@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{37661,
  author       = {{Rösler, Margit and Voit, Michael}},
  issn         = {{0022-2526}},
  journal      = {{Studies in Applied Mathematics}},
  keywords     = {{Applied Mathematics}},
  number       = {{4}},
  pages        = {{474--500}},
  publisher    = {{Wiley}},
  title        = {{{Beta Distributions and Sonine Integrals for Bessel Functions on Symmetric Cones}}},
  doi          = {{10.1111/sapm.12217}},
  volume       = {{141}},
  year         = {{2018}},
}

@article{34663,
  author       = {{Black, Tobias}},
  issn         = {{1553-524X}},
  journal      = {{Discrete &amp; Continuous Dynamical Systems - B}},
  keywords     = {{Applied Mathematics, Discrete Mathematics and Combinatorics}},
  number       = {{4}},
  pages        = {{1253--1272}},
  publisher    = {{American Institute of Mathematical Sciences (AIMS)}},
  title        = {{{Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals}}},
  doi          = {{10.3934/dcdsb.2017061}},
  volume       = {{22}},
  year         = {{2017}},
}

@article{34631,
  author       = {{Hesse, Kerstin and Sloan, Ian H. and Womersley, Robert S.}},
  issn         = {{0029-599X}},
  journal      = {{Numerische Mathematik}},
  keywords     = {{Applied Mathematics, Computational Mathematics}},
  number       = {{3}},
  pages        = {{579--605}},
  publisher    = {{Springer Science and Business Media LLC}},
  title        = {{{Radial basis function approximation of noisy scattered data on the sphere}}},
  doi          = {{10.1007/s00211-017-0886-6}},
  volume       = {{137}},
  year         = {{2017}},
}

@article{45941,
  author       = {{Kovács, Balázs and Li, Buyang and Lubich, Christian and Power Guerra, Christian A.}},
  issn         = {{0029-599X}},
  journal      = {{Numerische Mathematik}},
  keywords     = {{Applied Mathematics, Computational Mathematics}},
  number       = {{3}},
  pages        = {{643--689}},
  publisher    = {{Springer Science and Business Media LLC}},
  title        = {{{Convergence of finite elements on an evolving surface driven by diffusion on the surface}}},
  doi          = {{10.1007/s00211-017-0888-4}},
  volume       = {{137}},
  year         = {{2017}},
}

@article{45942,
  author       = {{Kovács, Balázs and Lubich, Christian}},
  issn         = {{0029-599X}},
  journal      = {{Numerische Mathematik}},
  keywords     = {{Applied Mathematics, Computational Mathematics}},
  number       = {{2}},
  pages        = {{365--388}},
  publisher    = {{Springer Science and Business Media LLC}},
  title        = {{{Stability and convergence of time discretizations of quasi-linear evolution equations of Kato type}}},
  doi          = {{10.1007/s00211-017-0909-3}},
  volume       = {{138}},
  year         = {{2017}},
}

@article{45940,
  author       = {{Kovács, Balázs and Lubich, Christian}},
  issn         = {{0029-599X}},
  journal      = {{Numerische Mathematik}},
  keywords     = {{Applied Mathematics, Computational Mathematics}},
  number       = {{1}},
  pages        = {{91--117}},
  publisher    = {{Springer Science and Business Media LLC}},
  title        = {{{Stable and convergent fully discrete interior–exterior coupling of Maxwell’s equations}}},
  doi          = {{10.1007/s00211-017-0868-8}},
  volume       = {{137}},
  year         = {{2017}},
}

@article{45946,
  author       = {{Kovács, Balázs and Power Guerra, Christian Andreas}},
  issn         = {{0749-159X}},
  journal      = {{Numerical Methods for Partial Differential Equations}},
  keywords     = {{Applied Mathematics, Computational Mathematics, Numerical Analysis, Analysis}},
  number       = {{2}},
  pages        = {{518--554}},
  publisher    = {{Wiley}},
  title        = {{{Maximum norm stability and error estimates for the evolving surface finite element method}}},
  doi          = {{10.1002/num.22212}},
  volume       = {{34}},
  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{45945,
  author       = {{Kovács, Balázs and Power Guerra, Christian Andreas}},
  issn         = {{0749-159X}},
  journal      = {{Numerical Methods for Partial Differential Equations}},
  keywords     = {{Applied Mathematics, Computational Mathematics, Numerical Analysis, Analysis}},
  number       = {{2}},
  pages        = {{518--554}},
  publisher    = {{Wiley}},
  title        = {{{Maximum norm stability and error estimates for the evolving surface finite element method}}},
  doi          = {{10.1002/num.22212}},
  volume       = {{34}},
  year         = {{2017}},
}

@article{30259,
  author       = {{Wiens, Eugen and Homberg, Werner}},
  issn         = {{1877-7058}},
  journal      = {{Procedia Engineering}},
  keywords     = {{Applied Mathematics}},
  pages        = {{1755--1760}},
  publisher    = {{Elsevier BV}},
  title        = {{{Internal Flow-Turning – a new approach for the manufacture of tailored tubes with a constant external diameter}}},
  doi          = {{10.1016/j.proeng.2017.10.934}},
  volume       = {{207}},
  year         = {{2017}},
}

@article{34660,
  author       = {{Black, Tobias and Lankeit, Johannes and Mizukami, Masaaki}},
  issn         = {{0272-4960}},
  journal      = {{IMA Journal of Applied Mathematics}},
  keywords     = {{Applied Mathematics}},
  number       = {{5}},
  pages        = {{860--876}},
  publisher    = {{Oxford University Press (OUP)}},
  title        = {{{On the weakly competitive case in a two-species chemotaxis model}}},
  doi          = {{10.1093/imamat/hxw036}},
  volume       = {{81}},
  year         = {{2016}},
}

@article{34662,
  author       = {{Black, Tobias}},
  issn         = {{0022-247X}},
  journal      = {{Journal of Mathematical Analysis and Applications}},
  keywords     = {{Applied Mathematics, Analysis}},
  number       = {{1}},
  pages        = {{436--455}},
  publisher    = {{Elsevier BV}},
  title        = {{{Boundedness in a Keller–Segel system with external signal production}}},
  doi          = {{10.1016/j.jmaa.2016.08.049}},
  volume       = {{446}},
  year         = {{2016}},
}

@article{34659,
  author       = {{Black, Tobias}},
  issn         = {{0951-7715}},
  journal      = {{Nonlinearity}},
  keywords     = {{Applied Mathematics, General Physics and Astronomy, Mathematical Physics, Statistical and Nonlinear Physics}},
  number       = {{6}},
  pages        = {{1865--1886}},
  publisher    = {{IOP Publishing}},
  title        = {{{Blow-up of weak solutions to a chemotaxis system under influence of an external chemoattractant}}},
  doi          = {{10.1088/0951-7715/29/6/1865}},
  volume       = {{29}},
  year         = {{2016}},
}

@article{34661,
  author       = {{Black, Tobias}},
  issn         = {{1468-1218}},
  journal      = {{Nonlinear Analysis: Real World Applications}},
  keywords     = {{Applied Mathematics, Computational Mathematics, General Economics, Econometrics and Finance, General Engineering, General Medicine, Analysis}},
  pages        = {{593--609}},
  publisher    = {{Elsevier BV}},
  title        = {{{Sublinear signal production in a two-dimensional Keller–Segel–Stokes system}}},
  doi          = {{10.1016/j.nonrwa.2016.03.008}},
  volume       = {{31}},
  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{45936,
  author       = {{Kovács, Balázs and Power Guerra, Christian Andreas}},
  issn         = {{0749-159X}},
  journal      = {{Numerical Methods for Partial Differential Equations}},
  keywords     = {{Applied Mathematics, Computational Mathematics, Numerical Analysis, Analysis}},
  number       = {{4}},
  pages        = {{1200--1231}},
  publisher    = {{Wiley}},
  title        = {{{Error analysis for full discretizations of quasilinear parabolic problems on evolving surfaces}}},
  doi          = {{10.1002/num.22047}},
  volume       = {{32}},
  year         = {{2016}},
}

@article{45939,
  author       = {{Kovács, Balázs and Li, Buyang and Lubich, Christian}},
  issn         = {{0036-1429}},
  journal      = {{SIAM Journal on Numerical Analysis}},
  keywords     = {{Numerical Analysis, Applied Mathematics, Computational Mathematics}},
  number       = {{6}},
  pages        = {{3600--3624}},
  publisher    = {{Society for Industrial & Applied Mathematics (SIAM)}},
  title        = {{{A-Stable Time Discretizations Preserve Maximal Parabolic Regularity}}},
  doi          = {{10.1137/15m1040918}},
  volume       = {{54}},
  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{37663,
  author       = {{Rösler, Margit and Voit, Michael}},
  issn         = {{0022-247X}},
  journal      = {{Journal of Mathematical Analysis and Applications}},
  keywords     = {{Applied Mathematics, Analysis}},
  number       = {{1}},
  pages        = {{701--717}},
  publisher    = {{Elsevier BV}},
  title        = {{{A multivariate version of the disk convolution}}},
  doi          = {{10.1016/j.jmaa.2015.10.062}},
  volume       = {{435}},
  year         = {{2016}},
}