Global Arthur parameters via cuspidal excursion data

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Global Arthur parameters via cuspidal excursion data

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Recursos

Start with a cuspidal excursion parameter whose Frobenius classes all have Arthur's shape. This class shows how the paper turns that list of local shapes into one global commuting pair, an algebraic SL₂ φ_ν together with a Weil-group map τ_ν into its centralizer. It also gives you a test to judge whether a point of the global support produces that pair.

Why local shapes are not enough

Theorem 1.1 writes each Frobenius class as φ(diag(a^{d_x}, a^{−d_x}))·c_x. The factors c_x are chosen one place at a time, so nothing makes them a homomorphism, and inertia never appears in a list of Frobenius classes [0:07]. The same thing happens with matrices: their eigenvalues don't determine a representation, but joint invariants such as tr(g₁g₂) do. Excursion data keep exactly those tuple invariants.

Theorem 1.2 asks for σ_ν(w) = τ_ν(w)·φ_ν(diag(a^{deg w}, a^{−deg w})) for every w in W_U, inertia included, with τ_ν pure [0:50]. It does not claim ellipticity or a multiplicity formula.

The three moves

Global support [1:28]. A cusp vector f ≠ 0 gives, through shtuka cohomology, a finite-type scheme Spec R. Each of its points is a pair (b, e) with Ad(b(w))e = q^{deg w}e, and its tuple invariants equal those of σ_ν,ad. Because R has finite type, you can test the same support at as many places as you need.
Open-orbit criterion [2:06]. Take a place where H = Z_L(t) is connected. On the resonance space E_t, H has a unique open orbit, which lies in O_ν. One point whose e lands there at a single place is enough to build (φ_ν, τ_ν).
Contradiction [3:17]. Suppose every point lies in the boundary at every place. Then the boundary tests q_i vanish on fibers, so q_Π^ f = 0*. But a simple cuspidal Hecke quotient shows that q_Π^ f ≠ 0*. The two results contradict each other, so an open-orbit point must exist.

Check yourself [2:52]: suppose a point is in the boundary at three test places and in the open orbit at a fourth. It gives the enhancement, because one place is enough.

When you read any local-to-global argument, look for these same three moves. The local tests themselves come from a Frobenius-compatible Iwahori category and from spherical transfer [4:06].

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