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Статті в журналах з теми "Automorphic periods"

Автор: Grafiati

Опубліковано: 5 квітня 2025

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1

Jacquet, Hervé, Erez Lapid, and Jonathan Rogawski. "Periods of automorphic forms." Journal of the American Mathematical Society 12, no. 1 (1999): 173–240. http://dx.doi.org/10.1090/s0894-0347-99-00279-9.

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2

Frahm, Jan, and Feng Su. "Upper bounds for geodesic periods over rank one locally symmetric spaces." Forum Mathematicum 30, no. 5 (September 1, 2018): 1065–77. http://dx.doi.org/10.1515/forum-2017-0185.

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Анотація:

AbstractWe prove upper bounds for geodesic periods of automorphic forms over general rank one locally symmetric spaces. Such periods are integrals of automorphic forms restricted to special totally geodesic cycles of the ambient manifold and twisted with automorphic forms on the cycles. The upper bounds are in terms of the Laplace eigenvalues of the two automorphic forms, and they generalize previous results for real hyperbolic manifolds to the context of all rank one locally symmetric spaces.

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3

Zelditch, Steven. "geodesic periods of automorphic forms." Duke Mathematical Journal 56, no. 2 (April 1988): 295–344. http://dx.doi.org/10.1215/s0012-7094-88-05613-x.

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4

Yamana, Shunsuke. "Periods of residual automorphic forms." Journal of Functional Analysis 268, no. 5 (March 2015): 1078–104. http://dx.doi.org/10.1016/j.jfa.2014.11.009.

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5

Ichino, Atsushi, and Shunsuke Yamana. "Periods of automorphic forms: the case of." Compositio Mathematica 151, no. 4 (November 13, 2014): 665–712. http://dx.doi.org/10.1112/s0010437x14007362.

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Анотація:

Following Jacquet, Lapid and Rogawski, we define a regularized period of an automorphic form on $\text{GL}_{n+1}\times \text{GL}_{n}$ along the diagonal subgroup $\text{GL}_{n}$ and express it in terms of the Rankin–Selberg integral of Jacquet, Piatetski-Shapiro and Shalika. This extends the theory of Rankin–Selberg integrals to all automorphic forms on $\text{GL}_{n+1}\times \text{GL}_{n}$.

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6

Lee, Min Ho. "Mixed automorphic forms and differential equations." International Journal of Mathematics and Mathematical Sciences 13, no. 4 (1990): 661–68. http://dx.doi.org/10.1155/s0161171290000916.

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Анотація:

We construct mixed automorphic forms associated to a certain class of nonhomogeneous linear ordinary differential equations. We also establish an isomorphism between the space of mixed automorphic forms of the second kind modulo exact forms nd a certain parabolic cohomology explicitly in terms of the periods of mixed automorphic forms.

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7

Daughton, Austin. "A Hecke correspondence theorem for automorphic integrals with infinite log-polynomial sum period functions." International Journal of Number Theory 10, no. 07 (September 9, 2014): 1857–79. http://dx.doi.org/10.1142/s1793042114500596.

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We generalize the correspondence between Dirichlet series with finitely many poles that satisfy a functional equation and automorphic integrals with log-polynomial sum period functions. In particular, we extend the correspondence to hold for Dirichlet series with finitely many essential singularities. We also study Dirichlet series with infinitely many poles in a vertical strip. For Hecke groups with λ ≥ 2 and some weights, we prove a similar correspondence for these Dirichlet series. For this case, we provide a way to estimate automorphic integrals with infinite log-polynomial periods by automorphic integrals with finite log-polynomial periods.

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8

Yamana, Shunsuke. "PERIODS OF AUTOMORPHIC FORMS: THE TRILINEAR CASE." Journal of the Institute of Mathematics of Jussieu 17, no. 1 (October 26, 2015): 59–74. http://dx.doi.org/10.1017/s1474748015000377.

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Following Jacquet, Lapid and Rogawski, we regularize trilinear periods. We use the regularized trilinear periods to compute Fourier–Jacobi periods of residues of Eisenstein series on metaplectic groups, which has an application to the Gan–Gross–Prasad conjecture.

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9

ZYDOR, Michal. "Periods of automorphic forms over reductive subgroups." Annales scientifiques de l'École Normale Supérieure 55, no. 1 (2022): 141–83. http://dx.doi.org/10.24033/asens.2493.

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10

Sharp, Richard. "Closed Geodesics and Periods of Automorphic Forms." Advances in Mathematics 160, no. 2 (June 2001): 205–16. http://dx.doi.org/10.1006/aima.2001.1987.

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11

Lysenko, Sergey. "Geometric Waldspurger periods." Compositio Mathematica 144, no. 2 (March 2008): 377–438. http://dx.doi.org/10.1112/s0010437x07003156.

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AbstractLet X be a smooth projective curve. We consider the dual reductive pair $H=\mathrm {G\mathbb {O}}_{2m}$, $G=\mathrm {G\mathbb {S}p}_{2n}$ over X, where H splits on an étale two-sheeted covering $\pi :\tilde X\to X$. Let BunG (respectively, BunH) be the stack of G-torsors (respectively, H-torsors) on X. We study the functors FG and FH between the derived categories D(BunG) and D(BunH), which are analogs of the classical theta-lifting operators in the framework of the geometric Langlands program. Assume n=m=1 and H nonsplit, that is, $H=\pi _*{\mathbb {G}_m}$ with $\tilde X$ connected. We establish the geometric Langlands functoriality for this pair. Namely, we show that FG :D(BunH)→D(BunG) commutes with Hecke operators with respect to the corresponding map of Langlands L-groups LH→LG. As an application, we calculate Waldspurger periods of cuspidal automorphic sheaves on BunGL2 and Bessel periods of theta-lifts from $\mathrm {Bun}_{\mathrm {G\mathbb {O}}_4}$ to $\mathrm {Bun}_{\mathrm {G\mathbb {S}p}_4}$. Based on these calculations, we give three conjectural constructions of certain automorphic sheaves on $\mathrm {Bun}_{\mathrm {G\mathbb {S}p}_4}$ (one of them makes sense for ${\mathcal D}$-modules only).

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12

Grobner, Harald, and Michael Harris. "WHITTAKER PERIODS, MOTIVIC PERIODS, AND SPECIAL VALUES OF TENSOR PRODUCT -FUNCTIONS." Journal of the Institute of Mathematics of Jussieu 15, no. 4 (March 31, 2015): 711–69. http://dx.doi.org/10.1017/s1474748014000462.

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Анотація:

Let${\mathcal{K}}$be an imaginary quadratic field. Let${\rm\Pi}$and${\rm\Pi}^{\prime }$be irreducible generic cohomological automorphic representation of$\text{GL}(n)/{\mathcal{K}}$and$\text{GL}(n-1)/{\mathcal{K}}$, respectively. Each of them can be given two natural rational structures over number fields. One is defined by the rational structure on topological cohomology, and the other is given in terms of the Whittaker model. The ratio between these rational structures is called aWhittaker period. An argument presented by Mahnkopf and Raghuram shows that, at least if${\rm\Pi}$is cuspidal and the weights of${\rm\Pi}$and${\rm\Pi}^{\prime }$are in a standard relative position, the critical values of the Rankin–Selberg product$L(s,{\rm\Pi}\times {\rm\Pi}^{\prime })$are essentially algebraic multiples of the product of the Whittaker periods of${\rm\Pi}$and${\rm\Pi}^{\prime }$. We show that, under certain regularity and polarization hypotheses, the Whittaker period of a cuspidal${\rm\Pi}$can be given a motivic interpretation, and can also be related to a critical value of the adjoint$L$-function of related automorphic representations of unitary groups. The resulting expressions for critical values of the Rankin–Selberg and adjoint$L$-functions are compatible with Deligne’s conjecture.

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13

Anandavardhanan, U. K., and Dipendra Prasad. "A local-global question in automorphic forms." Compositio Mathematica 149, no. 6 (April 26, 2013): 959–95. http://dx.doi.org/10.1112/s0010437x12000772.

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AbstractIn this paper, we consider the $\mathrm{SL} (2)$ analogue of two well-known theorems about period integrals of automorphic forms on $\mathrm{GL} (2)$: one due to Harder–Langlands–Rapoport about non-vanishing of period integrals on ${\mathrm{GL} }_{2} ({ \mathbb{A} }_{F} )$ of cuspidal automorphic representations on ${\mathrm{GL} }_{2} ({ \mathbb{A} }_{E} )$ where $E$ is a quadratic extension of a number field $F$, and the other due to Waldspurger involving toric periods of automorphic forms on ${\mathrm{GL} }_{2} ({ \mathbb{A} }_{F} )$. In both these cases, now involving $\mathrm{SL} (2)$, we analyze period integrals on global$L$-packets; we prove that under certain conditions, a global automorphic $L$-packet which at each place of a number field has a distinguished representation, contains globally distinguished representations, and further, an automorphic representation which is locally distinguished is globally distinguished.

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14

Sugiyama, Shingo. "Regularized periods of automorphic forms on $GL(2)$." Tohoku Mathematical Journal 65, no. 3 (2013): 373–409. http://dx.doi.org/10.2748/tmj/1378991022.

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15

Lee, M. H. "Periods of mixed automorphic forms and differential equations." Applied Mathematics Letters 11, no. 4 (July 1998): 15–19. http://dx.doi.org/10.1016/s0893-9659(98)00049-4.

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16

Gon, Yasuro. "Dirichlet series constructed from periods of automorphic forms." Mathematische Zeitschrift 281, no. 3-4 (July 28, 2015): 747–73. http://dx.doi.org/10.1007/s00209-015-1506-8.

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17

Ichino, Atsushi, and Shunsuke Yamana. "Periods of automorphic forms: The case of (U n+1×U n ,U n )." Journal für die reine und angewandte Mathematik (Crelles Journal) 2019, no. 746 (January 1, 2019): 1–38. http://dx.doi.org/10.1515/crelle-2015-0107.

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Анотація:

Abstract Following Jacquet, Lapid and Rogawski, we define regularized periods of automorphic forms on \mathrm{U}_{n+1} \times \mathrm{U}_{n} along the diagonal subgroup {\mathrm{U}_{n}} and compute the regularized periods of cuspidal Eisenstein series and their residues. The formula for the periods of residues has an application to the Gan–Gross–Prasad conjecture.

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18

Zelditch, Steven. "Geodesics in homology classes and periods of automorphic forms." Advances in Mathematics 88, no. 1 (July 1991): 113–29. http://dx.doi.org/10.1016/0001-8708(91)90004-q.

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19

Harris, Michael. "Beilinson–Bernstein Localization Over and Periods of Automorphic Forms." International Mathematics Research Notices 2013, no. 9 (March 27, 2012): 2000–2053. http://dx.doi.org/10.1093/imrn/rns101.

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20

Van Order, Jeanine. "$p$-Adic Interpolation of Automorphic Periods for GL$_2$." Documenta Mathematica 22 (2017): 1467–99. http://dx.doi.org/10.4171/dm/601.

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21

Venkatesh, Akshay. "Cohomology of arithmetic groups and periods of automorphic forms." Japanese Journal of Mathematics 12, no. 1 (December 21, 2016): 1–32. http://dx.doi.org/10.1007/s11537-016-1488-2.

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22

Flicker, Yuval Z. "Automorphic Forms with Anisotropic Periods on a Unitary Group." Journal of Algebra 220, no. 2 (October 1999): 636–63. http://dx.doi.org/10.1006/jabr.1998.7928.

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23

Flicker, Yuval Z. "Stable Bi-Period Summation Formula and Transfer Factors." Canadian Journal of Mathematics 51, no. 4 (August 1, 1999): 771–91. http://dx.doi.org/10.4153/cjm-1999-033-9.

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AbstractThis paper starts by introducing a bi-periodic summation formula for automorphic forms on a group G(E), with periods by a subgroup G(F), where E/F is a quadratic extension of number fields. The split case, where E = F ⊕ F, is that of the standard trace formula. Then it introduces a notion of stable bi-conjugacy, and stabilizes the geometric side of the bi-period summation formula. Thus weighted sums in the stable bi-conjugacy class are expressed in terms of stable bi-orbital integrals. These stable integrals are on the same endoscopic groups H which occur in the case of standard conjugacy.The spectral side of the bi-period summation formula involves periods, namely integrals over the group of F-adele points of G, of cusp forms on the group of E-adele points on the group G. Our stabilization suggests that such cusp forms—with non vanishing periods—and the resulting bi-period distributions associated to “periodic” automorphic forms, are related to analogous bi-period distributions associated to “periodic” automorphic forms on the endoscopic symmetric spaces H(E)/H(F). This offers a sharpening of the theory of liftings, where periods play a key role.The stabilization depends on the “fundamental lemma”, which conjectures that the unit elements of the Hecke algebras on G and H have matching orbital integrals. Even in stating this conjecture, one needs to introduce a “transfer factor”. A generalization of the standard transfer factor to the bi-periodic case is introduced. The generalization depends on a new definition of the factors even in the standard case.Finally, the fundamental lemma is verified for SL(2).

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24

Shimura, Goro. "On the fundamental periods of automorphic forms of arithmetic type." Inventiones Mathematicae 102, no. 1 (December 1990): 399–428. http://dx.doi.org/10.1007/bf01233433.

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25

Wang, Bryan Peng Jun. "Global theta lifting and automorphic periods associated to nilpotent orbits." Journal of Number Theory 271 (June 2025): 122–49. https://doi.org/10.1016/j.jnt.2024.11.004.

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26

Gan, Wee Teck, and Atsushi Ichino. "On endoscopy and the refined Gross–Prasad conjecture for (SO5, SO4)." Journal of the Institute of Mathematics of Jussieu 10, no. 2 (August 5, 2010): 235–324. http://dx.doi.org/10.1017/s1474748010000198.

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Анотація:

AbstractWe prove an explicit formula for periods of certain automorphic forms on SO5 × SO4 along the diagonal subgroup SO4 in terms of L-values. Our formula also involves a quantity from the theory of endoscopy, as predicted by the refined Gross–Prasad conjecture.

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27

Ginzburg, David, DiHua Jiang, and David Soudry. "Periods of automorphic forms, poles of L-functions and functorial lifting." Science China Mathematics 53, no. 9 (June 28, 2010): 2215–38. http://dx.doi.org/10.1007/s11425-010-4020-9.

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28

Harris, Michael. "Beilinson–Bernstein Localization Over ℚ and Periods of Automorphic Forms: Erratum". International Mathematics Research Notices 2020, № 3 (23 березня 2018): 957–60. http://dx.doi.org/10.1093/imrn/rny043.

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29

FLICKER, YUVAL Z. "CUSP FORMS ON GSp(4) WITH SO(4)-PERIODS." International Journal of Number Theory 07, no. 04 (June 2011): 855–919. http://dx.doi.org/10.1142/s1793042111004186.

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Анотація:

The Saito–Kurokawa lifting of automorphic representations from PGL(2) to the projective symplectic group of similitudes PGSp(4) of genus 2 is studied using the Fourier summation formula (an instance of the "relative trace formula"), thus characterizing the image as the representations with a nonzero period for the special orthogonal group SO(4, E/F) associated to a quadratic extension E of the global base field F, and a nonzero Fourier coefficient for a generic character of the unipotent radical of the Siegel parabolic subgroup. The image is nongeneric and almost everywhere nontempered, violating a naive generalization of the Ramanujan conjecture. Technical advances here concern the development of the summation formula and matching of relative orbital integrals.

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30

Jiang, Dihua, та Chenyan Wu. "Periods and (χ, b)-factors of cuspidal automorphic forms of symplectic groups". Israel Journal of Mathematics 225, № 1 (21 березня 2018): 267–320. http://dx.doi.org/10.1007/s11856-018-1658-4.

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31

Pal, Aprameyo, and Carlos de Vera‐Piquero. "AUTOMORPHIC SL 2 ‐PERIODS AND THE SUBCONVEXITY PROBLEM FOR GL 2 × GL 3." Mathematika 66, no. 4 (July 3, 2020): 855–99. http://dx.doi.org/10.1112/mtk.12047.

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32

Xue, Hang. "Refined global Gan–Gross–Prasad conjecture for Fourier–Jacobi periods on symplectic groups." Compositio Mathematica 153, no. 1 (January 2017): 68–131. http://dx.doi.org/10.1112/s0010437x16007752.

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Анотація:

In this paper, we propose a conjectural identity between the Fourier–Jacobi periods on symplectic groups and the central value of certain Rankin–Selberg $L$-functions. This identity can be viewed as a refinement to the global Gan–Gross–Prasad conjecture for $\text{Sp}(2n)\times \text{Mp}(2m)$. To support this conjectural identity, we show that when $n=m$ and $n=m\pm 1$, it can be deduced from the Ichino–Ikeda conjecture in some cases via theta correspondences. As a corollary, the conjectural identity holds when $n=m=1$ or when $n=2$, $m=1$ and the automorphic representation on the bigger group is endoscopic.

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33

Gourevitch, Dmitry, Henrik P. A. Gustafsson, Axel Kleinschmidt, Daniel Persson, and Siddhartha Sahi. "A reduction principle for Fourier coefficients of automorphic forms." Mathematische Zeitschrift 300, no. 3 (October 15, 2021): 2679–717. http://dx.doi.org/10.1007/s00209-021-02784-w.

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AbstractWe consider a special class of unipotent periods for automorphic forms on a finite cover of a reductive adelic group $$\mathbf {G}(\mathbb {A}_\mathbb {K})$$ G ( A K ) , which we refer to as Fourier coefficients associated to the data of a ‘Whittaker pair’. We describe a quasi-order on Fourier coefficients, and an algorithm that gives an explicit formula for any coefficient in terms of integrals and sums involving higher coefficients. The maximal elements for the quasi-order are ‘Levi-distinguished’ Fourier coefficients, which correspond to taking the constant term along the unipotent radical of a parabolic subgroup, and then further taking a Fourier coefficient with respect to a $${\mathbb K}$$ K -distinguished nilpotent orbit in the Levi quotient. Thus one can express any Fourier coefficient, including the form itself, in terms of higher Levi-distinguished coefficients. In companion papers we use this result to determine explicit Fourier expansions of minimal and next-to-minimal automorphic forms on split simply-laced reductive groups, and to obtain Euler product decompositions of certain Fourier coefficients.

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34

Sato, Fumihiro. "Zeta functions of prehomogeneous vector spaces with coefficients related to periods of automorphic forms." Proceedings Mathematical Sciences 104, no. 1 (February 1994): 99–135. http://dx.doi.org/10.1007/bf02830877.

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35

Shimura, Goro. "On the critical values of certain Dirichlet series and the periods of automorphic forms." Inventiones Mathematicae 94, no. 2 (June 1988): 245–305. http://dx.doi.org/10.1007/bf01394326.

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36

Furusawa, Masaaki, and Kazuki Morimoto. "On the Gross–Prasad conjecture with its refinement for (SO(5), SO(2)) and the generalized Böcherer conjecture." Compositio Mathematica 160, no. 9 (September 2024): 2115–202. http://dx.doi.org/10.1112/s0010437x24007267.

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Анотація:

We investigate the Gross–Prasad conjecture and its refinement for the Bessel periods in the case of $(\mathrm {SO}(5), \mathrm {SO}(2))$ . In particular, by combining several theta correspondences, we prove the Ichino–Ikeda-type formula for any tempered irreducible cuspidal automorphic representation. As a corollary of our formula, we prove an explicit formula relating certain weighted averages of Fourier coefficients of holomorphic Siegel cusp forms of degree two, which are Hecke eigenforms, to central special values of $L$ -functions. The formula is regarded as a natural generalization of the Böcherer conjecture to the non-trivial toroidal character case.

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37

Ichino, Atsushi, and Tamutsu Ikeda. "On the Periods of Automorphic Forms on Special Orthogonal Groups and the Gross–Prasad Conjecture." Geometric and Functional Analysis 19, no. 5 (December 15, 2009): 1378–425. http://dx.doi.org/10.1007/s00039-009-0040-4.

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38

DOLAN, LOUISE, and CHIARA R. NAPPI. "THE RAMOND–RAMOND SELF-DUAL FIVE-FORM'S PARTITION FUNCTION ON T10." Modern Physics Letters A 15, no. 19 (June 21, 2000): 1261–73. http://dx.doi.org/10.1142/s0217732300001547.

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Анотація:

In view of the recent interest in formulating a quantum theory of Ramond–Ramond p-forms, we exhibit an [Formula: see text] invariant partition function for the chiral four-form of Type IIB string theory on the ten-torus. We follow the strategy used to derive a modular invariant partition function for the chiral two-form of the M-theory five-brane. We also generalize the calculation to self-dual quantum fields in space–time dimension 2p = 2 + 4k, and display the [Formula: see text] automorphic forms for odd p > 1. We relate our explicit calculation to a computation of the B-cycle periods, which are discussed in the work of Witten.

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39

Akahori, Jiro, Norio Konno, Iwao Sato, and Yuma Tamura. "Absolute Zeta functions and periodicity of quantum walks on cycles." Quantum Information & Computation 24, no. 11/12 (October 2024): 901–16. http://dx.doi.org/10.26421/qic24.11-12-1.

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The quantum walk is a quantum counterpart of the classical random walk. On the other hand, absolute zeta functions can be considered as zeta functions over $\mathbf{F}_1$. This study presents a connection between quantum walks and absolute zeta functions. In this paper, we focus on Hadamard walks and $3$-state Grover walks on cycle graphs. The Hadamard walks and the Grover walks are typical models of the quantum walks. We consider the periods and zeta functions of such quantum walks. Moreover, we derive the explicit forms of the absolute zeta functions of corresponding zeta functions. Also, it is shown that our zeta functions of quantum walks are absolute automorphic forms.

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40

Grobner, Harald, and Ronnie Sebastian. "Period relations for cusp forms of GSp4." Forum Mathematicum 30, no. 3 (May 1, 2018): 581–98. http://dx.doi.org/10.1515/forum-2017-0005.

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Анотація:

AbstractLet F be a totally real number field and let π be a cuspidal automorphic representation of {\mathrm{GSp_{4}}(\mathbb{A}_{F})}, which contributes irreducibly to coherent cohomology. If π has a Bessel model, we may attach a period {p(\pi)} to this datum. In the present paper, which is Part I in a series of two, we establish a relation of these Bessel periods {p(\pi)} and all of their twists {p(\pi\otimes\xi)} under arbitrary algebraic Hecke characters ξ. In the appendix, we show that {(\mathfrak{g},K)}-cohomological cusp forms of {\mathrm{GSp_{4}}(\mathbb{A}_{F})} all qualify to be of the above type – providing a large source of examples. We expect that these period relations for {\mathrm{GSp_{4}}(\mathbb{A}_{F})} will allow a conceptual, fine treatment of rationality relations of special values of the spin L-function, which we hope to report on in Part II of this paper.

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41

Gorgadze, S. F., Dao Vu Shi та A. V. Ermakova. "Synchronization of M-sequences based on fast Нadamard transform". Radiotehnika i èlektronika 69, № 2 (7 жовтня 2024): 122–36. http://dx.doi.org/10.31857/s0033849424020031.

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Анотація:

Options have been developed for constructing circulant matrices of any M-sequence (MS) based on automorphic multiplicative groups of the extended Galois field, constructed using an irreducible primitive polynomial, on the basis of which the original MS is formed. The result of this approach is the identification of new methods for transforming MS circulant matrices to a matrix of Walsh functions, ordered by the powers of the antiderivative element of the field. It is shown for the first time that, depending on the initial conditions of the transformation, a set of any number of any cyclic shifts of the MP, shifted relative to each other by one symbol, can be transformed to any rows of the ordered matrix of Walsh functions, following one another. This circumstance makes it possible to simplify the MS synchronization algorithm for a known range of its cyclic shifts, especially in the case of large periods of its repetition, and also to reduce the computational complexity of the processing algorithm when working in a truncated basis of Walsh–Hadamard functions.

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42

Orlova, Lyubov A., and Valentina S. Zykina. "Radiocarbon Dating of Buried Holocene Soils in Siberia." Radiocarbon 44, no. 1 (2002): 113–22. http://dx.doi.org/10.1017/s0033822200064717.

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We have constructed a detailed chronological description of soil formation and its environments with data obtained on radiocarbon ages, palynology, and pedology of the Holocene buried soils in the forest steppe of western and central Siberia. We studied a number of Holocene sections, which were located in different geomorphic situations. Radiocarbon dating of materials from several soil horizons, including soil organic matter (SOM), wood, peat, charcoal, and carbonates, revealed three climatic periods and five stages of soil formation in the second part of the Holocene. 14C ages of approximately 6355 BP, 6020 BP, and 5930 BP showed that the longest and most active stage is associated with the Holocene Climatic Optimum, when dark-grey soils were formed in the forest environment. The conditions of birch forest steppe favored formation of chernozem and associated meadow-chernozem and meadow soils. Subboreal time includes two stages of soil formation corresponding to lake regressions, which were less intense than those of the Holocene Optimum. The soils of that time are chernozem, grassland-chernozem, and saline types, interbedded with thin peat layers 14C dated to around 4555 B P, 4240 BP and 3480 BP, and 3170 B P. Subatlantic time includes two poorly developed hydromorphic paleosols formed within inshore parts of lakes and chernozem-type automorphic paleosol. The older horizon was formed during approximately 2500–1770 BP, and the younger one during approximately 1640–400 B P. The buried soils of the Subatlantic time period also attest to short episodes of lake regression. The climate changes show an evident trend: in the second part of the Atlantic time period it was warmer and drier than at present, and in the Subboreal and Subatlantic time periods the climate was cool and humid.

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43

Januszewski, Fabian. "On period relations for automorphic $L$-functions I." Transactions of the American Mathematical Society 371, no. 9 (September 24, 2018): 6547–80. http://dx.doi.org/10.1090/tran/7527.

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44

Lin, Jie. "Period relations for automorphic induction and applications, I." Comptes Rendus Mathematique 353, no. 2 (February 2015): 95–100. http://dx.doi.org/10.1016/j.crma.2014.10.016.

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45

Grigoryan, H. A. "AUTOMORPHISMS OF FREE BURNSIDE GROUPS OF PERIOD 3." Proceedings of the YSU A: Physical and Mathematical Sciences 53, no. 1 (248) (April 15, 2019): 13–16. http://dx.doi.org/10.46991/pysu:a/2019.53.1.013.

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46

Cavaliere, Richard A. "New results on automorphic integrals and their period functions." Transactions of the American Mathematical Society 301, no. 1 (January 1, 1987): 401. http://dx.doi.org/10.1090/s0002-9947-1987-0879581-8.

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47

Zhang, Wei. "Automorphic period and the central value of Rankin-Selberg L-function." Journal of the American Mathematical Society 27, no. 2 (January 27, 2014): 541–612. http://dx.doi.org/10.1090/s0894-0347-2014-00784-0.

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48

Hawkins, John H., and Marvin I. Knopp. "A Hecke correspondence theorem for automorphic integrals with rational period functions." Illinois Journal of Mathematics 36, no. 2 (June 1992): 178–207. http://dx.doi.org/10.1215/ijm/1255987529.

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49

MANAENKOV, A. S., P. M. PODGAETSKAYA, and V. S. POPOV. "IMPACT OF FOREST SHELTER BELTS ON THE DEVELOPMENT OF SPRING WHEAT IN THE NEAR-EDGE ZONE OF CROPS." Ser-5_2023_4 78, no. 4 (2023) (September 18, 2023): 97–106. http://dx.doi.org/10.55959/10.55959/msu0579-9414.5.78.4.9.

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The system of shelterbelt forests is a long-lasting ecological framework of an agrarian territory. Their main function is to prevent the degradation of arable soils, improve the microclimate of fields, and preserve the stability and biological diversity of landscape. However, the presence of forest belts complicates field cultivation, and along their borders zones (strips) of depression in crop development could formed, reducing the productivity of lands. This impedes field-protective afforestation, and, consequently, the solution of the problem of reliable protection of land resources. The aim of the work is to establish the causes and regularities of formation of depressive zones in agrocenoses, and to determine the possibility and methods of suppressing their development. The research has been conducted for 8 years in the experimental-production system of 30 to 53-year old 2 to 4-row forest belts of Betula pendula ROTH, Pinus silvestris L., Ulmus laevis PALL. and other species on the automorphic chestnut soil of the Kulunda steppe (the Altai territory) by the generally accepted methods. It was found that the depressive zone in agrocenoses is the least wide for relatively sparse forest belts of birch and pine, and also for forest belts with marginal rows of xerophytic shrubs (3-7 m in total on windward and leeward sides), and the largest (up to 25-30 m) for elm and poplar (Populus laurifolia LEDEB.), i. e. for plantations of hydrophylous tall species with dense crowns. It is 1-3 m wider on the leeward side of forest belts, where more snow is deposited in winter and soil moisture is better in spring. The increase in height and density of stands, influencing the length and intensity of day-time soil shading, stimulates the expansion of the zone. Crop depression is more pronounced in wet years. The increased amount of atmospheric precipitation during the cold season, as well as at the beginning of the growing season and during the reaping season till the onset of stable cold weather also contributes to it. The development of depressive zones is suppressed by abundant precipitation during the period of active growth of field crops. Thus, the formation of depressive zone in agrocenoses of shelterbelt forests depends on many factors. Under arid conditions, the most effective factors are the need for soil moisture and moisture availability for the stand. The most active expansion of the tree root system in the field and the suppression of crops occur in wet years and during the periods with high soil moisture in the absence or weakened competition of field crops. To reduce damage to their productivity, it is necessary to implement a set of coordinated organizational, silvicultural and agrotechnological measures aimed at increasing moisture availability and limiting the expansion of tree root system of forest belts in the field.

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50

Atabekyan, V. S., H. T. Aslanyan, H. A. Grigoryan, and A. E. Grigoryan. "ANALOGUES OF NIELSEN'S AND MAGNUS'S THEOREMS FOR FREE BURNSIDE GROUPS OF PERIOD 3." Proceedings of the YSU A: Physical and Mathematical Sciences 51, no. 3 (244) (December 15, 2017): 217–23. http://dx.doi.org/10.46991/pysu:a/2017.51.3.217.

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We prove that the free Burnside groups $B(m,3)$ of period 3 and rank $m\geq1$ have Magnus's property, that is if in $B(m,3)$ the normal closures of $r$ and $s$ coincide, then $r$ is conjugate to $s$ or $s^{-1}$. We also prove that any automorphism of $B(m,3)$ induced by a Nielsen automorphism of the free group $F_m$ of rank $m$. We show that the kernel of the natural homomorphism $\text{Aut}(B(2,3)) \rightarrow GL_2(\mathbb{Z}_3)$ is the group of inner automorphisms of $B(2,3)$.

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