Differential forms on elliptic curves are classified by their pole structure. Holomorphic differential forms have no poles and are called first-kind forms. Meromorphic differential forms are divided into two different classes, based on the presence of single poles. They are referred to as second kind forms with no single poles, and third kind forms otherwise. Period integrals of first and second kind forms (periods and quasi-periods) are related to the classical complete elliptic integrals of the first and second kind and satisfy an irreducible second-order Picard–Fuchs differential equation. Integrals of third kind forms instead satisfy a first-order inhomogeneous differential equation with inhomogeneities built from periods and quasi-periods. These are all well-known facts from the classical theory of elliptic integrals.
In my talk, I show that precisely these three types of period integrals arise in certain Feynman integrals. Their properties are crucial for deriving canonical differential equations. Moreover, I demonstrate how third-kind integrals are related to new transcendental functions—often referred to as G-functions or epsilon-functions—that appear in the epsilon-factorization procedure. At higher loop orders, these structures generalize to third kind forms and their corresponding periods on K3 surfaces and Calabi–Yau manifolds, where the associated simple poles may be associated to points or elliptic curves, depending on the underlying Feynman graph.