2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/173282We show that finite fields over which there is a curve of a given genus g with its Jacobian having a small exponent, are very rare. This extends a recent result of W. Duke in the case g=1. We also show when g=1 or g=2 that our bounds are best possible.V3. Small corrections; references updatedNumber Theory11G20; 14H40text