2,103 research outputs found

    Elementary abelian subgroups: from algebraic groups to finite groups

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    We describe a new approach for classifying conjugacy classes of elementary abelian subgroups in simple algebraic groups over an algebraically closed field, and understanding the normaliser and centraliser structure of these. For toral subgroups, we give an effective classification algorithm. For non-toral elementary abelian subgroups, we focus on algebraic groups of exceptional type with a view to future applications, and in this case we provide tables explicitly describing the subgroups and their local structure. We then describe how to transfer results to the corresponding finite groups of Lie type using the Lang-Steinberg Theorem; this will be used in forthcoming work to complete the classification of elementary abelian p-subgroups for torsion primes p in finite groups of exceptional Lie type. Such classification results are important for determining the maximal p-local subgroups and p-radical subgroups, both of which play a crucial role in modular representation theory

    Symmetries of Riemann surfaces and magnetic monopoles

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    This thesis studies, broadly, the role of symmetry in elucidating structure. In particular, I investigate the role that automorphisms of algebraic curves play in three specific contexts; determining the orbits of theta characteristics, influencing the geometry of the highly-symmetric Bring’s curve, and in constructing magnetic monopole solutions. On theta characteristics, I show how to turn questions on the existence of invariant characteristics into questions of group cohomology, compute comprehensive tables of orbit decompositions for curves of genus 9 or less, and prove results on the existence of infinite families of curves with invariant characteristics. On Bring’s curve, I identify key points with geometric significance on the curve, completely determine the structure of the quotients by subgroups of automorphisms, finding new elliptic curves in the process, and identify the unique invariant theta characteristic on the curve. With respect to monopoles, I elucidate the role that the Hitchin conditions play in determining monopole spectral curves, the relation between these conditions and the automorphism group of the curve, and I develop the theory of computing Nahm data of symmetric monopoles. As such I classify all 3-monopoles whose Nahm data may be solved for in terms of elliptic functions

    On the essential torsion finiteness of abelian varieties over torsion fields

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    The classical Mordell-Weil theorem implies that an abelian variety AA over a number field KK has only finitely many KK-rational torsion points. This finitude of torsion still holds even over the cyclotomic extension Kcyc=KQabK^{\rm cyc}=K\mathbb{Q}^{\mathrm{ab}} by a result of Ribet. In this article, we consider the finiteness of torsion points of an abelian variety AA over the infinite algebraic extension KBK_B obtained by adjoining the coordinates of all torsion points of an abelian variety BB. Assuming the Mumford-Tate conjecture, and up to a finite extension of the base field KK, we give a necessary and sufficient condition for the finiteness of A(KB)torsA(K_B)_{\rm tors} in terms of Mumford--Tate groups. We give a complete answer when both abelian varieties have dimension both three, or when both have complex multiplication.Comment: 35 page

    Realizing a Fake Projective Plane as a Degree 25 Surface in P5\mathbb P^5

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    Fake projective planes are smooth complex surfaces of general type with Betti numbers equal to that of the usual projective plane. Recent explicit constructions of fake projective planes embed them via their bicanonical embedding in P9\mathbb P^9. In this paper, we study Keum's fake projective plane (a=7,p=2,{7},D327)(a=7, p=2, \{7\}, D_3 2_7) and use the equations of \cite{Borisov} to construct an embedding of fake projective plane in P5\mathbb P^5. We also simplify the 84 cubic equations defining the fake projective plane in P9\mathbb P^9.Comment: 11 pages, 1 table. Mathematica, Magma, and Macaulay2 code and key equations from the paper are included in separate files for convenienc

    Minimal PD-sets for codes associated with the graphs Qm2, m even

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    Please read abstract in the article.The National Research Foundation of South Africahttp://link.springer.com/journal/2002021-12-08hj2021Mathematics and Applied Mathematic

    KK-theoretic counterexamples to Ravenel's telescope conjecture

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    At each prime pp and height n+1≄2n+1 \ge 2, we prove that the telescopic and chromatic localizations of spectra differ. Specifically, for Z\mathbb{Z} acting by Adams operations on BP⟹n⟩\mathrm{BP}\langle n \rangle, we prove that the T(n+1)T(n+1)-localized algebraic KK-theory of BP⟹n⟩hZ\mathrm{BP}\langle n \rangle^{h\mathbb{Z}} is not K(n+1)K(n+1)-local. We also show that Galois hyperdescent, A1\mathbb{A}^1-invariance, and nil-invariance fail for the K(n+1)K(n+1)-localized algebraic KK-theory of K(n)K(n)-local E∞\mathbb{E}_{\infty}-rings. In the case n=1n=1 and p≄7p \ge 7 we make complete computations of T(2)∗K(R)T(2)_*\mathrm{K}(R), for RR certain finite Galois extensions of the K(1)K(1)-local sphere. We show for p≄5p\geq 5 that the algebraic KK-theory of the K(1)K(1)-local sphere is asymptotically L2fL_2^{f}-local.Comment: 100 pages. Comments very welcom

    Hodge-Tate stacks and non-abelian pp-adic Hodge theory of v-perfect complexes on rigid spaces

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    Let XX be a quasi-compact quasi-separated pp-adic formal scheme that is smooth either over a perfectoid Zp\mathbb{Z}_p-algebra or over some ring of integers of a complete discretely valued extension of Qp\mathbb{Q}_p with pp-finite residue field. We construct a fully faithful functor from perfect complexes on the Hodge-Tate stack of XX up to isogeny to perfect complexes on the v-site of the generic fibre of XX. Moreover, we describe perfect complexes on the Hodge-Tate stack in terms of certain derived categories of Higgs, resp. Higgs-Sen modules. This leads to a derived pp-adic Simpson functor. We deduce new results about the pp-adic Simpson correspondence in both cases

    Generating Polynomials of Exponential Random Graphs

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    The theory of random graphs describes the interplay between probability and graph theory: it is the study of the stochastic process by which graphs form and evolve. In 1959, Erdős and Rényi defined the foundational model of random graphs on n vertices, denoted G(n, p) ([ER84]). Subsequently, Frank and Strauss (1986) added a Markov twist to this story by describing a topological structure on random graphs that encodes dependencies between local pairs of vertices ([FS86]). The general model that describes this framework is called the exponential random graph model (ERGM). In the past, determining when a probability distribution has strong negative dependence has proven to be difficult ([Pem00, BBL09]). The negative dependence of a probability distribution is characterized by properties of its corresponding generating polynomial ([BBL09]). This thesis bridges the theory of exponential random graphs with the geometry of their generating polynomials, namely, when and how they satisfy the stable or Lorentzian properties ([Wag09, BBL09, BH20, AGV21]). We provide necessary and sufficient conditions as well as full characterizations of the parameter space for when this model has a stable or Lorentzian generating polynomial. This is done using a well-developed dictionary between probability distributions and their corresponding multiaffine generating polynomials. In particular, we characterize when the generating polynomial of a random graph model with a large symmetry group is irreducible. We assert that the edge parameter of the exponential random graph model does not affect stability and that the triangle and k-star parameters are necessarily related if the model is stable or Lorentzian. We also provide full Lorentzian and stable characterizations for the model on K3 and a Lorentzian characterization for specializations of the model on K4

    Left braces of size 8 p

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    We describe all left braces aof size 8 p for an odd p diferent from 3 and 7.Postprint (author's final draft
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