Restricted Ramification Research Articles

Correction to “Isomorphisms of Galois groups of number fields with restricted ramification”

Correction to “Isomorphisms of Galois groups of number fields with restricted ramification”

On the p-Rank of Class Groups of p-Extensions

Abstract We prove a local–global principle for the embedding problems of global fields with restricted ramification. By this local–global principle, for a global field $k$, we use only the local information to give a presentation of the maximal pro-$p$ Galois group of $k$ with restricted ramification, when some Galois cohomological conditions are satisfied. For a Galois $p$-extension $K/k$, we use our presentation result for $k$ to study the structure of pro-$p$ Galois groups of $K$. Then for $k={{\mathbb{Q}}}$ and $k={{\mathbb{F}}}_{q}(t)$ with $p\nmid q$, we give upper and lower bounds for the rank of $p$-torsion subgroup of the class group of $K$, and these bounds depend on the structure of the Galois group and the inertia subgroups of $K/k$. Finally, we study the $p$-rank of class groups of cyclic $p$-extensions of ${{\mathbb{Q}}}$ and the $2$-rank of class groups of multiquadratic extensions of ${{\mathbb{Q}}}$, for a fixed ramification type.

Brauer-Manin obstructions on hyperelliptic curves

We describe a practical algorithm for computing Brauer-Manin obstructions to the existence of rational points on hyperelliptic curves defined over number fields. This offers advantages over descent based methods in that its correctness does not rely on rigorous class and unit group computations of large degree number fields. We report on experiments showing it to be a very effective tool for deciding existence of rational points: Among a random samples of curves over Q of genus at least 5 we were able to decide existence of rational points for over 99% of curves. We also demonstrate its effectiveness for high genus curves, giving an example of a genus 50 hyperelliptic curve with a Brauer-Manin obstruction to the Hasse Principle. The main theoretical development allowing for this algorithm is an extension of the descent theory for abelian torsors to a framework of torsors with restricted ramification.

Isomorphisms of Galois groups of number fields with restricted ramification

AbstractLet K be a number field and S a set of primes of K. We write for the maximal extension of K unramified outside S and for its Galois group. In this paper, we answer the following question under some assumptions: “For , let be a number field, a (sufficiently large) set of primes of and an isomorphism. Is σ induced by a unique isomorphism between and ?” Here, the main assumption is about the Dirichlet density of .

The Neukirch–Uchida theorem with restricted ramification

Abstract Let K be a number field and S a set of primes of K. We write K S / K {K_{S}/K} for the maximal extension of K unramified outside S and G K , S {G_{K,S}} for its Galois group. In this paper, we prove the following generalization of the Neukirch–Uchida theorem under some assumptions: “For i = 1 , 2 {i=1,2} , let K i {K_{i}} be a number field and S i {S_{i}} a set of primes of K i {K_{i}} . If G K 1 , S 1 {G_{K_{1},S_{1}}} and G K 2 , S 2 {G_{K_{2},S_{2}}} are isomorphic, then K 1 {K_{1}} and K 2 {K_{2}} are isomorphic.” Here the main assumption is that the Dirichlet density of S i {S_{i}} is not zero for at least one i. A key step of the proof is to recover group-theoretically the l-adic cyclotomic character of an open subgroup of G K , S {G_{K,S}} for some prime number l.

On pro-p-extensions of number fields with restricted ramification over intermediate [formula omitted]-extensions

We consider pro-p-extensions of a number field in which the ramification and decomposition are restricted over an intermediate Zp-extension. For such a maximal pro-p-extension under a certain restriction condition, we obtain a Koch type presentation of the pro-p Galois group by generators and relations. As its applications, we also obtain an extension of triple cubic residue symbols, a new example of arithmetical mild pro-p group, and another proof of a criterion of finiteness of certain Iwasawa modules.

Level compatibility in the passage from modular symbols to cup products

For a positive integer M and an odd prime p there exists a map $$\varpi _M$$ from the first homology group of the modular curve $$X_1(M)$$ with $${\mathbb {Z}}_p$$ -coefficients to a second Galois cohomology group over $${\mathbb {Q}}(\mu _M)$$ with restricted ramification and $${\mathbb {Z}}_p(2)$$ -coefficients. This map takes Manin symbols to certain cup products of cyclotomic M-units. It has previously been shown that if $$p\mid M$$ and $$p\ge 5$$ , then $$\varpi _{Mp}$$ and $$\varpi _M$$ are compatible via the map of homology induced by the quotient $$X_1(Mp)\rightarrow X_1(M)$$ and corestriction from $${\mathbb {Q}}(\mu _{Mp})$$ to $${\mathbb {Q}}(\mu _M)$$ . We show that for a prime $$\ell \not \mid M$$ , $$\ell \ne p\ge 5$$ , the maps $$\varpi _{M\ell }$$ and $$\varpi _M$$ are again compatible under a certain combination of the two standard degeneracy maps from level $$M\ell $$ to level M and corestriction.

On the Shafarevich group of restricted ramification extensions of number fields in the tame case

Let $K$ be a number field and $S$ a finite set of places of $K$. We study the kernels $\Sha_S$ of maps $H^2(G_S,\fq_p) \rightarrow \oplus_{v\in S} H^2(\G_v,\fq_p)$. There is a natural injection $\Sha_S \hookrightarrow \CyB_S$, into the dual $\CyB_S$ of a certain readily computable Kummer group $V_S$, which is always an isomorphism in the wild case. The tame case is much more mysterious. Our main result is that given a finite $X$ coprime to $p$, there exists a finite set of places $S$ coprime to $p$ such that $\Sha_{S\cup X} \stackrel{\simeq}{\hookrightarrow} \CyB_{S\cup X} \stackrel{\simeq}{\twoheadleftarrow} \CyB_X \hookleftarrow \Sha_X$. In particular, we show that in the tame case $\Sha_Y$ can {\it increase} with increasing $Y$. This is in contrast with the wild case where $\Sha_Y$ is nonincreasing in size with increasing $Y$.

On the Structure of the Galois Group of the Maximal Pro-$p$ Extension with Restricted Ramification over the Cyclotomic $\mathbb{Z}_p$-extension

Let $k_\infty$ be the cyclotomic $\mathbb{Z}_p$-extension of an algebraic number field $k$. We denote by $S$ a finite set of prime numbers which does not contain $p$, and $S(k_\infty)$ the set of primes of $k_\infty$ lying above $S$. In the present paper, we will study the structure of the Galois group $\mathcal{X}_S (k_\infty)$ of the maximal pro-$p$ extension unramified outside $S (k_\infty)$ over $k_\infty$. We mainly consider the question whether $\mathcal{X}_S (k_\infty)$ is a non-abelian free pro-$p$ group or not. In the former part, we treat the case when $k$ is an imaginary quadratic field and $S = \emptyset$ (here $p$ is an odd prime number which does not split in $k$). In the latter part, we treat the case when $k$ is a totally real field and $S \neq \emptyset$.

On pro-𝑝 link groups of number fields

As an analogue of a link group, we consider the Galois group of the maximal pro- p p -extension of a number field with restricted ramification which is cyclotomically ramified at p p , i.e., tamely ramified over the intermediate cyclotomic Z p \mathbb Z_p -extension of the number field. In some basic cases, such a pro- p p Galois group also has a Koch type presentation described by linking numbers and mod 2 2 Milnor numbers (Rédei symbols) of primes. Then the pro- 2 2 Fox derivative yields a calculation of Iwasawa polynomials analogous to Alexander polynomials.

On mod 3 triple Milnor invariants and triple cubic residue symbols in the Eisenstein number field

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 Redei’s 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 Redei’s dihedral extension over $$\mathbb {Q}$$ . We also give a cohomological interpretation of our symbols by triple Massey products in Galois cohomology.

Tame pro-2 Galois groups and the basic ℤ₂-extension

For a number field, we consider the Galois group of the maximal tamely ramified pro-2-extension with restricted ramification. Providing a general criterion for the metacyclicity of such Galois groups in terms of 2-ranks and 4-ranks of ray class groups, we classify all finite sets of odd prime numbers such that the maximal pro-2-extension unramified outside the set has prometacyclic Galois group over the Z 2 \mathbb Z_2 -extension of the rationals. The list of such sets yields new affirmative examples of Greenberg’s conjecture.

A Hermite–Minkowski type theorem of varieties over finite fields

We show the finiteness of étale coverings of a variety over a finite field with given degree whose ramification bounded along an effective Cartier divisor. The proof is an application of P. Delgine's theorem (H. Esnault and M. Kerz, 2012) [4] on a finiteness of l-adic sheaves with restricted ramification. By applying our result to a smooth curve over a finite field, we obtain a function field analogue of the classical Hermite–Minkowski theorem.

Higher Massey products in the cohomology of mild pro-p-groups

Translating results due to J. Labute into group cohomological language, A. Schmidt proved that a finitely presented pro-p-group G is mild and hence of cohomological dimension cdG=2 if H1(G,Fp)=U⊕V as Fp-vector space and the cup-product H1(G,Fp)⊗H1(G,Fp)→H2(G,Fp) maps U⊗V surjectively onto H2(G,Fp) and is identically zero on V⊗V. This has led to important results in the study of p-extensions of number fields with restricted ramification, in particular in the case of tame ramification. In this paper, we extend Labute's theory of mild pro-p-groups with respect to weighted Zassenhaus filtrations and prove a generalization of the above result for higher Massey products which allows to construct mild pro-p-groups with defining relations of arbitrary degree. We apply these results to one-relator pro-p-groups and obtain some new evidence of an open question due to Serre.

On 2-extensions of the rationals with restricted ramification

On 2-extensions of the rationals with restricted ramification

Quaternion extensions with restricted ramification

In any normal number field having $Q_8$, the quaternion group of order $8$, as Galois group over the rationals, at least two finite primes must ramify. The classical example by Dedekind of such a field is extraordinary in that it is totally real and only

Characterization of Arithmetical Equivalence of Number Fields by Galois Groups with Restricted Ramification

We will give a characterization of arithmetical equivalence of number fields in terms of certain associated families of Galois groups with restricted ramification.

Nekovár duality over p-adic Lie extensions of global fields

Tate duality is a Pontryagin duality between the i th Galois cohomology group of the absolute Galois group of a local field with coefficents in a finite module and the (2-i) th cohomology group of the Tate twist of the Pontryagin dual of the module. Poitou-Tate duality has a similar formulation, but the duality now takes place between Galois cohomology groups of a global field with restricted ramification and compactly-supported cohomology groups. Nekovár proved analogues of these in which the module in question is a finitely generated module T over a complete commutative local Noetherian ring R with a commuting Galois action, or a bounded complex thereof, and the Pontryagin dual is replaced with the Grothendieck dual T^* , which is a bounded complex of the same form. The cochain complexes computing the Galois cohomology groups of T and T^*(1) are then Grothendieck dual to each other in the derived category of finitely generated R -modules. Given a p -adic Lie extension of the ground field, we extend these to dualities between Galois cochain complexes of induced modules of T and T^*(1) in the derived category of finitely generated modules over the possibly noncommutative Iwasawa algebra with R -coefficients.

Rational Functions with a Unique Critical Point

Over an algebraically closed field of positive characteristic, there exist rational functions with only one critical point. We give an elementary characterization of these functions in terms of their continued fraction expansions. Then we use this tool to discern some of the basic geometry of the space of unicritical rational functions, as well as its quotients by the SL(2)-actions of conjugation and postcomposition. We also give an application to dynamical systems with restricted ramification defined over non-Archimedean fields of positive residue characteristic.

Heun equations coming from geometry

We give a list of Heun equations which are Picard-Fuchs associated to families of algebraic varieties. Our list is based on the classification of families of elliptic curves with four singular fibers done by Herfurtner. We also show that pull-backs of hypergeometric functions by rational Belyi functions with restricted ramification data give rise to Heun equations.