Most factorization algorithms in use today fit in one of two camps: sieve-based methods based on congruences of squares, and algorithms based on decompositions of algebraic groups. In this article, we trace the common thread connecting the latter.
In this note, I’ll discuss why square-root cancellation is so typical in problems in number theory and give a quick survey of important sums known or widely conjectured to satisfy bounds of this form.
In 1946, S. Bochner published the paper Formal Lie Groups, in which he noted that several classical theorems (due to Sophus Lie) concerning infinitesimal transformations on Lie groups continue to hold when the (convergent) power series locally representing the group law was replaced by a suitable formal analogue. It was not long before this formalism found far-reaching uses in algebraic number theory and algebraic topology.
Unfortunately, few students see more than two or three explicit (i.e. closed form) group laws before stumbling into the deep end of abstract nonsense. In this article, we’ll see in a rigorous sense why this must be the case, providing along the way a complete classification of polynomial and rational formal group laws (over any reduced ring).