中国科学院数学与系统科学研究院期刊网

15 January 2021, Volume 37 Issue 1
    

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  • Bing Yong XIE
    Acta Mathematica Sinica. 2021, 37(1): 1-34. https://doi.org/10.1007/s10114-019-8046-9
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    In this paper we study the derivatives of Frobenius and the derivatives of Hodge-Tate weights for families of Galois representations with triangulations. We generalize the Fontaine-Mazur L-invariant and use it to build a formula which is a generalization of the Colmez-Greenberg-Stevens formula. For the purpose of proving this formula we show two auxiliary results called projection vanishing property and "projection vanishing implying L-invariants" property.

  • Hao Yu HU
    Acta Mathematica Sinica. 2021, 37(1): 35-58. https://doi.org/10.1007/s10114-019-8175-1
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    In this article, we investigate the shift of Abbes and Saito's ramification filtrations of the absolute Galois group of a complete discrete valuation field of positive characteristic under a purely inseparable extension. We also study a functoriality property for characteristic forms.

  • Yong Quan HU
    Acta Mathematica Sinica. 2021, 37(1): 59-72. https://doi.org/10.1007/s10114-020-8213-z
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    In the p-adic local Langlands correspondence for GL2(Qp), the following theorem of Berger and Breuil has played an important role:the locally algebraic representations of GL2(Qp) associated to crystabelline Galois representations admit a unique unitary completion. In this note, we give a new proof of the weaker statement that the locally algebraic representations admit at most one unitary completion and such a completion is automatically admissible. Our proof is purely representation theoretic, involving neither (φ, Γ)-module techniques nor global methods. When F is a finite extension of Qp, we also get a simpler proof of a theorem of Vignéras for the existence of integral structures for (locally algebraic) special series and for (smooth) tamely ramified principal series.

  • En Lin YANG, Yi Geng ZHAO
    Acta Mathematica Sinica. 2021, 37(1): 73-94. https://doi.org/10.1007/s10114-019-8356-y
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    We propose a conjecture on the relative twist formula of l-adic sheaves, which can be viewed as a generalization of Kato—Saito's conjecture. We verify this conjecture under some transversal assumptions. We also define a relative cohomological characteristic class and prove that its formation is compatible with proper push-forward. A conjectural relation is also given between the relative twist formula and the relative cohomological characteristic class.

  • Da Sheng WEI
    Acta Mathematica Sinica. 2021, 37(1): 95-103. https://doi.org/10.1007/s10114-021-8193-7
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    Let X be a toric variety over a number field k with k[X]×=k×. Let WX be a closed subset of codimension at least 2. We prove that X \ W satisfies strong approximation with algebraic Brauer-Manin obstruction.

  • Xin WAN
    Acta Mathematica Sinica. 2021, 37(1): 104-120. https://doi.org/10.1007/s10114-021-8355-7
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    We prove an anticyclotomic Iwasawa main conjecture proposed by Perrin-Riou for Heegner points for semi-stable elliptic curves E over a quadratic imaginary field K satisfying a certain generalized Heegner hypothesis, at an ordinary prime p. It states that the square of the index of the anticyclotomic family of Heegner points in E equals the characteristic ideal of the torsion part of its Bloch—Kato Selmer group (see Theorem 1.3 for precise statement). As a byproduct we also prove the equality in the Greenberg-Iwasawa main conjecture for certain Rankin-Selberg product (Theorem 1.7) under some local conditions, and an improvement of Skinner's result on a converse of Gross—Zagier and Kolyvagin theorem (Corollary 1.11).

  • Yi Wen DING
    Acta Mathematica Sinica. 2021, 37(1): 121-141. https://doi.org/10.1007/s10114-020-8396-3
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    We study the adjunction property of the Jacquet-Emerton functor in certain neighborhoods of critical points in the eigencurve. As an application, we construct two-variable p-adic L-functions around critical points via Emerton's representation theoretic approach.

  • Bin XU
    Acta Mathematica Sinica. 2021, 37(1): 142-172. https://doi.org/10.1007/s10114-020-8422-5
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    Let F be a p-adic field of characteristic 0. We study a twisted local descent construction for the metaplectic groups Sp2n(F), and also its relation to the corresponding local descent construction for odd special orthogonal groups via local theta correspondence. In consequence, we show that this descent construction gives irreducible supercuspidal genuine representations of Sp2n(F) parametrized by a simple local L-parameter φτ corresponding to an irreducible supercuspidal representation τ of GL2n(F) of symplectic type, and the genericity of the representations constructed can be indicated by a local epsilon factor condition. In particular, this local descent construction recovers the local Shimura correspondence for supercuspidal representations.

  • Shan Wen WANG
    Acta Mathematica Sinica. 2021, 37(1): 173-204. https://doi.org/10.1007/s10114-020-8414-5
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    This article is the second article on the generalization of Kato's Euler system. The main subject of this article is to construct a family of Kato's Euler systems over the cuspidal eigencurve, which interpolate the Kato's Euler systems associated to the modular forms parametrized by the cuspidal eigencurve. We also explain how to use this family of Kato's Euler system to construct a family of distributions on Zp over the cuspidal eigencurve; this distribution gives us a two-variable p-adic L function which interpolate the p-adic L function of modular forms.

  • Jiang Wei XUE, Chia Fu YU
    Acta Mathematica Sinica. 2021, 37(1): 205-228. https://doi.org/10.1007/s10114-020-8415-4
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    This paper contains two parts toward studying abelian varieties from the classification point of view. In a series of papers[Doc. Math., 21, 1607-1643 (2016)],[Taiwanese J. Math., 20(4), 723-741 (2016)], etc., the current authors and T. C. Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over finite fields. In this paper, we give an explicit formula for the size of the isogeny class of simple abelian surfaces with real Weil number q. This establishes a key step that extends our previous explicit calculation of superspecial abelian surfaces to those of supersingular abelian surfaces. The second part is to introduce the notion of genera and idealcomplexes of abelian varieties with additional structures in a general setting. The purpose is to generalize the previous work by the second named author[Forum Math., 22(3), 565-582 (2010)] on abelian varieties with additional structures to similitude classes, which establishes more results on the connection between geometrically defined and arithmetically defined masses for further investigations.