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Register a new userThe External Fundamental Group of an Algebraic Number Field
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Authors
T. M. Gendron
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We associate to every algebraic number field a hyperbolic surface lamination and an external fundamental group: the latter a generalization of the fundamental germ that necessarily contains external (not first order definable) elements. The external fundamental group of the rationals is a split extension of the absolute Galois group, that conjecturally contains a subgroup whose abelianization is isomorphic to the idele class group.
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Let $X$ be a smooth projective connected curve of genus $gge 2$ defined over an algebraically closed field $k$ of characteristic $p>0$. Let $G$ be a finite group, $P$ a Sylow $p$-subgroup of $G$ and $N_G(P)$ its normalizer in $G$. We show that if there exists an etale Galois cover $Yto X$ with group $N_G(P)$, then $G$ is the Galois group wan etale Galois cover $mathcal{Y}tomathcal{X}$, where the genus of $mathcal{X}$ depends on the order of $G$, the number of Sylow $p$-subgroups of $G$ and $g$. Suppose that $G$ is an extension of a group $H$ of order prime to $p$ by a $p$-group $P$ and $X$ is defined over a finite field $mathbb{F}_q$ large enough to contain the $|H|$-th roots of unity. We show that integral idempotent relations in the group ring $mathbb{C}[H]$ imply similar relations among the corresponding generalized Hasse-Witt invariants.
Let $f(x)$ be an irreducible polynomial with integer coefficients of degree at least two. Hooley proved that the roots of the congruence equation $f(x)equiv 0mod n$ is uniformly distributed. as a parallel of Hooleys theorem under ideal theoretical setting, we prove the uniformity of the distribution of residues of an algebraic number modulo degree one ideals. Then using this result we show that the roots of a system of polynomial congruences are uniformly distributed. Finally, the distribution of digits of n-adic expansions of an algebraic number is discussed.
We introduce mod 3 triple Milnor invariants and triple cubic residue symbols for certain primes of the Eisenstein number field $mathbb{Q}(sqrt{-3})$, following the analogies between knots and primes. Our triple symbol generalizes both the cubic residue symbol and R{e}deis triple symbol, and describes the decomposition law of a prime in a mod 3 Heisenberg extension of degree 27 over $mathbb{Q}(sqrt{-3})$ with restricted ramification, which we construct concretely in the form similar to R{e}deis dihedral extension over $mathbb{Q}$. We also give a cohomological interpretation of our symbols by triple Massey products in Galois cohomology.
Let $Pi$ be the fundamental group of a smooth variety X over $F_p$. Given a non-Archimedean place $lambda$ of the field of algebraic numbers which is prime to p, consider the $lambda$-adic pro-semisimple completion of $Pi$ as an object of the groupoid whose objects are pro-semisimple groups and whose morphisms are isomorphisms up to conjugation by elements of the neutral connected component. We prove that this object does not depend on $lambda$. If dim X=1 we also prove a crystalline generalization of this fact. We deduce this from the Langlands conjecture for function fields (proved by L. Lafforgue) and its crystalline analog (proved by T. Abe) using a reconstruction theorem in the spirit of Kazhdan-Larsen-Varshavsky. We also formulate two related conjectures, each of which is a reciprocity law involving a sum over all $l$-adic cohomology theories (including the crystalline theory for $l=p$).
We employ methods from homotopy theory to define new obstructions to solutions of embedding problems. By using these novel obstructions we study embedding problems with non-solvable kernel. We apply these obstructions to study the unramified inverse Galois problem. That is, we show that our methods can be used to determine that certain groups cannot be realized as the Galois groups of unramified extensions of certain number fields. To demonstrate the power of our methods, we give an infinite family of totally imaginary quadratic number fields such that $text{Aut}(text{PSL}(2,q^2))$ for $q$ an odd prime power, cannot be realized as an unramified Galois group over $K,$ but its maximal solvable quotient can. To prove this result, we determine the ring structure of the etale cohomology ring $H^*(text{Spec }mathcal{O}_K;mathbb{Z}/ 2mathbb{Z})$ where $mathcal{O}_K$ is the ring of integers of an arbitrary totally imaginary number field $K.$
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