The ionospheric sounding data obtained by the MARSIS (Mars Advanced Radar for Subsurface and Ionospheric Sounding) instrument on the Mars Express spacecraft show that the dayside ionosphere has considerable structure over regions of strong crustal magnetic fields. This structure is typically seen as a hyperbola-shaped trace in a display of echo intensity versus apparent altitude and time. The hyperbola shapes are consistent with oblique reflections from regions of enhanced electron density that are fixed with respect to Mars. Comparisons with the Cain et al. (2003) model for the crustal magnetic field of Mars show that the apexes of the hyperbolas, which identify the closest approach to the regions of enhanced electron density, usually coincide with regions where the crustal magnetic field is strong and nearly vertical. The electron density enhancements, which extend as much as 50 km above the surrounding ionosphere, are believed to arise from increases in the scale height of the ionosphere, possibly due to heating of the ionosphere by solar wind electrons that reach the base of the ionosphere along the nearly vertical (open) magnetic field lines. Statistical analyses of the apparent altitudes of the apexes of the hyperbolas, as well as analyses of repeated passes over the same region, indicate that the electron density enhancements usually consist of horizontal cylinder-like structures rather than isolated hemispherical structures. In many cases the axes of the cylindrical density structures are aligned with the symmetry axes of adjacent cylindrical magnetic field structures with opposite polarity.