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In this paper and its sequel, we develop a technique for finding thedistribution of $\ell^{\infty}$-Selmer groups in degree $\ell$ twist familiesof Galois modules over number fields. Given an elliptic curve E over a numberfield satisfying certain technical conditions, this technique can be used toshow that 100% of the quadratic twists of E have rank at most 1. Given a prime$\ell$ and a number field F not containing $\mu_{2\ell}$, this method alsoshows that the $\ell^{\infty}$-class groups in the family of degree $\ell$cyclic extensions of F have a distribution consistent with theCohen-Lenstra-Gerth heuristics.  For this work, we develop the theory of the fixed point Selmer group, whichserves as the base layer of the $\ell^{\infty}$-Selmer group. This first papergives a technique for finding the distribution of $\ell^{\infty}$-Selmer groupsin certain families of twists where the fixed point Selmer group is stable. Inthe sequel paper, we will give a technique for controlling fixed point Selmergroups.

The distribution of $\ell^\infty$-Selmer groups in degree $\ell$ twist families I