LAUSR

Let p be a prime. We study the structure of and the inclusion relations among the terms in the monomial lattice in the modular symmetric power representations of GL 2 ( F p ) $\text {GL}_{2}(\mathbb {F}_{p})$ . We also determine the structure of certain related quotients of the symmetric power representations which arise when studying the reductions of local Galois representations of slope at most p . In particular, we show that these quotients are periodic and depend only on the congruence class modulo p ( p − 1). Many of our results are stated in terms of the sizes of various sums of digits in base p -expansions and in terms of the vanishing or non-vanishing of certain binomial coefficients modulo p .