheaf associated to the presheaf U ~-, Kj(F(U, Cv)) of Quillen K-groups. The collection of Zariski cohomology groups H~(X, ~Fj) will be referred to as the K-cohomology of X. The groups contain a great deal of information about the geometry of X. For example, Bloch's formula [11] says that Hi(X, ~ ) is isomorphic to th...
متن کاملThe inverse Galois problem over formal power series fields
Introduction The inverse Galois problem asks whether every finite group G occurs as a Galois group over the field Q of rational numbers. We then say that G is realizable over Q. This problem goes back to Hilbert [Hil] who realized Sn and An over Q. Many more groups have been realized over Q since 1892. For example, Shafarevich [Sha] finished in 1958 the work started by Scholz 1936 [Slz] and Rei...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2006