Let $X = \mathbb{P}^1 \smallsetminus \{0,1,\infty\}$ be the thrice-punctured over a ring of $S$-integers $\mathcal{O}_{K,S}$ in a number field~$K$. For any quotient $\pi_1^{\mathrm{mot}}(X,0) \twoheadrightarrow \Pi$ of Deligne--Goncharov's motivic fundamental group there is an associated Selmer scheme which parametrises $\Pi$-torsors with a mixed Tate motive structure. We give several descriptions of the motivic Selmer scheme which make it amenable to computations, using $\mathbb{G}_m$-equivariant cocycles of algebraic groups, Lie algebras, and complete Hopf algebras. We prove that the Selmer scheme is isomorphic to an affine space $\mathbb{A}^N_{\mathbb{Q}}$, and we construct coordinates realising this isomorphism. This is a key ingredient for making the motivic Chabauty--Kim method explicit in a general setting, without restrictions on the base field or the choice of fundamental group quotient.
Let $C$ be a smooth projective geometrically connected curve over a field $k$ with a $k$-rational point. Let $J$ be the Jacobian variety of $C$. For an integer $r\geq 1$ and a positive integer $n$ prime to the characteristic of $k$, we prove that the Galois symbol map \[ K(k;J,\mathbb{G}_{m},\ldots,\mathbb{G}_{m})/n \t...
Let $\mathfrak L$ be a Frobenius maximal parabolic subalgebra of $\mathfrak{sl}_n$. For any $F\in\mathfrak L^*$ for which the Kirillov form $B_F(x,y)=F([x,y])$ is non-degenerate, let $\widehat F$ denote the associated principal element. We prove that the multiplicities of the eigenvalues of $\operatorname{ad}_{\widehat...
A. Giaquinto, J. Irving, A. Lauve et al.· 2 citations
Fix a flat and projective morphism X→Σ$X\rightarrow \Sigma$ of schemes. We show, first, that any set of P1${\mathbb {P}}^1$ ‐fibrations on X$X$ defines a set of simple roots, a set of simple coroots and a Cartan matrix C$C$ . Second, X$X$ is an étale F${\mathcal {F}}$ ‐bundle over some projective Σ$\Sigma$ ‐scheme, whe...
I. Grojnowski, N. Shepherd-Barron· Proceedings of the London Ma...· 0 citations
Let $\mathscr{B}$ be the Bruhat--Tits building of $\mathrm{PGL}_n(F)$, where $F$ is a non-Archimedean local field. We introduce geometric $k$-geodesics in $\mathscr{B}$ by means of CAT(0) convexity and combinatorial $k$-geodesics by a local successor relation on pointed $k$-facets. We prove that the two notions coincid...
Let $K_\infty/K$ be a $p$-adic Lie extension of a $p$-adic field $K$. We study the subring of pro-analytic vectors in the de Rham period ring $\mathbf{B}_{\mathrm{dR}}^+(K_\infty)$. We show that the pro-analytic subring admits a Galois-equivariant isomorphism with a formal power series ring $\widehat{K}_{\infty}^{\math...
Let $f: X\rightarrow S$ be a faithful affine map. In this article we construct a group homomorphism $\theta_{{fppf}}: \text{Br}(f) \longrightarrow H^1_{{fppf}}\!\left(S, (f_*\mathcal{O}_X^\times / \mathcal{O}_S^\times)_{{fppf}}\right)$, extending the formerly defined morphism over an \'etale site $\theta_{et}$. In doin...
Sourayan Banerjee· 0 citations
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