Artin's conjecture on primitive roots: Difference between revisions - Wikipedia

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Specifically, they said there exists a constant <math>C_E</math> for a given point of infinite order <math>P</math> in the set of rational points <math>E(\mathbb{Q})</math> such that the number <math>N(P)</math> of primes (<math>p\leq x</math>) for which the reduction of the point <math>P\pmod p</math> denoted by <math>\bar{P}</math> generates the whole set of points in <math>\mathbb{F_p}</math> in <math>E</math>, denoted by <math>\bar{E}(\mathbb{F_p})</math>, is given by <math>N(P)\sim C_E\left ( \frac{x}{\log x} \right )</math>.<ref>{{Cite journal |last=Lang and 2 Trotter |date=1977 |title=Primitive points on Elliptic Curves |journal=Bull. Amer. Math. Soc. |volume=83 |issue=2 |pages=289–292|doi=10.1090/S0002-9904-1977-14310-3 |doi-access=free |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-83/issue-2/Primitive-points-on-elliptic-curves/bams/1183538693.pdf }}</ref> Here we exclude the primes which divide the denominators of the coordinates of <math>P</math>.

Gupta and Murty proved... the Lang and Trotter conjecture for <math>E/\mathbb{Q}</math> with complex multiplication under the Generalized Riemann Hypothesis, for primes splitting in the relevant imaginary quadratic field, i. e., they proved some asymptotic formula for only half of the prime numbers without actually verifying any particular case of Lang and Trotter conjecture which is stated for all prime numbers and which probably is totally false! Moreover Gupta and Murty proved that the main term of their asymptotic formula is positive for some particular classes of elliptic curves with complex multiplication. If one consideres the group P generated by several independently linear points P1, P2,...,Pg in <math>E(\mathbb{Q})</math>, for g sufficiently large (i.e. g>20), then indeed Goupta and Murty obtainded an asymptotic formula for all prime numbers with positive main term, for some particular classes of elliptic curves with complex multiplication, and one could ask what are the minimum subsets of P1, P2,...,Pg, for which such asymptotic formula exists. This result could be considered as an analogous to Artin's primitive root conjecture!

<ref>{{Cite journal |last=Gupta and Murty |date=1987 |title=Primitive points on elliptic curves |url=https://eudml.org/doc/89763 |journal=Compositio Mathematica |volume=58 |pages=13–44}}</ref>