Some New Elliptic Integrals

In 1981 James Davenport surmised that if an algebraic function $f(x,t)$ is not integrable (with respect to $x$) by elementary means when $t$ is an independent variable, then there are most finitely many complex numbers $\tau$ such that $f(x,\tau)$ is integrable by elementary means. In 2020 Umberto Zannier and I obtained a couple of counterexamples and in broad principle classified all of them with algebraic coefficients (they are necessarily somewhat rare). In this talk I will review our work, sketch our recent discovery of yet more counterexamples (they are not unrelated to Ribet curves), and give a more precise description of all elliptic counterexamples.