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Register a new userFinite non-metabelian Schur sigma-Galois groups of class field towers
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Daniel C. Mayer
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For each odd prime p>=5, there exist finite p-groups G with derived quotient G/D(G)=C(p)xC(p) and nearly constant transfer kernel type k(G)=(1,2,...,2) having two fixed points. It is proved that, for p=7, this type k(G) with the simplest possible case of logarithmic abelian quotient invariants t(G)=(11111,111,21,21,21,21,21,21) of the eight maximal subgroups is realized by exactly 98 non-metabelian Schur sigma-groups S of order 7^11 with fixed derived length dl(S)=3 and metabelianizations S/D(D(S)) of order 7^7. For p=5, the type k(G) with t(G)=(2111,111,21,21,21,21) leads to infinitely many non-metabelian Schur sigma-groups S of order at least 5^14 with unbounded derived length dl(S)>=3 and metabelianizations S/D(D(S)) of fixed order 5^7. These results admit the conclusion that d=-159592 is the first known discriminant of an imaginary quadratic field with 7-class field tower of precise length L=3, and d=-90868 is a discriminant of an imaginary quadratic field with 5-class field tower of length L>=3, whose exact length remains unknown.
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It is shown that the real class field towers are always finite. The proof is based on Castelnuovos theory of the algebraic surfaces and a functor from such surfaces to the Etesi C*-algebras.
We describe examples motivated by the work of Serre and Abhyankar.
Let $K$ be the function field of a smooth, irreducible curve defined over $overline{mathbb{Q}}$. Let $fin K[x]$ be of the form $f(x)=x^q+c$ where $q = p^{r}, r ge 1,$ is a power of the prime number $p$, and let $betain overline{K}$. For all $ninmathbb{N}cup{infty}$, the Galois groups $G_n(beta)=mathop{rm{Gal}}(K(f^{-n}(beta))/K(beta))$ embed into $[C_q]^n$, the $n$-fold wreath product of the cyclic group $C_q$. We show that if $f$ is not isotrivial, then $[[C_q]^infty:G_infty(beta)]<infty$ unless $beta$ is postcritical or periodic. We are also able to prove that if $f_1(x)=x^q+c_1$ and $f_2(x)=x^q+c_2$ are two such distinct polynomials, then the fields $bigcup_{n=1}^infty K(f_1^{-n}(beta))$ and $bigcup_{n=1}^infty K(f_2^{-n}(beta))$ are disjoint over a finite extension of $K$.
In previous work, the authors confirmed the speculation of J. G. Thompson that certain multiquadratic fields are generated by specified character values of sufficiently large alternating groups $A_n$. Here we address the natural generalization of this speculation to the finite general linear groups $mathrm{GL}_mleft(mathbb{F}_qright)$ and $mathrm{SL}_2left(mathbb{F}_qright)$.
As a consequence of the classification of finite simple groups, the classification of permutation groups of prime degree is complete, apart from the question of when the natural degree $(q^n-1)/(q-1)$ of ${rm L}_n(q)$ is prime. We present heuristic arguments and computational evidence to support a conjecture that for each prime $nge 3$ there are infinitely many primes of this form, even if one restricts to prime values of $q$.
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