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25847 results in Abstract analysis

Page 121 of 1293

  • Operator equalities and Characterizations of Orthogonality in Pre-Hilbert C*-Modules

  • Part of
  • Rasoul Eskandari, M. S. Moslehian, Dan Popovici
  • Journal:
    Proceedings of the Edinburgh Mathematical Society / Volume 64 / Issue 3 / August 2021
    Published online by Cambridge University Press:
    17 June 2021, pp. 594-614

  • In the first part of the paper, we use states on $C^{*}$-algebras in order to establish some equivalent statements to equality in the triangle inequality, as well as to the parallelogram identity for elements of a pre-Hilbert $C^{*}$-module. We also characterize the equality case in the triangle inequality for adjointable operators on a Hilbert $C^{*}$-module. Then we give certain necessary and sufficient conditions to the Pythagoras identity for two vectors in a pre-Hilbert $C^{*}$-module under the assumption that their inner product has a negative real part. We introduce the concept of Pythagoras orthogonality and discuss its properties. We describe this notion for Hilbert space operators in terms of the parallelogram law and some limit conditions. We present several examples in order to illustrate the relationship between the Birkhoff–James, Roberts, and Pythagoras orthogonalities, and the usual orthogonality in the framework of Hilbert $C^{*}$-modules.

  • GEOMETRIC QUADRATIC CHABAUTY

  • Part of
  • Bas Edixhoven, Guido Lido
  • Journal:
    Journal of the Institute of Mathematics of Jussieu / Volume 22 / Issue 1 / January 2023
    Published online by Cambridge University Press:
    17 June 2021, pp. 279-333
    Print publication:
    January 2023
    • Article
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  • Since Faltings proved Mordell’s conjecture in [16] in 1983, we have known that the sets of rational points on curves of genus at least $2$ are finite. Determining these sets in individual cases is still an unsolved problem. Chabauty’s method (1941) [10] is to intersect, for a prime number p, in the p-adic Lie group of p-adic points of the Jacobian, the closure of the Mordell–Weil group with the p-adic points of the curve. Under the condition that the Mordell–Weil rank is less than the genus, Chabauty’s method, in combination with other methods such as the Mordell–Weil sieve, has been applied successfully to determine all rational points in many cases.

    Minhyong Kim’s nonabelian Chabauty programme aims to remove the condition on the rank. The simplest case, called quadratic Chabauty, was developed by Balakrishnan, Besser, Dogra, Müller, Tuitman and Vonk, and applied in a tour de force to the so-called cursed curve (rank and genus both $3$).

    This article aims to make the quadratic Chabauty method small and geometric again, by describing it in terms of only ‘simple algebraic geometry’ (line bundles over the Jacobian and models over the integers).

Page 121 of 1293