Notes on the minimal number of ramified primes in some l-extensions of Q

Nomura, Akito

doi:10.1007/s00013-008-2586-z

Cited by 3 publications

(2 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Thus the minimal ramification problem has an affirmative solution for odd order ℓ-groups G of nilpotency class 2. A. Nomura [6] has refined Plans' result and proved that the minimal ramification problem has an affirmative solution for 3-groups of order ≤ 3 5 .…”

Section: Introductionmentioning

confidence: 94%

On the minimal ramification problem for ℓ-groups

Kisilevsky¹,

Sonn²

2010

Compositio Math.

View full text Add to dashboard Cite

Let be a prime number. It is not known whether every finite -group of rank n 1 can be realized as a Galois group over Q with no more than n ramified primes. We prove that this can be done for the (minimal) family of finite -groups which contains all the cyclic groups of -power order and is closed under direct products, (regular) wreath products and rank-preserving homomorphic images. This family contains the Sylow -subgroups of the symmetric groups and of the classical groups over finite fields of characteristic not . On the other hand, it does not contain all finite -groups.

show abstract

Section: Introductionmentioning

confidence: 94%

On the minimal ramification problem for ℓ-groups

Kisilevsky¹,

Sonn²

2010

Compositio Math.

View full text Add to dashboard Cite

show abstract

“…For solvable groups G, one can use Approach II, to obtain upper bounds on m(G) and for some subclasses of solvable groups, the full conjecture, see [2,15,16,20,23,24]. For example, Kisilevsky, Neftin, and Sonn [15] establish the conjecture for semi-abelian p-groups.…”

Section: The Minimal Ramification Problemmentioning

confidence: 99%

Sieves and the Minimal Ramification Problem

Bary-Soroker¹,

Schlank²

2018

J. Inst. Math. Jussieu

View full text Add to dashboard Cite

The minimal ramification problem may be considered as a quantitative version of the inverse Galois problem. For a nontrivial finite group $G$, let $m(G)$ be the minimal integer $m$ for which there exists a $G$-Galois extension $N/\mathbb{Q}$ that is ramified at exactly $m$ primes (including the infinite one). So, the problem is to compute or to bound $m(G)$.In this paper, we bound the ramification of extensions $N/\mathbb{Q}$ obtained as a specialization of a branched covering $\unicode[STIX]{x1D719}:C\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$. This leads to novel upper bounds on $m(G)$, for finite groups $G$ that are realizable as the Galois group of a branched covering. Some instances of our general results are: $$\begin{eqnarray}1\leqslant m(S_{k})\leqslant 4\quad \text{and}\quad n\leqslant m(S_{k}^{n})\leqslant n+4,\end{eqnarray}$$ for all $n,k>0$. Here $S_{k}$ denotes the symmetric group on $k$ letters, and $S_{k}^{n}$ is the direct product of $n$ copies of $S_{k}$. We also get the correct asymptotic of $m(G^{n})$, as $n\rightarrow \infty$ for a certain class of groups $G$.Our methods are based on sieve theory results, in particular on the Green–Tao–Ziegler theorem on prime values of linear forms in two variables, on the theory of specialization in arithmetic geometry, and on finite group theory.

show abstract

On semiabelian p-groups

Neftin

2011

Journal of Algebra

View full text Add to dashboard Cite

Notes on the minimal number of ramified primes in some l-extensions of Q

Cited by 3 publications

References 5 publications

On the minimal ramification problem for ℓ-groups

On the minimal ramification problem for ℓ-groups

Sieves and the Minimal Ramification Problem

On semiabelian p-groups

Contact Info

Product

Resources

About