Newton Map
The Newton map \(\nu : B(G) \to (X_*(T)_\mathbb{Q})^\Gamma / W\) extracts a "slope vector" from a \(\sigma\)-conjugacy class, refining the Newton polygon of \(\mathrm{GL}_n\)-isocrystals to arbitrary reductive groups. Together with the Kottwitz map, it gives the complete combinatorial description of \(B(G)\).
Setup
Let \(F\) be a non-archimedean local field, \(\breve F\) the completion of \(F^{\mathrm{unr}}\) with Frobenius \(\sigma\), \(G\) a connected reductive group over \(F\), and \(T \subseteq G\) a maximal torus. Let \(\Gamma = \mathrm{Gal}(\bar F/F)\), \(W = W(G, T)\).
Definition
For \(b \in G(\breve F)\), define its Newton point as follows:
- Some power \(\sigma^n(b) \cdot \sigma^{n-1}(b) \cdots b\) lies in a compact subgroup; thus there is a unique homomorphism (after passing to a finite extension) \[ \nu_b : \mathbb{D} \to G_{\breve F} \] from the "pro-torus" \(\mathbb{D} = \mathrm{Spec}\, \breve F[\mathbb{Q}]\) to \(G_{\breve F}\), characterised by \(b \cdot \sigma(\nu_b) \cdot b^{-1} = \nu_b\) and an integrality condition.
- Restrict \(\nu_b\) to a rational cocharacter of \(T\) (after conjugating into \(T\)): \[ \nu_b \in X_*(T)_\mathbb{Q}. \]
- Project to a dominant element modulo Weyl-group action and Galois-invariance: \[ \nu([b]) \in (X_*(T)_\mathbb{Q})^\Gamma / W. \]
Equivalent: slope homomorphism
For \(G = \mathrm{GL}_n\): the Newton point of an isocrystal \((V, F)\) is its Newton polygon, a multiset of rational slopes summing to \(\dim V\), recorded as a non-increasing sequence \((\nu_1 \ge \nu_2 \ge \cdots \ge \nu_n)\) with \(\nu_i \in \mathbb{Q}\).
For \(G\) general: the Newton point is the analogue, valued in a dominant rational cocharacter chamber.
Properties
- \(\nu\) is well-defined on \(\sigma\)-conjugacy classes: \(\nu(b) = \nu(g b \sigma(g)^{-1})\).
- \(\nu\) is lower semicontinuous in the right sense (in families).
- The image of \(\nu\) is a partially ordered set with dominance order: \(\nu \le \nu'\) iff \(\nu' - \nu\) is a non-negative rational combination of positive coroots.
Basic vs.\ non-basic
\([b] \in B(G)\) is basic if the Newton point \(\nu([b])\) is central in \(G\), i.e.\ \(\nu([b]) \in X_*(Z(G))_\mathbb{Q}\) (a "constant" slope vector).
The basic locus consists of the \(\sigma\)-conjugacy classes "closest to the identity" — analogues of "supersingular" isocrystals for \(\mathrm{GL}_n\).
Examples
\(\mathrm{GL}_n\): Newton polygons
\(B(\mathrm{GL}_n)\) = Newton polygons in \([0, n]\) (where the \(y\)-coordinate is at height = sum of slopes so far, at \(x = 0, 1, \ldots, n\)).
- Basic strata: slope vector \((s, s, \ldots, s)\) for \(s = k/n\), \(k \in \mathbb{Z}\); these are the "supersingular" isocrystals.
- Largest stratum (Hodge-Newton "ordinary"): slopes \((1, 1, \ldots, 1, 0, 0, \ldots, 0)\).
\(\mathrm{SL}_n\)
\(B(\mathrm{SL}_n)\) corresponds to Newton polygons in \(\mathrm{GL}_n\) with slope sum 0. Newton point in \((X_*(T)_\mathbb{Q})^\Gamma /W\) with \(X_* = \{(a_i) : \sum a_i = 0\}\).
Tori
For a torus \(T\) over \(F\): \(B(T) = X_*(T)^\Gamma\) (via the Kottwitz map), and \(\nu : B(T) \to X_*(T)_\mathbb{Q}^\Gamma\) is just the inclusion.
Newton stratification
For each Newton point \(\bar\nu \in (X_*(T)_\mathbb{Q})^\Gamma / W\): the preimage \(\nu^{-1}(\bar\nu)\) is the Newton stratum of \(B(G)\) at \(\bar\nu\). Newton strata stratify \(B(G)\), with closure relations given by the dominance order.
For Shimura varieties: the Newton stratification of the reduction modulo \(p\) corresponds (via Rapoport-Zink uniformisation) to a Newton stratification of \(B(G_{\mathrm{cong}})\).
Hodge-Newton decomposition
For an element \(b \in G(\breve F)\) with \(\nu(b)\) lying on a wall of the dominant chamber, there is a canonical Hodge-Newton decomposition of \(b\) into a Levi piece (slope corresponding to the wall) and a basic-like piece in the Levi. This is the structural result behind the Kottwitz conjecture on Shimura varieties.
Role in Langlands
- Local Shtuka theory: Newton points classify isoshtukas with \(G\)-structure.
- Affine Deligne-Lusztig varieties \(X_w(b)\): non-emptiness and dimension formulas depend on the Newton point of \(b\) and the affine Weyl group element \(w\).
- Local Langlands at non-tempered: Arthur parameters with non- trivial \(\mathrm{SL}_2\)-factor correspond to non-basic Newton points (via Jacobson-Morozov).
- Reduction of Shimura varieties: the Newton stratification of \(\mathrm{Sh}_K \pmod p\) is indexed by \(B(G)\) modulo Kottwitz's admissibility conditions.
References
- Kottwitz, Isocrystals with additional structure I, II (Compositio 1985, 1997).
- Rapoport–Richartz, On the classification and specialization of F-isocrystals with additional structure (Compositio 1996).
- Hartl, Period spaces in p-adic Hodge theory (book draft).
- Görtz, Affine Deligne-Lusztig varieties and the geometry of Shimura varieties (notes, 2009).