Abel’s theorem

Abel’s theorem characterizes when a degree-zero divisor on a compact Riemann surface is the divisor of a meromorphic function: its Abel–Jacobi image must vanish modulo periods.

Abel–Jacobi map: A map sending degree-zero divisors on a compact Riemann surface to its Jacobian by integrating holomorphic differentials. It converts the divisor in Abel’s theorem into a point whose vanishing detects whether the divisor is principal.
Compact Riemann surface: A compact one-dimensional complex manifold, equivalently a smooth projective algebraic curve over the complex numbers. The theorem concerns divisors and integrals on this geometric setting.
Niels Henrik Abel: A Norwegian mathematician whose work shaped the theory of elliptic functions, algebraic equations, and infinite series. Abel established the theorem’s original form while studying algebraic integrals.
Linear equivalence of divisors: An equivalence relation in which two divisors differ by the divisor of a meromorphic function. Abel’s theorem decides whether a degree-zero divisor is equivalent to zero.
Abel’s theorem on power series: A theorem asserting boundary convergence properties for a power series whose coefficients have a finite sum. It shares Abel’s name but concerns power-series limits, not integrals on algebraic curves.
Holomorphic differential: A complex differential form that is holomorphic everywhere on a Riemann surface. These differentials are integrated along paths to construct the Abel–Jacobi map.
Divisor (algebraic geometry): A finite formal integer combination of codimension-one subvarieties, or points on a curve. The theorem begins with a degree-zero divisor whose relation to a meromorphic function is in question.
Abel’s theorem on algebraic integrals: A nineteenth-century relation among sums of integrals of algebraic functions, reducing them to a finite set of independent integrals. This classical formulation expresses the integral relations that underpin the divisor criterion.
Principal divisor: A divisor that is the zero-and-pole divisor of a nonzero meromorphic function. The theorem characterizes principal divisors through their Abel–Jacobi images.
Abel’s binomial theorem: An identity expanding a power of a sum using generalized binomial coefficients and a parameter-dependent factor. This separate result by Abel has no divisor or period-lattice criterion.

Linked from 2 pages

Poncelet's closure theoremRelated: The elliptic-curve formulation translates polygon closure into a condition on divisor classes.