Gonadotropins are hormones synthesized and released by the anterior pituitary gland, which act on the testes and ovaries to increase the production of sex hormones and stimulate production of either sperm or ova.
The hypothalamus contains Gonadotropin-releasing hormone (GnRH). Intermittent released of GnRH is responsible for the biosynthesis and secretion of follicle-stimulating hormone (FSH) and luteinizing hormone (LH) from the anterior pituitary gland.
LH stimulates interstitial cells to release testosterone, which triggers spermatogenesis. The rising level of testosterone exerts negative feedback control on the hypothalamus and pituitary glands.
GnRH has been shown here in a negative feedback loop
This feedback loop has great interplay with the number receptor levels as you suggest.
(From a quick search) I found one paper (behind a paywall) that states the following information when studying rats:
We observed an acute reduction of both GnRH and GnRH-R mRNAs 24 h after the injection [of GnRH agonist (triptorelin)] (about 38% of control).
Transcription does not equal translation, but it seems that acute increased levels of GnRH cause a reduction and mRNA of GnRH and GnRH-R and therefore potential reduction (down-regulation) of protein levels of GnRH-R.