The Impact of the Infinite Primes on the Riemann Hypothesis for Characteristic p Valued L-series

David Goss

A chapter in Algebra, Arithmetic and Geometry with Applications, 2004, pp 357-380 from Springer

Abstract: Abstract In [12] we proposed an analog of the classical Riemann hypothesis for characteristic p valued L-series based on known results for $$ \zeta _{\mathbb{F}_r [\theta ]} (s) $$ and two assumptions that have subsequently proved to be incorrect. The first assumption is that we can ignore the trivial zeroes of characteristic p L-series in formulating our conjectures. Instead, we show here how the trivial zeroes influence nearby zeroes and so lead to counter-examples of the original Riemann hypothesis analog. We then sketch an approach to handling such “near-trivial” zeroes via Hensel’s and Krasner’s Lemmas. Moreover, we show that $$ \zeta _{\mathbb{F}_r [\theta ]} (s) $$ is not representative of general L-series as, surprisingly, all its zeroes are near-trivial, much as the Artin-Weil zeta-function of $$ \mathbb{P}^1 /\mathbb{F}_r $$ is not representative of general complex L-functions of curves. The second assumption in [12] is that certain Taylor expansions associated to L-series of number fields would exhibit complicated behavior with respect to their zeroes. We present a simple argument that this is not so, and, at the same time, characterize functional equations.

Keywords: Cusp Form; Algebraic Closure; Galois Extension; Riemann Hypothesis; Newton Polygon (search for similar items in EconPapers)
Date: 2004
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-18487-1_23

Ordering information: This item can be ordered from
http://www.springer.com/9783642184871

DOI: 10.1007/978-3-642-18487-1_23

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().