This paper is concerned with tile proOlenl or improving hue ~lma^u~ u~ under Stein＇s loss. By the partial Iwasawa coordinates of covariance matrix, the corresponding risk can be split into three parts. One can use th...This paper is concerned with tile proOlenl or improving hue ~lma^u~ u~ under Stein＇s loss. By the partial Iwasawa coordinates of covariance matrix, the corresponding risk can be split into three parts. One can use the information in the weighted matrix of weighted quadratic loss to improve one part of risk. However, this paper indirectly takes advantage of the information in the sample mean and reuses Iwasawa coordinates to improve the rest of risk. It is worth mentioning that the process above can be repeated. Finally, a Monte Carlo simulation study is carried out to verify the theoretical results.展开更多
This paper is concerned with a conjecture formulated by Coates et al in [4] which describes the relation between the integrality of the characteristic elements and their evaluations in the noncommutative Iwasawa theor...This paper is concerned with a conjecture formulated by Coates et al in [4] which describes the relation between the integrality of the characteristic elements and their evaluations in the noncommutative Iwasawa theory. We give an almost equivalent description of the conjecture and prove a certain Dart of it.展开更多
The field K ---- Q（x/-AT） is the only imaginary quadratic field with class number 1, in which the prime 2 splits, and we fix one of the primes p of K lying above 2. The modular elliptic curve X0 （49） has complex m...The field K ---- Q（x/-AT） is the only imaginary quadratic field with class number 1, in which the prime 2 splits, and we fix one of the primes p of K lying above 2. The modular elliptic curve X0 （49） has complex multiplication by the maximal order O of K, and we let E be the twist of Xo （49） by the quadratic extension K（v/M）/K, where M is any square free element of O with M -- i mod 4 and （M, 7） = 1. In the present note, we use surprisingly simple algebraic arguments to prove a sharp estimate for the rank of the MordeII-Weil group modulo torsion of E over the field F∞=k（Ep∞ ）, where Epic denotes the group of p∞-division points on E. Moreover, writing B for the twist of X0（49） by K（C/ET）/K, our Iwasawa-theoretic arguments also show that the weak form of the conjecture of Birch and Swinnerton-Dyer implies the non-vanishing at s =1 of the complex L-series of B over every finite layer of the unique Z2-extension of K unramified outside p. We hope to give a proof of this last non-vanishing assertion in a subsequent paper.展开更多
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...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).展开更多
In this paper the author derives a geometric characterization of totally isotropic Willmore two-spheres in S^(6), which also yields to a description of such surfaces in terms of the loop group language. Moreover, appl...In this paper the author derives a geometric characterization of totally isotropic Willmore two-spheres in S^(6), which also yields to a description of such surfaces in terms of the loop group language. Moreover, applying the loop group method, he also obtains an algorithm to construct totally isotropic Willmore two-spheres in S^(6). This allows him to derive new examples of geometric interests. He first obtains a new, totally isotropic Willmore two-sphere which is not S-Willmore(i.e., has no dual surface) in S^(6). This gives a negative answer to an open problem of Ejiri in 1988. In this way he also derives many new totally isotropic, branched Willmore two-spheres which are not S-Willmore in S^(6).展开更多
Let p be an odd prime and F∞ a p-adic Lie extension of a number field F with Galois group G. Suppose that G is a compact pro-p p-adic Lie group with no torsion and that it contains a closed normal subgroup H such tha...Let p be an odd prime and F∞ a p-adic Lie extension of a number field F with Galois group G. Suppose that G is a compact pro-p p-adic Lie group with no torsion and that it contains a closed normal subgroup H such that G/H≌Zp. Under various assumptions, we establish asymptotic upper bounds for the growth of p-exponents of the class groups in the said p-adic Lie extension. Our results generalize a previous result of Lei, where he established such an estimate under the assumption that H≌Zp.展开更多
A Nash group is said to be almost linear if it has a Nash representation with a finite kernel. Structures and basic properties of these groups are studied.
基金supported by the National Natural Science Foundation of China under Grant No.11371236the Graduate Student Innovation Foundation of Shanghai University of Finance and Economics(CXJJ-2015-440)
文摘This paper is concerned with tile proOlenl or improving hue ~lma^u~ u~ under Stein＇s loss. By the partial Iwasawa coordinates of covariance matrix, the corresponding risk can be split into three parts. One can use the information in the weighted matrix of weighted quadratic loss to improve one part of risk. However, this paper indirectly takes advantage of the information in the sample mean and reuses Iwasawa coordinates to improve the rest of risk. It is worth mentioning that the process above can be repeated. Finally, a Monte Carlo simulation study is carried out to verify the theoretical results.
文摘This paper is concerned with a conjecture formulated by Coates et al in [4] which describes the relation between the integrality of the characteristic elements and their evaluations in the noncommutative Iwasawa theory. We give an almost equivalent description of the conjecture and prove a certain Dart of it.
文摘The field K ---- Q（x/-AT） is the only imaginary quadratic field with class number 1, in which the prime 2 splits, and we fix one of the primes p of K lying above 2. The modular elliptic curve X0 （49） has complex multiplication by the maximal order O of K, and we let E be the twist of Xo （49） by the quadratic extension K（v/M）/K, where M is any square free element of O with M -- i mod 4 and （M, 7） = 1. In the present note, we use surprisingly simple algebraic arguments to prove a sharp estimate for the rank of the MordeII-Weil group modulo torsion of E over the field F∞=k（Ep∞ ）, where Epic denotes the group of p∞-division points on E. Moreover, writing B for the twist of X0（49） by K（C/ET）/K, our Iwasawa-theoretic arguments also show that the weak form of the conjecture of Birch and Swinnerton-Dyer implies the non-vanishing at s =1 of the complex L-series of B over every finite layer of the unique Z2-extension of K unramified outside p. We hope to give a proof of this last non-vanishing assertion in a subsequent paper.
基金the Chinese Academy of Science(Grant No.Y729025EE1)NSFC(Grant Nos.11688101,11621061)an NSFC grant associated to the recruitment Program of Global Experts。
文摘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).
基金supported by the National Natural Science Foundation of China(Nos.11971107,11571255).
文摘In this paper the author derives a geometric characterization of totally isotropic Willmore two-spheres in S^(6), which also yields to a description of such surfaces in terms of the loop group language. Moreover, applying the loop group method, he also obtains an algorithm to construct totally isotropic Willmore two-spheres in S^(6). This allows him to derive new examples of geometric interests. He first obtains a new, totally isotropic Willmore two-sphere which is not S-Willmore(i.e., has no dual surface) in S^(6). This gives a negative answer to an open problem of Ejiri in 1988. In this way he also derives many new totally isotropic, branched Willmore two-spheres which are not S-Willmore in S^(6).
基金Supported by National Natural Science Foundation of China(Grant Nos.11550110172 and 11771164)
文摘Let p be an odd prime and F∞ a p-adic Lie extension of a number field F with Galois group G. Suppose that G is a compact pro-p p-adic Lie group with no torsion and that it contains a closed normal subgroup H such that G/H≌Zp. Under various assumptions, we establish asymptotic upper bounds for the growth of p-exponents of the class groups in the said p-adic Lie extension. Our results generalize a previous result of Lei, where he established such an estimate under the assumption that H≌Zp.
基金supported by the National Natural Science Foundation of China(Nos.11222101,11321101)
文摘A Nash group is said to be almost linear if it has a Nash representation with a finite kernel. Structures and basic properties of these groups are studied.
