This article by Henrik Lenstra has an intriguing quote:

The key notion underlying the second algorithm is that of â€œinfrastructureâ€, a word coined by Shanks (see [11]) to describe a certain multiplicative structure that he detected within the period of the continued fraction expansion of âˆšd. It was subsequently shown (see [7]) that this period can be â€œembeddedâ€ in a circle group of â€œcircumferenceâ€ Rd, the embedding preserving the cyclical structure. In the modern terminology of Arakelov theory, one may describe that circle group as the kernel of the natural map Pic0Z[âˆšd] â†’ PicZ[âˆšd] from the group of â€œmetrized line bundles of degree 0â€ on the â€œarithmetic curveâ€ corresponding to Z[âˆšd] to the usual class group of invertible ideals. By means of Gaussâ€™s reduced binary quadratic forms one can do explicit computations in Pic0Z[âˆšd] and in its â€œcircleâ€ subgroup.

I’m a sucker for anything that related elementary topics (like Pell’s equation) to advanced topics that I don’t understand (like Arakelov theory).