We enhance the efficient congruencing method for estimating Vinogradov’s integral for moments of order 2$s$, with $${1\leqslant s\leqslant k^{2}-1}$$ . In this way, we prove the main conjecture for such even moments when $${1\leqslant s\leqslant \tfrac{1}{4}(k+1)^{2}}$$ , showing that the moments exhibit strongly diagonal behaviour in this range. There are improvements also for larger values of$s$, these finding application to the asymptotic formula in Waring’s problem.

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We prove trace class estimates for self-adjoint elliptic operators defined in $\R[n]$ or $\R[]_+$. These estimates are also applicable when a physical system is governed by a wave equation by employing separation of variables to obtain corresponding temporal and spatial Hamiltonians. It is shown, in one important example, that the resulting Hamiltonians are of trace class such that quantum statistics can be applied to the system.

We show that the product in the quantum K-ring of a generalized flag manifold G/P involves only finitely many powers of the Novikov variables. In contrast to previous approaches to this finiteness question, we exploit the finite difference module structure of quantum K-theory. At the core of the proof is a bound on the asymptotic growth of the J-function, which in turn comes from an analysis of the singularities of the zastava spaces studied in geometric representation theory. An appendix by H. Iritani establishes the equivalence between finiteness and a quadratic growth condition on certain shift operators.

We study a local multiplicity problem related to so-called generalized Shalika models. By establishing a local trace formula for these kind of models, we are able to prove a multiplicity formula for discrete series. As a result, we can show that these multiplicities are constant over every discrete Vogan L-packet and are related to local exterior square L-functions.