Koblitz’s Conjecture for Abelian Varieties

Ute Spreckels () and Andreas Stein ()
Additional contact information
Ute Spreckels: Carl von Ossietzky Universität Oldenburg, Institut für Mathematik
Andreas Stein: Carl von Ossietzky Universität Oldenburg, Institut für Mathematik

A chapter in Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory, 2017, pp 611-622 from Springer

Abstract: Abstract Consider a principally polarized abelian variety A of dimension d defined over a number field F. If 𝔭 $$\mathfrak p$$ is a prime ideal in F such that A has good reduction at p, let N 𝔭 $$N_{\mathfrak p}$$ be the order of A mod 𝔭 $$A\operatorname {mod}\mathfrak p$$ . We have formulae for the density p ℓ of primes 𝔭 $$\mathfrak p$$ such that N 𝔭 $$N_{\mathfrak p}$$ is divisible by a fixed prime number ℓ in two cases: A is a CM abelian variety and the CM-field is contained in F, or A has trivial endomorphism ring and its dimension is 2, 6 or odd. In both cases, we can prove that C A = ∏ ℓ 1 − p ℓ 1 − 1 / ℓ $$C_A=\prod _\ell \frac {1-p_\ell }{1-1/\ell }$$ is a positive constant. We conjecture that the number of primes 𝔭 $$\mathfrak p$$ with norm up to n such that N 𝔭 $$N_{\mathfrak p}$$ is prime is given by the formula C A n d log ( n ) 2 $$C_A\frac {n}{d\log (n)^2}$$ , generalizing a formula by N. Koblitz, conjectured in 1988 for elliptic curves. Numerical evidence that supports this conjectural formula is provided.

Keywords: Abelian varieties over finite fields; Galois representations; General symplectic group over a finite field; Serre’s open image theorem; Torsion points of abelian varieties; 11G10; 11N05; 11F80; 11G20 (search for similar items in EconPapers)
Date: 2017
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:978-3-319-70566-8_27

Ordering information: This item can be ordered from
http://www.springer.com/9783319705668

DOI: 10.1007/978-3-319-70566-8_27

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 ().