@article{53191,
  abstract     = {{<p>This paper is the first in a series of two dedicated to the study of period relations of the type <disp-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L left-parenthesis one half plus k comma normal upper Pi right-parenthesis element-of left-parenthesis 2 pi i right-parenthesis Superscript d dot k Baseline normal upper Omega Subscript left-parenthesis negative 1 right-parenthesis Sub Superscript k Subscript Baseline reverse-solidus bf upper Q left-parenthesis normal upper Pi right-parenthesis comma one half plus k critical comma">
  <mml:semantics>
    <mml:mrow>
      <mml:mi>L</mml:mi>
      <mml:mstyle scriptlevel="0">
        <mml:mrow class="MJX-TeXAtom-ORD">
          <mml:mo maxsize="1.623em" minsize="1.623em">(</mml:mo>
        </mml:mrow>
      </mml:mstyle>
      <mml:mfrac>
        <mml:mn>1</mml:mn>
        <mml:mn>2</mml:mn>
      </mml:mfrac>
      <mml:mo>+</mml:mo>
      <mml:mi>k</mml:mi>
      <mml:mo>,</mml:mo>
      <mml:mi mathvariant="normal">Π<!-- Π --></mml:mi>
      <mml:mstyle scriptlevel="0">
        <mml:mrow class="MJX-TeXAtom-ORD">
          <mml:mo maxsize="1.623em" minsize="1.623em">)</mml:mo>
        </mml:mrow>
      </mml:mstyle>
      <mml:mspace width="thickmathspace" />
      <mml:mo>∈<!-- ∈ --></mml:mo>
      <mml:mspace width="thickmathspace" />
      <mml:mo stretchy="false">(</mml:mo>
      <mml:mn>2</mml:mn>
      <mml:mi>π<!-- π --></mml:mi>
      <mml:mi>i</mml:mi>
      <mml:msup>
        <mml:mo stretchy="false">)</mml:mo>
        <mml:mrow class="MJX-TeXAtom-ORD">
          <mml:mi>d</mml:mi>
          <mml:mo>⋅<!-- ⋅ --></mml:mo>
          <mml:mi>k</mml:mi>
        </mml:mrow>
      </mml:msup>
      <mml:msub>
        <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi>
        <mml:mrow class="MJX-TeXAtom-ORD">
          <mml:mo stretchy="false">(</mml:mo>
          <mml:mo>−<!-- − --></mml:mo>
          <mml:mn>1</mml:mn>
          <mml:msup>
            <mml:mo stretchy="false">)</mml:mo>
            <mml:mi>k</mml:mi>
          </mml:msup>
        </mml:mrow>
      </mml:msub>
      <mml:mrow class="MJX-TeXAtom-ORD">
        <mml:mtext>\bf Q</mml:mtext>
      </mml:mrow>
      <mml:mo stretchy="false">(</mml:mo>
      <mml:mi mathvariant="normal">Π<!-- Π --></mml:mi>
      <mml:mo stretchy="false">)</mml:mo>
      <mml:mo>,</mml:mo>
      <mml:mspace width="1em" />
      <mml:mfrac>
        <mml:mn>1</mml:mn>
        <mml:mn>2</mml:mn>
      </mml:mfrac>
      <mml:mo>+</mml:mo>
      <mml:mi>k</mml:mi>
      <mml:mspace width="thickmathspace" />
      <mml:mtext>critical</mml:mtext>
      <mml:mo>,</mml:mo>
    </mml:mrow>
    <mml:annotation encoding="application/x-tex">\begin{equation*} L\Big (\frac {1}{2}+k,\Pi \Big )\;\in \;(2\pi i)^{d\cdot k}\Omega _{(-1)^k}\textrm {\bf Q}(\Pi ),\quad \frac {1}{2}+k\;\text {critical}, \end{equation*}</mml:annotation>
  </mml:semantics>
</mml:math>
</disp-formula>
 for certain automorphic representations <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Pi">
  <mml:semantics>
    <mml:mi mathvariant="normal">Π<!-- Π --></mml:mi>
    <mml:annotation encoding="application/x-tex">\Pi</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula> of a reductive group <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G period">
  <mml:semantics>
    <mml:mrow>
      <mml:mi>G</mml:mi>
      <mml:mo>.</mml:mo>
    </mml:mrow>
    <mml:annotation encoding="application/x-tex">G.</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula> In this paper we discuss the case <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G equals normal upper G normal upper L left-parenthesis n plus 1 right-parenthesis times normal upper G normal upper L left-parenthesis n right-parenthesis period">
  <mml:semantics>
    <mml:mrow>
      <mml:mi>G</mml:mi>
      <mml:mo>=</mml:mo>
      <mml:mrow class="MJX-TeXAtom-ORD">
        <mml:mi mathvariant="normal">G</mml:mi>
        <mml:mi mathvariant="normal">L</mml:mi>
      </mml:mrow>
      <mml:mo stretchy="false">(</mml:mo>
      <mml:mi>n</mml:mi>
      <mml:mo>+</mml:mo>
      <mml:mn>1</mml:mn>
      <mml:mo stretchy="false">)</mml:mo>
      <mml:mo>×<!-- × --></mml:mo>
      <mml:mrow class="MJX-TeXAtom-ORD">
        <mml:mi mathvariant="normal">G</mml:mi>
        <mml:mi mathvariant="normal">L</mml:mi>
      </mml:mrow>
      <mml:mo stretchy="false">(</mml:mo>
      <mml:mi>n</mml:mi>
      <mml:mo stretchy="false">)</mml:mo>
      <mml:mo>.</mml:mo>
    </mml:mrow>
    <mml:annotation encoding="application/x-tex">G=\mathrm {GL}(n+1)\times \mathrm {GL}(n).</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula> The case <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G equals normal upper G normal upper L left-parenthesis 2 n right-parenthesis">
  <mml:semantics>
    <mml:mrow>
      <mml:mi>G</mml:mi>
      <mml:mo>=</mml:mo>
      <mml:mrow class="MJX-TeXAtom-ORD">
        <mml:mi mathvariant="normal">G</mml:mi>
        <mml:mi mathvariant="normal">L</mml:mi>
      </mml:mrow>
      <mml:mo stretchy="false">(</mml:mo>
      <mml:mn>2</mml:mn>
      <mml:mi>n</mml:mi>
      <mml:mo stretchy="false">)</mml:mo>
    </mml:mrow>
    <mml:annotation encoding="application/x-tex">G=\mathrm {GL}(2n)</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula> is discussed in part two. Our method is representation theoretic and relies on the author’s recent results on global rational structures on automorphic representations. We show that the above period relations are intimately related to the field of definition of the global representation <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Pi">
  <mml:semantics>
    <mml:mi mathvariant="normal">Π<!-- Π --></mml:mi>
    <mml:annotation encoding="application/x-tex">\Pi</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula> under consideration. The new period relations we prove are in accordance with Deligne’s Conjecture on special values of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L">
  <mml:semantics>
    <mml:mi>L</mml:mi>
    <mml:annotation encoding="application/x-tex">L</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula>-functions, and the author expects this method to apply to other cases as well.</p>}},
  author       = {{Januszewski, Fabian}},
  issn         = {{0002-9947}},
  journal      = {{Transactions of the American Mathematical Society}},
  keywords     = {{Applied Mathematics, General Mathematics}},
  number       = {{9}},
  pages        = {{6547--6580}},
  publisher    = {{American Mathematical Society (AMS)}},
  title        = {{{On period relations for automorphic 𝐿-functions I}}},
  doi          = {{10.1090/tran/7527}},
  volume       = {{371}},
  year         = {{2018}},
}

@article{48321,
  author       = {{Wessel, Lena and Erath, Kirstin}},
  issn         = {{1863-9690}},
  journal      = {{ZDM}},
  keywords     = {{General Mathematics, Education}},
  number       = {{6}},
  pages        = {{1053--1064}},
  publisher    = {{Springer Science and Business Media LLC}},
  title        = {{{Theoretical frameworks for designing and analyzing language-responsive mathematics teaching–learning arrangements}}},
  doi          = {{10.1007/s11858-018-0980-y}},
  volume       = {{50}},
  year         = {{2018}},
}

@article{40050,
  author       = {{Baeumer, Boris and Luks, Tomasz and Meerschaert, Mark M.}},
  issn         = {{0025-584X}},
  journal      = {{Mathematische Nachrichten}},
  keywords     = {{General Mathematics}},
  number       = {{17-18}},
  pages        = {{2516--2535}},
  publisher    = {{Wiley}},
  title        = {{{Space‐time fractional Dirichlet problems}}},
  doi          = {{10.1002/mana.201700111}},
  volume       = {{291}},
  year         = {{2018}},
}

@article{31272,
  author       = {{Harris, Benjamin and Weich, Tobias}},
  issn         = {{0001-8708}},
  journal      = {{Advances in Mathematics}},
  keywords     = {{General Mathematics}},
  pages        = {{176--236}},
  publisher    = {{Elsevier BV}},
  title        = {{{Wave front sets of reductive Lie group representations III}}},
  doi          = {{10.1016/j.aim.2017.03.025}},
  volume       = {{313}},
  year         = {{2017}},
}

@article{31267,
  author       = {{Guillarmou, Colin and Hilgert, Joachim and Weich, Tobias}},
  issn         = {{0025-5831}},
  journal      = {{Mathematische Annalen}},
  keywords     = {{General Mathematics}},
  number       = {{3-4}},
  pages        = {{1231--1275}},
  publisher    = {{Springer Science and Business Media LLC}},
  title        = {{{Classical and quantum resonances for hyperbolic surfaces}}},
  doi          = {{10.1007/s00208-017-1576-5}},
  volume       = {{370}},
  year         = {{2017}},
}

@article{45943,
  author       = {{Kovács, Balázs}},
  issn         = {{0272-4979}},
  journal      = {{IMA Journal of Numerical Analysis}},
  keywords     = {{Applied Mathematics, Computational Mathematics, General Mathematics}},
  number       = {{1}},
  pages        = {{430--459}},
  publisher    = {{Oxford University Press (OUP)}},
  title        = {{{High-order evolving surface finite element method for parabolic problems on evolving surfaces}}},
  doi          = {{10.1093/imanum/drx013}},
  volume       = {{38}},
  year         = {{2017}},
}

@article{53189,
  author       = {{Januszewski, Fabian}},
  issn         = {{0025-5831}},
  journal      = {{Mathematische Annalen}},
  keywords     = {{General Mathematics}},
  number       = {{3-4}},
  pages        = {{1805--1881}},
  publisher    = {{Springer Science and Business Media LLC}},
  title        = {{{Rational structures on automorphic representations}}},
  doi          = {{10.1007/s00208-017-1567-6}},
  volume       = {{370}},
  year         = {{2017}},
}

@article{48323,
  author       = {{Schüler-Meyer, Alexander and Prediger, Susanne and Kuzu, Taha and Wessel, Lena and Redder, Angelika}},
  issn         = {{1571-0068}},
  journal      = {{International Journal of Science and Mathematics Education}},
  keywords     = {{General Mathematics, Education}},
  number       = {{2}},
  pages        = {{317--339}},
  publisher    = {{Springer Science and Business Media LLC}},
  title        = {{{Is Formal Language Proficiency in the Home Language Required to Profit from a Bilingual Teaching Intervention in Mathematics? A Mixed Methods Study on Fostering Multilingual Students’ Conceptual Understanding}}},
  doi          = {{10.1007/s10763-017-9857-8}},
  volume       = {{17}},
  year         = {{2017}},
}

@article{33260,
  abstract     = {{In this paper we continue the study of group representations which are counterexamples to the Ize conjecture. As in previous papers we find new infinite series of finite groups leading to such counterexamples. These new series are quite different from the previous ones, for example the group orders do not form an arithmetic progression. However, as before we find Lie groups which contain all these groups. This additional structure was observed, but not used in the previous studies of this problem. Here we also investigate the related bifurcations. To a large extent, these are closely related to the presence of mentioned compact Lie group containing the finite groups. This might give a tool to study the bifurcations related to all low dimensional counterexamples of the Ize conjecture. It also gives an indication of where we can expect to find examples where the bifurcation behaviour is different from what we have seen in the known examples.}},
  author       = {{Lauterbach, Reiner and Schwenker, Sören}},
  issn         = {{1468-9367}},
  journal      = {{Dynamical Systems}},
  keywords     = {{Computer Science Applications, General Mathematics}},
  number       = {{1}},
  pages        = {{117--147}},
  publisher    = {{Informa UK Limited}},
  title        = {{{Equivariant bifurcations in four-dimensional fixed point spaces}}},
  doi          = {{10.1080/14689367.2016.1219696}},
  volume       = {{32}},
  year         = {{2016}},
}

@article{45944,
  author       = {{Kovács, Balázs and Power Guerra, Christian Andreas}},
  issn         = {{0272-4979}},
  journal      = {{IMA Journal of Numerical Analysis}},
  keywords     = {{Applied Mathematics, Computational Mathematics, General Mathematics}},
  number       = {{1}},
  pages        = {{460--494}},
  publisher    = {{Oxford University Press (OUP)}},
  title        = {{{Higher order time discretizations with ALE finite elements for parabolic problems on evolving surfaces}}},
  doi          = {{10.1093/imanum/drw074}},
  volume       = {{38}},
  year         = {{2016}},
}

@article{45937,
  author       = {{Kovács, Balázs and Lubich, Christian}},
  issn         = {{0272-4979}},
  journal      = {{IMA Journal of Numerical Analysis}},
  keywords     = {{Applied Mathematics, Computational Mathematics, General Mathematics}},
  number       = {{1}},
  pages        = {{1--39}},
  publisher    = {{Oxford University Press (OUP)}},
  title        = {{{Numerical analysis of parabolic problems with dynamic boundary conditions}}},
  doi          = {{10.1093/imanum/drw015}},
  volume       = {{37}},
  year         = {{2016}},
}

@article{53188,
  author       = {{Januszewski, Fabian}},
  issn         = {{2195-4755}},
  journal      = {{Annales mathématiques du Québec; Special Issue in Honor of Glenn Stevens}},
  keywords     = {{General Mathematics}},
  number       = {{2}},
  pages        = {{453--489}},
  publisher    = {{Springer Science and Business Media LLC}},
  title        = {{{p-adic L-functions for Rankin–Selberg convolutions over number fields}}},
  doi          = {{10.1007/s40316-016-0061-y}},
  volume       = {{40}},
  year         = {{2016}},
}

@article{40066,
  author       = {{Bañuelos, Rodrigo and Bogdan, Krzysztof and Luks, Tomasz}},
  issn         = {{0024-6107}},
  journal      = {{Journal of the London Mathematical Society}},
  keywords     = {{General Mathematics}},
  number       = {{2}},
  pages        = {{462--478}},
  publisher    = {{Wiley}},
  title        = {{{Hardy–Stein identities and square functions for semigroups}}},
  doi          = {{10.1112/jlms/jdw042}},
  volume       = {{94}},
  year         = {{2016}},
}

@article{31291,
  abstract     = {{<jats:p>We consider a simple model of an open partially expanding map. Its trapped set <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0143385715000346_inline1" /><jats:tex-math>${\mathcal{K}}$</jats:tex-math></jats:alternatives></jats:inline-formula> in phase space is a fractal set. We first show that there is a well-defined discrete spectrum of Ruelle resonances which describes the asymptotic of correlation functions for large time and which is parametrized by the Fourier component <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0143385715000346_inline2" /><jats:tex-math>$\unicode[STIX]{x1D708}$</jats:tex-math></jats:alternatives></jats:inline-formula> in the neutral direction of the dynamics. We introduce a specific hypothesis on the dynamics that we call ‘minimal captivity’. This hypothesis is stable under perturbations and means that the dynamics is univalued in a neighborhood of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0143385715000346_inline3" /><jats:tex-math>${\mathcal{K}}$</jats:tex-math></jats:alternatives></jats:inline-formula>. Under this hypothesis we show the existence of an asymptotic spectral gap and a fractal Weyl law for the upper bound of density of Ruelle resonances in the semiclassical limit <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0143385715000346_inline4" /><jats:tex-math>$\unicode[STIX]{x1D708}\rightarrow \infty$</jats:tex-math></jats:alternatives></jats:inline-formula>. Some numerical computations with the truncated Gauss map and Bowen–Series maps illustrate these results.</jats:p>}},
  author       = {{ARNOLDI, JEAN FRANCOIS and FAURE, FRÉDÉRIC and Weich, Tobias}},
  issn         = {{0143-3857}},
  journal      = {{Ergodic Theory and Dynamical Systems}},
  keywords     = {{Applied Mathematics, General Mathematics}},
  number       = {{1}},
  pages        = {{1--58}},
  publisher    = {{Cambridge University Press (CUP)}},
  title        = {{{Asymptotic spectral gap and Weyl law for Ruelle resonances of open partially expanding maps}}},
  doi          = {{10.1017/etds.2015.34}},
  volume       = {{37}},
  year         = {{2015}},
}

@article{53185,
  author       = {{Januszewski, Fabian}},
  issn         = {{1073-7928}},
  journal      = {{International Mathematics Research Notices}},
  keywords     = {{General Mathematics}},
  number       = {{17}},
  pages        = {{7884--7949}},
  publisher    = {{Oxford University Press (OUP)}},
  title        = {{{On p-adic L-functions for GL(n) × GL (n-1) over totally real fields}}},
  doi          = {{10.1093/imrn/rnu181}},
  volume       = {{2015}},
  year         = {{2014}},
}

@article{37667,
  author       = {{Rösler, Margit and Remling, Heiko}},
  issn         = {{0021-9045}},
  journal      = {{Journal of Approximation Theory}},
  keywords     = {{Applied Mathematics, General Mathematics, Numerical Analysis, Analysis}},
  pages        = {{30--48}},
  publisher    = {{Elsevier BV}},
  title        = {{{Convolution algebras for Heckman–Opdam polynomials derived from compact Grassmannians}}},
  doi          = {{10.1016/j.jat.2014.07.005}},
  volume       = {{197}},
  year         = {{2014}},
}

@article{34845,
  abstract     = {{Computational Galois theory, in particular the problem of computing the Galois group of a given polynomial, is a very old problem. Currently, the best algorithmic solution is Stauduhar’s method. Computationally, one of the key challenges in the application of Stauduhar’s method is to find, for a given pair of groups H<G, a G-relative H-invariant, that is a multivariate polynomial F that is H-invariant, but not G-invariant. While generic, theoretical methods are known to find such F, in general they yield impractical answers. We give a general method for computing invariants of large degree which improves on previous known methods, as well as various special invariants that are derived from the structure of the groups. We then apply our new invariants to the task of computing the Galois groups of polynomials over the rational numbers, resulting in the first practical degree independent algorithm.}},
  author       = {{Fieker, Claus and Klüners, Jürgen}},
  issn         = {{1461-1570}},
  journal      = {{LMS Journal of Computation and Mathematics}},
  keywords     = {{Computational Theory and Mathematics, General Mathematics}},
  number       = {{1}},
  pages        = {{141--158}},
  publisher    = {{Wiley}},
  title        = {{{Computation of Galois groups of rational polynomials}}},
  doi          = {{10.1112/s1461157013000302}},
  volume       = {{17}},
  year         = {{2014}},
}

@article{42801,
  abstract     = {{We exhibit a practical algorithm for solving the constructive membership problem for discrete free subgroups of rank 2 in PSL₂(R) or SL₂(R). This algorithm, together with methods for checking whether a two-generator subgroup of PSL₂(R) or SL₂(R) is discrete and free, have been implemented in Magma for groups defined over real algebraic number fields.}},
  author       = {{Kirschmer, Markus and LEEDHAM-GREEN, CHARLES}},
  issn         = {{0017-0895}},
  journal      = {{Glasgow Mathematical Journal}},
  keywords     = {{General Mathematics}},
  number       = {{1}},
  pages        = {{173--180}},
  publisher    = {{Cambridge University Press (CUP)}},
  title        = {{{Computing with subgroups of the modular group }}},
  doi          = {{10.1017/s0017089514000202}},
  volume       = {{57}},
  year         = {{2014}},
}

@article{42794,
  abstract     = {{We exhibit a practical algorithm for solving the constructive membership problem for discrete free subgroups of rank 2 in PSL₂(R) or SL₂(R). This algorithm, together with methods for checking whether a two-generator subgroup of PSL₂(R) or SL₂(R) is discrete and free, have been implemented in Magma for groups defined over real algebraic number fields.}},
  author       = {{Eick, B. and Kirschmer, Markus and Leedham-Green, C.}},
  issn         = {{1461-1570}},
  journal      = {{LMS Journal of Computation and Mathematics}},
  keywords     = {{Computational Theory and Mathematics, General Mathematics}},
  number       = {{1}},
  pages        = {{345--359}},
  publisher    = {{Wiley}},
  title        = {{{The constructive membership problem for discrete free subgroups of rank 2 of SL₂(R)}}},
  doi          = {{10.1112/s1461157014000047}},
  volume       = {{17}},
  year         = {{2014}},
}

@article{53419,
  author       = {{Gundlach, Fabian}},
  issn         = {{0024-6107}},
  journal      = {{Journal of the London Mathematical Society}},
  keywords     = {{General Mathematics}},
  number       = {{2}},
  pages        = {{599--618}},
  publisher    = {{Wiley}},
  title        = {{{Integral Brauer-Manin obstructions for sums of two squares and a power}}},
  doi          = {{10.1112/jlms/jdt042}},
  volume       = {{88}},
  year         = {{2013}},
}