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Search: MSC category 11R58 ( Arithmetic theory of algebraic function fields [See also 14-XX] )

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1. CJM Online first

Cojocaru, Alina Carmen; Shulman, Andrew Michael
The Distribution of the First Elementary Divisor of the Reductions of a Generic Drinfeld Module of Arbitrary Rank
Let $\psi$ be a generic Drinfeld module of rank $r \geq 2$. We study the first elementary divisor $d_{1, \wp}(\psi)$ of the reduction of $\psi$ modulo a prime $\wp$, as $\wp$ varies. In particular, we prove the existence of the density of the primes $\wp$ for which $d_{1, \wp} (\psi)$ is fixed. For $r = 2$, we also study the second elementary divisor (the exponent) of the reduction of $\psi$ modulo $\wp$ and prove that, on average, it has a large norm. Our work is motivated by the study of J.-P. Serre of an elliptic curve analogue of Artin's Primitive Root Conjecture, and, moreover, by refinements to Serre's study developed by the first author and M.R. Murty.

Keywords:Drinfeld modules, density theorems
Categories:11R45, 11G09, 11R58

2. CJM 2010 (vol 62 pp. 787)

Landquist, E.; Rozenhart, P.; Scheidler, R.; Webster, J.; Wu, Q.
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 1998 (vol 50 pp. 1253)

López-Bautista, Pedro Ricardo; Villa-Salvador, Gabriel Daniel
Integral representation of $p$-class groups in ${\Bbb Z}_p$-extensions and the Jacobian variety
For an arbitrary finite Galois $p$-extension $L/K$ of $\zp$-cyclotomic number fields of $\CM$-type with Galois group $G = \Gal(L/K)$ such that the Iwasawa invariants $\mu_K^-$, $ \mu_L^-$ are zero, we obtain unconditionally and explicitly the Galois module structure of $\clases$, the minus part of the $p$-subgroup of the class group of $L$. For an arbitrary finite Galois $p$-extension $L/K$ of algebraic function fields of one variable over an algebraically closed field $k$ of characteristic $p$ as its exact field of constants with Galois group $G = \Gal(L/K)$ we obtain unconditionally and explicitly the Galois module structure of the $p$-torsion part of the Jacobian variety $J_L(p)$ associated to $L/k$.

Keywords:${\Bbb Z}_p$-extensions, Iwasawa's theory, class group, integral representation, fields of algebraic functions, Jacobian variety, Galois module structure
Categories:11R33, 11R23, 11R58, 14H40

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