ELDORADO, Tex.  A year and a half after the state authorities raided a fundamentalist Mormon ranch here, the first of a half-dozen leaders of the sect has gone on trial on charges that he had sex with an under-age girl who state officials assert was one of several wives assigned to him by church elders.

The trial, taking place in a makeshift courtroom in this rural hamlet, has opened a window on the polygamous world inside the Yearning for Zion ranch, the compound of log buildings built around a stone temple where the state contends that arranged marriages involving under-age girls were commonplace.

But the proceeding has also exposed the difficulties prosecutors face in proving sex crimes were committed at the ranch, since the young woman at the center of the case, like most of the female sect members, has not cooperated with the authorities.

The state’s case rests mainly on genetic evidence that links the defendant, Raymond M. Jessop, 38, to a child born in 2005 to the 16-year-old girl he was living with along with several other women.