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1. CMB 2012 (vol 56 pp. 695)
Carmichael meets Chebotarev For any finite Galois extension $K$ of $\mathbb Q$
and any conjugacy class $C$ in $\operatorname {Gal}(K/\mathbb Q)$,
we show that there exist infinitely many Carmichael numbers
composed solely of primes for which the associated class of Frobenius
automorphisms is $C$. This result implies that for every natural
number $n$ there are infinitely many Carmichael numbers of the form
$a^2+nb^2$ with $a,b\in\mathbb Z $.
Keywords:Carmichael numbers, Chebotarev density theorem Categories:11N25, 11R45 |
2. CMB 2005 (vol 48 pp. 16)
On the Surjectivity of the Galois Representations Associated to Non-CM Elliptic Curves Let $ E $ be an elliptic curve defined over
$\Q,$ of conductor $N$ and without complex multiplication. For any
positive integer $l$, let $\phi_l$ be the Galois representation
associated to the $l$-division points of~$E$. From a celebrated
1972 result of Serre we know that $\phi_l$ is surjective for any
sufficiently large prime $l$. In this paper we find conditional
and unconditional upper bounds in terms of $N$ for the primes $l$
for which $\phi_l$ is {\emph{not}} surjective.
Categories:11G05, 11N36, 11R45 |
3. CMB 2004 (vol 47 pp. 431)
A Note on $4$-Rank Densities For certain real quadratic number fields, we prove density results concerning
$4$-ranks of tame kernels. We also discuss a relationship between $4$-ranks of
tame kernels and %% $4$-class ranks of narrow ideal class groups. Additionally,
we give a product formula for a local Hilbert symbol.
Categories:11R70, 19F99, 11R11, 11R45 |
4. CMB 2002 (vol 45 pp. 86)
On Cyclic Fields of Odd Prime Degree $p$ with Infinite Hilbert $p$-Class Field Towers Let $k$ be a cyclic extension of odd prime degree $p$ of the field of
rational numbers. If $t$ denotes the number of primes that ramify in $k$,
it is known that the Hilbert $p$-class field tower of $k$ is infinite if
$t>3+2\sqrt p$. For each $t>2+\sqrt p$, this paper shows that a positive
proportion of such fields $k$ have infinite Hilbert $p$-class field towers.
Categories:11R29, 11R37, 11R45 |