On counting cuspidal automorphic representations for GSp(4)

We find the number s k ⁢ ( p , Ω ) s_{k}(p,\Omega) of cuspidal automorphic representations of GSp ⁢ ( 4 , A Q ) \mathrm{GSp}(4,\mathbb{A}_{\mathbb{Q}}) with trivial central character such that the archimedean component is a holomorphic discrete series representation of weight k ≥ 3 k\geq 3 , and the non-archimedean component at 𝑝 is an Iwahori-spherical representation of type Ω and unramified otherwise. Using the automorphic Plancherel density theorem, we show how a limit version of our formula for s k ⁢ ( p , Ω ) s_{k}(p,\Omega) generalizes to the vector-valued case and a finite number of ramified places.

Galois representations associated to holomorphic limits of discrete series

Generalizing previous results of Deligne–Serre and Taylor, Galois representations are attached to cuspidal automorphic representations of unitary groups whose Archimedean component is a holomorphic limit of discrete series. The main ingredient is a construction of congruences, using the Hasse invariant, that is independent of$q$-expansions.

The semigroup of nonempty finite subsets of rationals

LetQbe the additive group of rational numbers and letℛbe the additive semigroup of all nonempty finite subsets ofQ. ForX∈ℛ, defineAXto be the basis of〈X−min(X)〉andBXthe basis of〈max(X)−X〉. In the greatest semilattice decomposition ofℛ, let𝒜(X)denote the archimedean component containingX. In this paper we examine the structure ofℛand determine its greatest semilattice decomposition. In particular, we show that forX,Y∈ℛ,𝒜(X)=𝒜(Y)if and only ifAX=AYandBX=BY. Furthermore, ifXis a non-singleton, then the idempotent-free𝒜(X)is isomorphic to the direct product of a power joined subsemigroup and the groupQ.

The semigroup of nonempty finite subsets of integers

LetZbe the additive group of integers andgthe semigroup consisting of all nonempty finite subsets ofZwith respect to the operation defined byA+B={a+b:a∈A, b∈B}, A,B∈g.ForX∈g, defineAXto be the basis of〈X−min(X)〉andBXthe basis of〈max(X)−X〉. In the greatest semilattice decomposition ofg, letα(X)denote the archimedean component containingXand defineα0(X)={Y∈α(X):min(Y)=0}. In this paper we examine the structure ofgand determine its greatest semilattice decomposition. In particular, we show that forX,Y∈g,α(X)=α(Y)if and only ifAX=AYandBX=BY. Furthermore, ifX∈gis a non-singleton, then the idempotent-freeα(X)is isomorphic to the direct product of the (idempotent-free) power joined subsemigroupα0(X)and the groupZ.