Generic Manin–Mumford

Abstract: The unlikely intersection, or Manin–Mumford, principle asserts that an algebraic variety containing many special points must itself contain special subvarieties. An elementary illustration is the following theorem of Ihara, Serre, and Tate: if a polynomial equation $F(X,Y)=0$ has infinitely many solutions in roots of unity, then either $X^rY^s-w$ or $X^r-w Y^s$ divides $F$, where $w$ is a root of unity. This may be viewed as the first instance of the general Manin–Mumford theorem describing subvarieties of semi-abelian varieties containing many torsion points.

Motivated by recent work of Binyamini, Kiro, and Pila, we ask: what happens if one replaces the roots of unity by a different collection of algebraic numbers?

I will discuss a general criterion guaranteeing a Manin–Mumford type theorem for a set $S$ of algebraic numbers. Roughly speaking, if the elements of $S$ are sufficiently generic in the Galois-theoretic sense, then any algebraic variety containing many points of $S^n$ must contain an obvious special subvariety, cut out by equations of the form $x_i=x_j$ or $x_i=a$ with $a$ in $S$. As applications, we recover a recent theorem of Binyamini, Kiro, and Pila on roots of Laguerre polynomials, obtain analogous results for several other classical polynomial families, and prove that the phenomenon holds almost surely for roots of random polynomials.

number theory

Audience: researchers in the topic

London number theory seminar

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