Berger Tobias, Klosin Krzysztof
School of Mathematics and Statistics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield, S3 7RH UK.
Department of Mathematics, Queens College, City University of New York, 65-30 Kissena Blvd, Queens, NY 11367 USA.
Res Number Theory. 2021;7(3):41. doi: 10.1007/s40993-021-00265-x. Epub 2021 Jun 4.
We prove (under certain assumptions) the irreducibility of the limit of a sequence of irreducible essentially self-dual Galois representations (as approaches 2 in a -adic sense) which mod reduce (after semi-simplifying) to with irreducible, two-dimensional of determinant , where is the mod cyclotomic character. More precisely, we assume that are crystalline (with a particular choice of weights) and Siegel-ordinary at . Such representations arise in the study of -adic families of Siegel modular forms and properties of their limits as appear to be important in the context of the Paramodular Conjecture. The result is deduced from the finiteness of two Selmer groups whose order is controlled by -adic -values of an elliptic modular form (giving rise to ) which we assume are non-zero.
我们（在某些假设下）证明了一列不可约本质自对偶伽罗瓦表示的极限（在(p)-adic意义下当(p)趋于2时）的不可约性，这些表示模(p)（在半简化后）约化为(\overline{\rho})，其中(\overline{\rho})是不可约的、二维的且行列式为(\overline{\omega})，这里(\overline{\omega})是模(p)分圆特征。更确切地说，我们假设(\rho_p)是晶态的（具有特定的权重选择）且在(p)处是西格尔-普通的。这样的表示出现在西格尔模形式的(p)-adic族的研究中，并且它们极限的性质在准模猜想的背景下似乎很重要。该结果是从两个塞尔默群的有限性推导出来的，这两个塞尔默群的阶由一个椭圆模形式（产生(\overline{\rho})）的(p)-adic (L)-值控制，我们假设这些值非零。
