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1. CJM 2012 (vol 65 pp. 1201)
Application of the Strong Artin Conjecture to the Class Number Problem We construct unconditionally several families of number fields with
the largest possible class numbers. They are number fields of degree 4
and 5 whose Galois closures have the Galois group $A_4, S_4$ and
$S_5$. We first construct families of number fields with smallest
regulators, and by using the strong Artin conjecture and applying zero
density result of Kowalski-Michel, we choose subfamilies of
$L$-functions which are zero free close to 1.
For these subfamilies, the $L$-functions have the extremal value at
$s=1$, and by the class number formula, we obtain the extreme class
numbers.
Keywords:class number, strong Artin conjecture Categories:11R29, 11M41 |
2. CJM 2010 (vol 62 pp. 787)
An Explicit Treatment of Cubic Function Fields with Applications We give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for non-singularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description of the signature of any rational place in a cubic extension that involves only the defining curve and the order of the base field. All these quantities only require simple polynomial arithmetic as well as a few square-free polynomial factorizations and, in some cases, square and cube root extraction modulo an irreducible polynomial. We also illustrate why and how signature computation plays an important role in computing the class number of the function field. This in turn has applications to the study of zeros of zeta functions of function fields.
Keywords:cubic function field, discriminant, non-singularity, integral basis, genus, signature of a place, class number Categories:14H05, 11R58, 14H45, 11G20, 11G30, 11R16, 11R29 |
3. CJM 2001 (vol 53 pp. 1194)
Explicit Upper Bounds for Residues of Dedekind Zeta Functions and Values of $L$-Functions at $s=1$, and Explicit Lower Bounds for Relative Class Numbers of $\CM$-Fields |
Explicit Upper Bounds for Residues of Dedekind Zeta Functions and Values of $L$-Functions at $s=1$, and Explicit Lower Bounds for Relative Class Numbers of $\CM$-Fields We provide the reader with a uniform approach for obtaining various
useful explicit upper bounds on residues of Dedekind zeta functions of
numbers fields and on absolute values of values at $s=1$ of $L$-series
associated with primitive characters on ray class groups of number
fields. To make it quite clear to the reader how useful such bounds
are when dealing with class number problems for $\CM$-fields, we
deduce an upper bound for the root discriminants of the normal
$\CM$-fields with (relative) class number one.
Keywords:Dedekind zeta functions, $L$-functions, relative class numbers, $\CM$-fields Categories:11R42, 11R29 |
4. CJM 2001 (vol 53 pp. 33)
Merit Factors of Polynomials Formed by Jacobi Symbols We give explicit formulas for the $L_4$ norm (or equivalently for the
merit factors) of various sequences of polynomials related to the
polynomials
$$
f(z) := \sum_{n=0}^{N-1} \leg{n}{N} z^n.
$$
and
$$
f_t(z) = \sum_{n=0}^{N-1} \leg{n+t}{N} z^n.
$$
where $(\frac{\cdot}{N})$ is the Jacobi symbol.
Two cases of particular interest are when $N = pq$ is a product of two
primes and $p = q+2$ or $p = q+4$. This extends work of H{\o}holdt,
Jensen and Jensen and of the authors.
This study arises from a number of conjectures of Erd\H{o}s,
Littlewood and others that concern the norms of polynomials with
$-1,1$ coefficients on the disc. The current best examples are of the
above form when $N$ is prime and it is natural to see what happens for
composite~$N$.
Keywords:Character polynomial, Class Number, $-1,1$ coefficients, Merit factor, Fekete polynomials, Turyn Polynomials, Littlewood polynomials, Twin Primes, Jacobi Symbols Categories:11J54, 11B83, 12-04 |
5. CJM 1998 (vol 50 pp. 794)
Upper bounds on $|L(1,\chi)|$ and applications We give upper bounds on the modulus of the values at $s=1$ of
Artin $L$-functions of abelian extensions unramified at all
the infinite places. We also explain how we can compute better
upper bounds and explain how useful such computed bounds are
when dealing with class number problems for $\CM$-fields. For
example, we will reduce the determination of all the
non-abelian normal $\CM$-fields of degree $24$ with Galois
group $\SL_2(F_3)$ (the special linear group over the finite
field with three elements) which have class number one to the
computation of the class numbers of $23$ such $\CM$-fields.
Keywords:Dedekind zeta function, Dirichlet series, $\CM$-field, relative class number Categories:11M20, 11R42, 11Y35, 11R29 |