Recent zbMATH articles in MSC 11Shttps://zbmath.org/atom/cc/11S2024-02-28T19:32:02.718555ZWerkzeugA family of irreducible supersingular representations of \(\mathrm{GL}_2(F)\) for some ramified \(p\)-adic fieldshttps://zbmath.org/1527.110422024-02-28T19:32:02.718555Z"Schein, Michael M."https://zbmath.org/authors/?q=ai:schein.michael-mSuppose that \(F/\mathbb{Q}_p\) is a finite extension with residue field \(\mathbb{F}_{p^2}\). Let \(\rho\) stand for a generic continuous two-dimensional representation of \(\mathrm{Gal}(\overline{\mathbb{Q}}_p/F)\). In the paper under the review, the author provides a construction of an explicit family of diagrams giving rise to irreducible supersingular representations of \(\mathrm{GL}_2(F)\) such that the \(K\)-socle of \(V\) is of the form \[\bigoplus_{\sigma \in \mathcal{D}(\rho)} \sigma,\] where \(\mathcal{D}(\rho)\) denotes the set of Serre weights associated to \(\rho\) by the weight part of Serre's modularity conjecture.
We note that the genericity assumption implies \(p > 2\) and that the ramification degree of \(F\) over \(\mathbb{Q}_p\) is less than or equal to \((p-1)/2\).
Reviewer: Ivan MatiÄ‡ (Osijek)Clusters, inertia, and root numbershttps://zbmath.org/1527.110472024-02-28T19:32:02.718555Z"Bisatt, Matthew"https://zbmath.org/authors/?q=ai:bisatt.matthewLet \(K\) be a finite field extension of \(\mathbb Q_p\) where \(p\) is an odd prime, and let \(v\) be a valuation of the algebraic closure \(\overline K\). Let \(f\) be a polynomial over \(K\), and let \(R\) be the finite set of zeros of \(f\) in \(\overline K\). Let \(I\) be the inertia subgroup of \(\mathrm{Gal}(K(R)/K)\). A nonempty subset \(\mathfrak s\) of \(R\) is called a \textit{cluster} if \(\mathfrak s = D\cap R\) where \(D=\lbrace x \in \overline K : v(x-z)\ge d\rbrace\) for some \(z\in \overline K\) and \(d\in \mathbb Q\). \textit{The cluster picture of a square-free polynomial \(f\)} is the collection of all clusters of the zeros of \(f\). Cluster pictures can be defined more abstractly in a combinatorial fashion as well; see Definition 2.1. In [\textit{W. Jaco} et al., ``On minimal ideal triangulations of cusped hyperbolic 3-manifolds'', Preprint, \url{arXiv:1808.02836}], Dokchitser, Dokchitser, Maistret, and Morgan demonstrated that the following properties of a hyperelliptic curve \(C : y^2=f(x)\) can be studied via the configuration of the zeros of \(f\) in terms of the values of the valuation: the semistability, the special fibre of its minimal regular model, the conductor of its Jacobain, and the Galois action on its first etale cohomology. The author of the paper under review tries to remove the dependency of their work on the polynomial itself and the inertia action on its roots, and explore their functionality.
The main results of the paper are as follows.
\begin{itemize}
\item[(1)] Assume that \(I\) is tame. Then, the cluster picture of a square-free polynomial \(f\) with the valuation of its leading coefficients determines \(\mathrm H^1_{\mathrm{et}}(C/\overline K,\mathbb Q_\ell)\otimes \mathbb C\) as an \(I\)-representation;
\item[(2)] If \(\deg f=3\) and \(p \ge 5\), then the cluster picture of \(f\) uniquely determines the Kodaira type of the elliptic curve;
\item[(3)] Given a combinatorially defined cluster picture \(\Sigma\), there is a polynomial \(f\in K[x]\) for sufficiently large \(p\) whose cluster picture is isomorphic to \(\Sigma\) if there is a cyclic autormorphism of \(\Sigma\) satisfying certain technical conditions.
\end{itemize}
Reviewer: Sungkon Chang (Savannah)Computing normalizers of tiled orders in \(M_n(k)\)https://zbmath.org/1527.110572024-02-28T19:32:02.718555Z"Babei, Angelica"https://zbmath.org/authors/?q=ai:babei.angelicaIf \(k\) is a non-Archimedean local field with \(R\) its valuation ring, then an order \(\Gamma\) in \(M_n(K)\) is a full \(R\)-lattice that is a subring containing the multiplicative identity in \(M_n(K)\) with \(\Gamma \otimes_R k = M_n(k).\) The order is tiled if it contains a conjugate of the ring \(diag(R,R,\dots,R).\) The normalizer of \(\Gamma\) is \({\mathcal N}(\Gamma) = \{ \xi \in GL_n(k) \ | \ \xi \Gamma \xi^{-1} = \Gamma \}.\) The author gives a five part algorithm to determine the normalizers of tiled orders in matrix algebras.
For the entire collection see [Zbl 1416.11009].
Reviewer: Steven T. Dougherty (Scranton)Generalized zeta integrals on certain real prehomogeneous vector spaceshttps://zbmath.org/1527.110852024-02-28T19:32:02.718555Z"Li, Wen-Wei"https://zbmath.org/authors/?q=ai:li.wen-wei.1|li.wen-wei|li.wenwei|li.wen-wei.2Let $R$ be the real number field. A reductive prehomogeneous vector space over $R$ is a triplet $(G,\rho,X)$ where $G$ is a connected reductive $R$-group, $X\neq\{0\}$ is a finite-dimensional $R$-vector space, and $\rho:G\to GL(X)$ is a homomorphism of algebraic groups such that $X$ has a Zariski dense open $G$-orbit $X^+$. These vector spaces are a rich source of zeta integrals with meromorphic continuation and functional equation. The author of the paper under review subsumes the archimedean zeta integrals of Godement-Jacquet, those of Sato-Shintani (in the spherical case) [\textit{M. Sato} and \textit{T. Shintani}, Ann. Math. (2) 100, 131--170 (1974; Zbl 0309.10014)], and the previous works of Bopp-Rubenthaler to prove his main results. He define local zeta integrals that involve the integration of Schwartz-Bruhat functions on $X$ against generalized matrix coefficients of admissible representations of $G(R)$, twisted by complex powers of relative invariants. He establishes the convergence of these integrals in some range, the meromorphic continuation, as well as a functional equation in terms of abstract $\gamma$-factors. The hard part of the paper is The proof of functional equations which is based on Knop's results on Capelli operators [\textit{F. Knop}, NATO ASI Ser., Ser. C, Math. Phys. Sci. 514, 301--317 (1998; Zbl 0915.20021)].
Reviewer: Manouchehr Misaghian (Prairie View)On unitarizability and Arthur packetshttps://zbmath.org/1527.220262024-02-28T19:32:02.718555Z"TadiÄ‡, Marko"https://zbmath.org/authors/?q=ai:tadic.markoThe author studies the unitary dual of \(p\)-adic classical groups and a key problem is to understand the unitarity of irreducible representations of so-called critical type, which are related to the isolated representations in the unitary dual. The paper under review constructs a two-parameter family of irreducible representations of critical type and proves their unitarity by showing that they belong to some discrete elementary Arthur packets. In general, the author also makes a conjecture that an irreducible representation of critical type is unitary if and only if it belongs to some Arthur packet. He proves the conjecture when the corank \(\le 3\).
Reviewer: Bin Xu (Beijing)Solving higher-order \(p\)-adic polynomial equations via Newton-Raphson methodhttps://zbmath.org/1527.650312024-02-28T19:32:02.718555Z"Rabago, Julius Fergy T."https://zbmath.org/authors/?q=ai:rabago.julius-fergy-tiongsonSummary: We consider the root-finding problem \(f(x) = 0\), \(f \in \mathbb Z_p[x]\), and seek a root in \(\mathbb Z_p\) of this equation through a \(p\)-adic analogue of the Newton-Raphson method. We show in particular that, under appropriate assumptions, the sequence of approximants generated by the iterative formula of the Newton-Raphson method converges to a unique root of \(f\) in \(\mathbb Z_p\). Also, we give the rate of convergence of this method in the \(p\)-adic setting. Our work generalizes previous results concerning \(q\)-th roots of \(p\)-adic numbers due to \textit{K. Mohamed} and \textit{Z. Tahar} [Filomat 27, No. 3, 429--434 (2013; Zbl 1382.11086)] and \textit{P. S. Ignacio} et al. [``Computation of square and cube roots of \(p\)-adic numbers via Newton-Raphson method'', J. Math. Res. 5, No. 2, 31--38 (2016; \url{doi:10.5539/jmr.v5n2p31})].