= Abel theorem for divisors
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= Abel's theorem
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A degree-zero <complex analytic divisor> $E$ on a <compact Riemann surface> is principal exactly when its <Abel-Jacobi map of a compact Riemann surface> is zero. Normalize a <differential of the third kind> with residues equal to the coefficients of $E$. The <Riemann bilinear relations for a compact surface> express its $b$-periods as $2\pi i$ times the Abel integrals. If those integrals are in the period lattice, subtract a suitable integral multiple of the normalized <holomorphic differential forms>; all periods then lie in $2\pi i\mathbb Z$, so exponentiation gives the required <meromorphic function>. Conversely, the logarithmic differential of a <meromorphic function> has integer residues and periods in $2\pi i\mathbb Z$, giving a zero Abel class by the same calculation.
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