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Co-periods and central symmetric cube L-values

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Pith's one-line read The paper proves that co-periods on the cubic metaplectic cover of GL(2) control the central value of the symmetric cube L-function, with local multiplicity one and an exact unramified formula.

desk verdict Strong new local results on co-periods for the cubic metaplectic cover, but the unramified L-value identity rests on an unproved adaptation of Hsieh's decomposition. read the letter →

arxiv 2507.15279 v2 pith:3NFTUDVD submitted 2025-07-21 math.NT

Li Cai , Yangyu Fan , Dongming She This is my paper · ORCID
classification math.NT MSC 11F6711F7022E50
keywords co-periodsymmetriccubeL-functionKazhdan-PattersonmetaplecticcoverexceptionalthetaseriesmultiplicityoneIchino-IkedaconjectureEisensteinunramifiedcalculation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies the co-period integral of an automorphic form on $\mathrm{GL}(2)$ against two exceptional $\theta$ series on the cubic Kazhdan-Patterson metaplectic cover of $\mathrm{GL}(2)$. Its aim is to establish that these co-periods are the period-theoretic control of the central value of the symmetric cube $L$-function $L(1/2, \sigma, \mathrm{Sym}^3)$, in analogy with how triple-product periods control triple-product $L$-values. Locally it proves that the space of invariant trilinear forms is one-dimensional, and at unramified places it computes the local integral exactly as $L(1/2, \sigma_v, \mathrm{Sym}^3)/L(1, \sigma_v, \mathrm{ad})$. Globally it gives an Euler decomposition for Eisenstein co-periods and proposes an Ichino-Ikeda type conjecture for cuspidal forms, together with a first verification in the Eisenstein case. If correct, the paper supplies a new way to detect nonvanishing of central symmetric cube $L$-values and yields many cuspidal representations with prescribed local components and nonvanishing $L(1/2, \sigma, \mathrm{Sym}^3)$.

What carries the argument

The carrying object is the cubic Kazhdan-Patterson metaplectic cover $\widetilde{G}$ of $\mathrm{GL}_2$, a central extension by the third roots of unity built from a Kubota 2-cocycle. On it live the exceptional representations $\sigma_{\pm} = V_0(\chi_{\pm})$, realized as residues of Eisenstein series, whose Whittaker models and spherical Whittaker functions are known explicitly. The proof passes through three linked mechanisms: the restriction theory of these representations to maximal compact-modulo-center subgroups, which produces the local multiplicity one; the Euler decomposition of the trilinear matrix-coefficient integral into a product of local co-period integrals, adapted from the analogous decomposition in the triple-product setting; and the substitution of the explicit spherical Whittaker function, which converts the unramified local integral into the ratio $L(1/2,\sigma_v,\mathrm{Sym}^3)/L(1,\sigma_v,\mathrm{ad})$.

What would settle it

Take an unramified principal series over a nonarchimedean local field with $v \nmid 3$ and with the exceptional characters satisfying $\chi_{+,1}\chi_{-,1}(\varpi^3) = -|\varpi|$, compute the matrix-coefficient integral $I$ directly for spherical vectors, and compare it with the asserted second branch $L(1,\sigma_v,\mathrm{Sym}^3)/(L(1/2,\sigma_v,\mathrm{Sym}^3)L(1,\sigma_v,\mathrm{ad}))$; a mismatch would falsify Theorem 2.24 and the conjecture it supports.

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Extended reading notes

Core claim

The central discovery is that for the cubic cover, the local co-period functional is one-dimensional: at each place $v$ with $v \nmid 3$ or $\sigma_v$ non-supercuspidal, $\dim \mathrm{Hom}_{G(F_v)}(\sigma_v \otimes \sigma_{+,v} \otimes \sigma_{-,v}, \mathbb{C}) = 1$. For unramified data, the integration of matrix coefficients equals $L(1/2,\sigma_v,\mathrm{Sym}^3)/L(1,\sigma_v,\mathrm{ad})$ up to a fixed normalization of the measure. Globally, the paper decomposes the regularized Eisenstein co-period into local factors and conjectures an Ichino-Ikeda type formula for cuspidal $\sigma$: the product of the co-period and its dual equals a rational constant times $L(1/2,\sigma,\mathrm{Sym}^3)/L(1,\sigma,\mathrm{ad})$ times the product of normalized local integrals. The Eisenstein case verifies this formula in the range $-1/6 < \lambda(\theta) < 1/6$.

Load-bearing premise

The load-bearing premise is that two results proved for reductive groups carry over verbatim to the covering group: the Euler decomposition of the trilinear period into local integrals (used for the unramified computation and the Eisenstein formula) and the Burger-Sarnak type lemma used to lift local components to global cuspidal forms; if either transfer fails, the corresponding conclusion collapses.

Editorial extensions

If this is right

  • At every local place away from 3, and for all non-supercuspidal components, the co-period functional is the unique invariant trilinear form, so there is no hidden multiplicity when passing from local to global periods.
  • At unramified places, nonvanishing of the normalized local integral is equivalent to nonvanishing of the local central symmetric cube factor, so the local ratio directly detects whether $L(1/2,\sigma_v,\mathrm{Sym}^3) \neq 0$.
  • The Eisenstein verification in the range $-1/6 < \lambda(\theta) < 1/6$ gives the first complete instance of the conjectured Euler-product formula, including the exact rational constant.
  • The local multiplicity one theorem implies that any prescribed finite set of supercuspidal local components away from 3 occurs in some cuspidal automorphic representation $\sigma$ with $L(1/2,\sigma,\mathrm{Sym}^3) \neq 0$, and hence there are infinitely many such $\sigma$.
  • If Conjecture 1.5 is true, the co-period $P$ is nonzero exactly when the central symmetric cube value is nonzero, giving a period-integral criterion for nonvanishing of $L(1/2,\sigma,\mathrm{Sym}^3)$ for all cuspidal $\sigma$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same restriction and decomposition techniques should extend to higher covers of $\mathrm{GL}_2$ and to $\mathrm{GL}_r$, where analogous co-periods would govern higher symmetric-power $L$-values; the paper already notes the method applies to general $n$-th covers.
  • The conjecture leaves the rational constant $C$ undetermined; the Eisenstein computation is a natural place to fix $C$ explicitly, which would turn the conjectural period relation into a sharp arithmetic identity.
  • The missing supercuspidal case at places dividing 3 is the next test of the method; the paper's upper bound reduces it to a Mackey analysis of the exceptional representations restricted to the pro-$p$ Iwahori subgroup.
  • The relation between nonvanishing of co-periods and central symmetric cube values suggests that metaplectic periods fit into the general conjectural description of period integrals for covering-group Hamiltonian spaces, rather than being isolated examples.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 4 minor

Summary. The paper studies co-period integrals attached to an automorphic form on GL(2) and two exceptional theta series on the cubic Kazhdan-Patterson metaplectic cover of GL(2). In the local aspect, it proves multiplicity one for the relevant Hom-space under a caveat (v not dividing 3, or sigma_v non-supercuspidal), constructs a distinguished local trilinear period as an integral of matrix coefficients, and computes this integral for unramified spherical data, obtaining the ratio L(1/2, sigma_v, Sym^3)/L(1, sigma_v, ad). In the global aspect, it proposes an Ichino-Ikeda type conjecture for cuspidal forms, proves an Euler-type decomposition for Eisenstein series, verifies the conjectural formula for Eisenstein data in a range of parameters, and derives the existence of cuspidal automorphic representations with prescribed supercuspidal local components and nonvanishing central symmetric cube L-values.

Significance. If correct, the paper would be the first systematic local and global treatment of co-periods for the cubic metaplectic cover, with an explicit unramified local identity that links the metaplectic co-period to the Langlands-Shahidi symmetric cube L-factor. The detailed restriction calculations for exceptional representations in Section 2.3, the explicit computation of spherical Whittaker functions, and the Eisenstein verification are substantial and go substantially beyond a mere conjecture. The paper also explicitly names its main conditional steps, which is helpful. However, the central local decomposition formulas are not proved in the text; they are asserted as adaptations of a result in the reductive GL(2) setting, and the unramified identity and the global Eisenstein check depend directly on them. The global existence theorem similarly relies on an unproved extension of Prasad's Burger-Sarnak lemma to a covering-group pair. These points are load-bearing, and the manuscript needs a major revision to close them.

major comments (5)
  1. [§2.5, Propositions 2.22 and 2.23] The proofs of Propositions 2.22 and 2.23 consist of the single sentence 'Adapting the argument in [9, Proposition 5.2]' (pp. 21-22). This is the load-bearing step for Theorems 2.21 and 2.24: the asserted identity I = ζ_F(1) ∑_t Ψ_t Ψ^∨_t (and its analogue I = ζ_F(1) Ψ Ψ^∨) involves the covering group, the summation over fZ*\eT', the support restriction Ψ_t = 0 unless t ∈ eT', the constant ζ_F(1), and the specific pairings in Whittaker and induced models. None of these features is present in the reductive GL(2) triple-product setting of [9], so the adaptation is not routine. Since the unramified calculation and the Eisenstein verification both pass through this identity, the central claim of the paper is conditional on this decomposition. The authors should either prove it directly or give a detailed verification of the adaptation.
  2. [§2.6, Proposition 2.27] The archimedean analogue is treated either by the same one-line 'adapting [9, Proposition 5.2]' or by a deformation argument whose steps are only sketched. In particular, the claim that the ratio I/(ΨΨ^∨) is holomorphic in a small strip and constant because it equals ζ_F(1) when χδ^s is unitary requires control over possible zeros and poles of ΨΨ^∨; the sentence 'Choose test vectors' does not establish this. Since Theorem 3.6 uses Proposition 2.27 at archimedean places, the global Eisenstein verification inherits this gap.
  3. [§1, Theorem 1.3] The proof of Theorem 1.3 invokes Prasad's Burger-Sarnak principle [21, Lemma 1] for the pair (∆µ3\(eG(A)×eG(A)), GL2(A)), with only the footnote that the proof carries over verbatim. This is not a proof: the cited lemma is stated for reductive groups, and the metaplectic cover and the diagonal embedding change the relevant invariant functionals, period integrals, and the relation between global nonvanishing and local Hom-spaces. Because Theorem 1.3 is one of the main advertised applications (nonvanishing of central symmetric cube L-values with prescribed supercuspidal components), the lemma must be proved or a reference containing the metaplectic case must be supplied.
  4. [Theorem 2.24] There is a discrepancy between the statement and the proof. The theorem statement displays factors involving χ_{+,1}χ_{-,2}(ϖ^3), while the computation in the proof (p. 23) uses χ_{+,1}χ_{-,1}(ϖ^3) in the same positions and the final dichotomy is stated for χ_{+,1}χ_{-,1}(ϖ^3) = |ϖ| or −|ϖ|. If the indices in the theorem are not a typo, the derivation does not prove the displayed formula; if they are a typo, it should be corrected. This point matters because the theorem is the local manifestation of the symmetric cube L-value ratio.
  5. [Abstract and Theorem 1.1] The abstract and introduction state that the Hom-space 'is always of one dimension', but Theorem 1.1 is proved only when v ∤ 3 or σ_v is non-supercuspidal, and the text says the supercuspidal case at v | 3 is still believed but not proved. The advertised claim should be qualified to match the theorem; otherwise the abstract overstates the result.
minor comments (4)
  1. [§2.1] There is a typo: 'Kutota' should be 'Kubota' in the description of the Kubota 2-cocycle.
  2. [§3.2] In the discussion of the regularized period, 'the digonal G(A)' should be 'the diagonal G(A)'.
  3. [References] In reference [3], the page span '413-339' appears to be a typo; it should likely be corrected to the actual page range of the paper.
  4. [§3.1] The additive characters ψ± used in the Whittaker-Fourier expansion of φ± are introduced only informally; for clarity, specify their relation to the fixed additive character ψ and to the exceptional characters χ±.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the co-period-to-Sym^3-L-value identity is obtained by explicit spherical Whittaker computation against independently defined Langlands-Shahidi L-factors, not by fitting or self-citation.

full rationale

I examined the derivation chain for Theorem 1.2 / Theorem 2.24, the global conjecture (Conjecture 1.5), and the Eisenstein verification (Theorem 3.6). The central unramified identity is computed, not imposed: spherical vectors are evaluated using the explicit Whittaker function of Kazhdan-Patterson [13, Theorem I.4.2], and the resulting product of four Euler factors is identified, via the Satake-parameter definitions supplied by Shahidi's Langlands-Shahidi method, with L(1/2, sigma, Sym^3)/L(1, sigma, ad). The local L-factors are defined independently of the co-period integral, so the matching is a theorem rather than a definitional equality. The normalization I^sharp dividing by L(1/2, Sym^3)/L(1, ad) is a convention; Conjecture 1.5 is posed as a conjecture, not derived from the normalization. Theorem 3.6 unfolds the Eisenstein co-period and reduces it to the same local computation; no equation is defined in terms of the advertised L-value. I also checked for self-citation chains. The paper's load-bearing external refs [9,13,15,21] are not authored by Cai-Fan-She, and the only 'adaptation' statements are to Hsieh's Proposition 5.2 and to Prasad's Burger-Sarnak lemma. These are genuine correctness risks: Propositions 2.22, 2.23, and 2.27 are asserted by one-line 'adapting the argument in [9, Proposition 5.2]' statements, and Theorem 1.3 assumes the Burger-Sarnak principle extends verbatim to the covering-group diagonal embedding. But missing or borrowed proof is not circularity: those results, if correct, are independent external inputs, and the paper does not reduce its target L-value identity to itself or to a fitted parameter. No circular step is exhibited, so the score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters fitted to data. The central claims rest on standard metaplectic theory (Kazhdan-Patterson), domain assumptions about the base field, and two load-bearing unproved adaptations of existing arguments.

assumptions (6)
  • standard math The Kubota 2-cocycle defines a cubic metaplectic cover of GL2, with a preferred set-theoretic splitting s that is a homomorphism on G(F) and N(A).
    Setup in Section 1; follows Kazhdan-Patterson [13].
  • standard math Exceptional representations V0(chi) exist and have the Jacquet modules, Whittaker functionals and gauge estimates stated in [13, Sections I.2-I.4].
    Used throughout Sections 2 and 3, e.g., Proposition 2.2 and Theorem 3.3.
  • domain assumption The base field F contains all 3rd roots of unity and an embedding epsilon: mu3 to C^x is fixed; all results are for the Kazhdan-Patterson cubic cover with arbitrary Kubota cocycle parameter c.
    Sets the scope of the paper; without mu3 subset F the exceptional theta series would not exist.
  • ad hoc to paper The decomposition formula of Hsieh [9, Proposition 5.2] adapts verbatim to the covering-group setting in Propositions 2.22, 2.23 and 2.27.
    The paper states the decomposition by 'adapting' without carrying out the adaptation; this is load-bearing for the local-global formula.
  • ad hoc to paper Prasad's Burger-Sarnak type lemma [21, Lemma 1] extends from reductive groups to the diagonal embedding of GL2(A) in Delta mu3 setminus eG(A) times eG(A).
    Invoked in Theorem 1.3 with the comment 'the proof carries over verbatim', but no proof is given.
  • standard math The mixed truncation operator of Yamana [33] regularizes the divergent co-period integral and has the stated asymptotics.
    Used in Section 3.2 to define the regularized period for Eisenstein series; taken from [33].

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Cite this review

Pith. "Pith review of Co-periods and central symmetric cube L-values." pith.science (2026). https://pith.science/paper/3NFTUDVD

@misc{pith2026250715279,
  author       = {Pith},
  title        = {Pith review of: Co-periods and central symmetric cube L-values},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3NFTUDVD}},
  note         = {Machine review of arXiv:2507.15279}
}
abstract

In this article, we study the co-period integral attached to an automorphic form on $\GL(2)$ and two exceptional theta series on the cubic Kazhdan-Patterson cover of $\GL(2)$. In the local aspect, we show the $\Hom$-space is always of one dimension and conduct the unramified calculations. In the global aspect, we give the Euler decomposition for the co-period integrals of Eisenstein series and propose an Ichino-Ikeda type conjecture relating the co-period integrals of cuspidal forms to the central critical value of symmetric cube $L$-functions. We also deduce from the local multiplicity one result that there exist cuspidal automorphic forms with prescribed local components and non-vanishing central symmetric cube $L$-values.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

33 extracted references · 32 canonical work pages

  1. [9]

    Hsieh,Hida families andp-adic triple productL-functions

    M-L. Hsieh,Hida families andp-adic triple productL-functions. Amer. J. Math. 143 (2021), no. 2, 411-532

    work page 2021

  2. [1]

    Bump,Automorphic forms and representations

    D. Bump,Automorphic forms and representations. Cambridge Stud. Adv. Math., 55 Cambridge University Press, Cambridge, 1997. xiv+574 pp. ISBN:0-521-55098-X

    work page 1997

  3. [2]

    Bump and D.Ginzburg,Symmetric square L-functions onGL(r), Ann

    D. Bump and D.Ginzburg,Symmetric square L-functions onGL(r), Ann. of Math. (2) 136 (1992), 137-205

    work page 1992

  4. [3]

    D. Bump, D. Ginzburg and J. Hoffstein,Symmetric cube L-functions, Invent. Math. 125 (1996), 413-339

    work page 1996

  5. [4]

    Ben-Zvi, Y

    D. Ben-Zvi, Y. Sakellaridis, A. Venkatesh,Relative Langlands duality.arXiv:2409.04677, preprint

    arXiv

  6. [5]

    Casselman,The restriction of a representation ofGL2(K) to GL2(O)

    W. Casselman,The restriction of a representation ofGL2(K) to GL2(O). Math. Ann. 206 (1973), 311-318

    work page 1973

  7. [6]

    Fratila, D

    D. Fratila, D. Prasad, Homological duality for covering groups of reductive p-adic groups. Pure Appl. Math.Q.18 (2022),no.5.1867-1950

    work page 2022

  8. [7]

    Ginzburg, D

    D. Ginzburg, D. Jiang and S. Rallis,Nonvanishing of the central critical value of the third symmetric power L-functions, Forum Math. 13 (2001), 109-132

    work page 2001

Show all 33 references
  1. [8]

    Hoffstein, J

    J. Hoffstein, J. Jung and M. Lee,Non-vanishing of symmetric cube L-functions. J. London Math. Soc., 107(2023): 153-188

    2023

  2. [10]

    Ichino,Trilinear forms and the central values of triple product L-functions

    A. Ichino,Trilinear forms and the central values of triple product L-functions. Duke Mathematical Journal 145.2 (2008): 281-307

    2008

  3. [11]

    A. C. Kable, The tensor product of exceptional representations on the general linear group. Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 5, 741-769

    2001

  4. [12]

    Kaplan, The characterization of theta-distinguished representations ofGL(n)

    E. Kaplan, The characterization of theta-distinguished representations ofGL(n). Israel Journal of Mathe- matics 222(2017), 551-598

    2017

  5. [13]

    Kazhdan and S.J

    D.A. Kazhdan and S.J. Patterson,Metaplectic forms. Inst. Hautes Études Sci. Publ. Math. No. 59 (1984), 35-142. 33

    1984

  6. [14]

    H. Kim, F. Shahidi, Symmetric cube L-Functions for GL(2) are entire. Annals of Mathematics, 150(2) (1999): 645-662

    1999

  7. [15]

    Loke,Trilinear forms ofgl(2)

    H.Y. Loke,Trilinear forms ofgl(2). Pacific Journal of Mathematics 197.1 (2001): 119-144

    2001

  8. [16]

    Michel and A

    P. Michel and A. Venkatesh,The subconvexity problem forGL2. Publ. Math. Inst. Hautes Études Sci. No. 111 (2010), 171-271

    2010

  9. [17]

    Newton and J

    J. Newton and J. Thorne,Symmetric power functoriality for holomorphic modular forms. Publ. Math. Inst. Hautes Études Sci. 134 (2021), 1-116

    2021

  10. [18]

    Newton and J

    J. Newton and J. Thorne,Symmetric power functoriality for holomorphic modular forms.Publ. Math. Inst. Hautes Études Sci. 134 (2021), 117-152

    2021

  11. [19]

    S. J. Patterson,The level structure of exceptional metaplectic representations. Dedicated to Martin Kneser on the occasion of his 70th birthday. J. Reine Angew. Math. 494 (1998), 85-100

    1998

  12. [20]

    Prasad, Trilinear forms for representations ofGL(2) and local ϵ-factors

    D. Prasad, Trilinear forms for representations ofGL(2) and local ϵ-factors. Compositio Math. 75 (1990), no. 1, 1-46

    1990

  13. [21]

    Prasad,Relating invariant linear form and local epsilon factors via global methods

    D. Prasad,Relating invariant linear form and local epsilon factors via global methods. With an appendix by Hiroshi Saito.Duke Math. J. 138 (2007), no. 2, 233-261

    2007

  14. [22]

    Prasad, Ext-analogues of branching laws

    D. Prasad, Ext-analogues of branching laws. Proceedings of the International Congress of Mathematicians Rio de Janeiro 2018. Vol. II. Invited lectures, 1367-1392, World Sci. Publ., Hackensack, NJ, 2018

    2018

  15. [23]

    Raghuram, Kirillov theory for GL2(D) where D is a division algebra over a non- Archimedean local field

    D.Prasad and A. Raghuram, Kirillov theory for GL2(D) where D is a division algebra over a non- Archimedean local field. Duke Math. J. 104 (2000), no. 1, 19-44

    2000

  16. [24]

    Pate and D

    P. Pate and D. Prasad,Multiplicity formula for restriction of representations offGL2(F ) to fSL2(F ). Proc. Amer. Math. Soc. 144(2016),no.2,903-908

    2016

  17. [25]

    Patel and D

    P. Patel and D. Prasad, Restriction of representations of metaplectic GL2(F ) to tori. Israell. Math.225 (2018),no.2.525-551

    2018

  18. [26]

    Takloo-Bighash,Bessel models forGSp(4)

    D.Prasad and R. Takloo-Bighash,Bessel models forGSp(4). J. Reine Angew. Math. 655 (2011), 189-243

    2011

  19. [27]

    Schmidt,Some remarks on local newforms forGL(2)

    R. Schmidt,Some remarks on local newforms forGL(2). J. Ramanujan Math. Soc. 17 (2002), no. 2, 115-147

    2002

  20. [28]

    Sakellaridis and A

    Y. Sakellaridis and A. Venkatesh,Periods and harmonic analysis on spherical varieties, Astérisque No. 396 (2017), viii+360 pp

    2017

  21. [29]

    Shahidi,On the Ramanujan conjecture and finiteness of poles for certainL-functions

    F. Shahidi,On the Ramanujan conjecture and finiteness of poles for certainL-functions. Annals of Mathe- matics, 127(3), 547-584

  22. [30]

    Shahidi,Third symmetric powerL-functions for GL(2)

    F. Shahidi,Third symmetric powerL-functions for GL(2). Compositio Math. 70(1989), 245-273

    1989

  23. [31]

    Takeda,The twisted symmetric squareL-function of GL(r), Duke Math

    S. Takeda,The twisted symmetric squareL-function of GL(r), Duke Math. J. 163, no. 1 (2014), 175-266

    2014

  24. [32]

    Yamana,Local symmetric squareL-factors of representations of general linear groups, Pacific Journal of Math., 286(2017), no

    S. Yamana,Local symmetric squareL-factors of representations of general linear groups, Pacific Journal of Math., 286(2017), no. 1, 215-256

    2017

  25. [33]

    Yamana, Periods of automorphic forms: the trilinear case

    S. Yamana, Periods of automorphic forms: the trilinear case. Journal of the Institute of Mathematics of Jussieu. 17(2018) no. 1, 59-74. Academy for Multidisciplinary Studies, Beijing National Center for Applied Mathematics, Capi- tal Normal University, Beijing, 100048, People’...

    2018

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