Number Theory Seminar

In recent work, Feng–Yun–Zhang constructed higher theta series for unitary groups valued in cycles on the moduli stack of Hermitian shtukas, and proved that these theta series are modular after restriction to the generic fiber and passing to l-adic cohomology. Motivated by ideas from the relative Langlands program, in joint work with Yujie Xu we generalize their construction to a natural class of "anomaly-free" reductive dual pairs, and prove generic modularity on l-adic cohomology. For the most classical version, one could hope to avoid many of the difficulties involved by studying a "universal" case, but there is a key problem: it is "anomalous," and so does not fit into the relative Langlands picture. Nevertheless, it is possible to prove analogous statements for the universal, or more generally anomalous, cases. We will study how this works, and suggest an interpretation.