## Counting points modulo p for some finitely generated subgroups of algebraic group (1982)

Venue: | Bull. London Math. Soc |

Citations: | 7 - 0 self |

### BibTeX

@ARTICLE{Matthews82countingpoints,

author = {C. R. Matthews},

title = {Counting points modulo p for some finitely generated subgroups of algebraic group},

journal = {Bull. London Math. Soc},

year = {1982},

volume = {14},

pages = {149--154}

}

### OpenURL

### Abstract

We begin by explaining the basic idea of this paper in a simple case. We write np for the order of 2 modulo the prime p, so that np is the number of powers of 2 which are distinct mod p. We have the elementary bounds logp < £ np ^ p-1.

### Citations

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An Introduction to the Theory of Numbers
- Hardy, Wright
- 1980
(Show Context)
Citation Context ...s therefore bounded by co{\a\), the number of prime factors of |a|. We have the classical estimate co(n) <^ logn/loglogn (already a consequence of the elementary estimates of prime number theory, cf. =-=[1]-=- Chapter XXII). Since we clearly have this gives the required result. log|fl| ^ 2h(y), We now suppose chosen a finite set s$ of generators for F, such that ,stf~ l = ,<&. We define the length l(y) of ... |

151 |
Free subgroups in linear groups
- Tits
- 1972
(Show Context)
Citation Context ...(x) taken to be x*~' ; we find again the result given in the Introduction, in this broader context: we have for almost all primes p (in the sense of density). In general, results of Wolf [8] and Tits =-=[7]-=- show that one of two possibilities holds: the rate of growth L(x) may lie between constant multiples of x d for some integer d, or between constant multiples of e cx for some positive c. The first of... |

76 |
A note on curvature and fundamental group
- Milnor
- 1968
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Citation Context ... element y ofV which is not in S' lies in F (p) for at most C'/(y)/log/(y) primes p. We define the counting function L(x) as \{y e F: l{y) ^ x}|. This is the rate of growth of F, as studied by Milnor =-=[4]-=-, [5] and others. We now formulate the idea of our method.s1 52 C. R. MATTHEWS THEOREM 1. Suppose given a positive function F which tends to infinity with x, and such that F/logF is increasing. Defini... |

46 |
Growth of finitely generated solvable groups and curvature of Riemannian manifolds
- Wolf
- 1968
(Show Context)
Citation Context ...x] +1. With F(x) taken to be x*~' ; we find again the result given in the Introduction, in this broader context: we have for almost all primes p (in the sense of density). In general, results of Wolf =-=[8]-=- and Tits [7] show that one of two possibilities holds: the rate of growth L(x) may lie between constant multiples of x d for some integer d, or between constant multiples of e cx for some positive c.... |

44 |
Growth of finitely generated solvable groups
- Milnor
- 1968
(Show Context)
Citation Context ...ent y ofV which is not in S' lies in F (p) for at most C'/(y)/log/(y) primes p. We define the counting function L(x) as \{y e F: l{y) ^ x}|. This is the rate of growth of F, as studied by Milnor [4], =-=[5]-=- and others. We now formulate the idea of our method.s1 52 C. R. MATTHEWS THEOREM 1. Suppose given a positive function F which tends to infinity with x, and such that F/logF is increasing. Defining E(... |

27 |
On Artin’s conjecture and Euclid’s algorithm in global fields
- Lenstra
- 1977
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Citation Context ...ounds of the kind given above, for a broad range of questions in the direction of Artin's conjecture. For a survey of many of the variants which have been considered, the reader may consult the paper =-=[3]-=- of Lenstra. From our point of view some general formulations in terms of algebraic groups are suggested; the results we obtain apply to affine algebraic groups or abelian varieties. We suppose first ... |

20 |
Quasi-fonctions et hauteurs sur les variétés abéliennes
- Néron
- 1965
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Citation Context ...raightforward carrying-over to abelian varieties of the idea of the previous section. We shall suppose some familiarity on the part of the reader with the theory of heights (see for example the paper =-=[6]-=- of Neron) and of abelian varieties. Suppose that A is an abelian variety defined over Q, and that 0 is a symmetric embedding of A in P", also defined over Q. The naive height function associated to c... |

11 |
Application of Sieve Methods to the Theory of Numbers,'' Cambridge Univ
- Hooley
- 1976
(Show Context)
Citation Context ...ct mod p. We have the elementary bounds logp <£ np ^ p-1 . The conjecture of E. Artin on primitive roots asserts that the upper bound is attained for a set of primes with positive density (see Hooley =-=[2]-=- for a discussion of conditional proofs in this case). The lower bound may be improved to P*" < np for almost all primes p, in the sense of density, for any e > 0. To see this, let E(x) count the prim... |