Local Heights of CM Points

2012 ◽  
Author(s):  
Xinyi Yuan ◽  
Shou-Wu Zhang ◽  
Wei Zhang
Keyword(s):  
Kernel Function ◽  
Quadratic Extension ◽  
Multiplicity Function ◽  
Schwartz Function ◽  
Shimura Curve ◽  
Fourth Case ◽  
Cm Points ◽  
Ordinary Case ◽  
Archimedean Place

This chapter computes the local heights and compares them with the derivatives computed before. It checks the theorem place by place and takes into account all the assumptions on the Schwartz function. According to the reduction of the Shimura curve, the situation is divided to the following four cases: archimedean case, supersingular case, superspecial case, and ordinary case. The treatments in different cases are similar in spirit, except that the fourth case is slightly different. The supersingular case is divided into two subcases: unramified case and ramified case. The chapter also describes local heights of CM points at any archimedean place v. The discussion covers the multiplicity function, the kernel function, unramified quadratic extension, ramified quadratic extension, ordinary components, supersingular components, and superspecial components.

scholarly journals Twisted Gross–Zagier Theorems

2009 ◽  
Vol 61 (4) ◽  
pp. 828-887 ◽  
Author(s):  
Benjamin Howard
Keyword(s):  
Complex Multiplication ◽  
Modular Curves ◽  
Shimura Curve ◽  
Modular Curve ◽  
Hida Theory ◽  
Companion Article ◽  
Cm Points ◽  
Hida Families ◽  
Derivatives Of

Abstract.The theorems of Gross–Zagier and Zhang relate the Néron–Tate heights of complex multiplication points on the modular curve X0(N) (and on Shimura curve analogues) with the central derivatives of automorphic L-function. We extend these results to include certain CM points on modular curves of the form X (Ⲅ0(M ) ∩ Ⲅ1(S)) (and on Shimura curve analogues). These results are motivated by applications to Hida theory that can be found in the companion article “Central derivatives of L -functions in Hida families”,Math. Ann. 399(2007), 803–818.

scholarly journals ANDRÉ–OORT CONJECTURE AND NONVANISHING OF CENTRAL -VALUES OVER HILBERT CLASS FIELDS

2016 ◽  
Vol 4 ◽  
Author(s):  
ASHAY A. BURUNGALE ◽  
HARUZO HIDA
Keyword(s):  
Modular Forms ◽  
Class Group ◽  
Complex Multiplication ◽  
Quadratic Extension ◽  
Real Field ◽  
Shimura Variety ◽  
Group Characters ◽  
Hilbert Modular ◽  
Cm Points ◽  
Definition Of

Let $F/\mathbf{Q}$ be a totally real field and $K/F$ a complex multiplication (CM) quadratic extension. Let $f$ be a cuspidal Hilbert modular new form over $F$. Let ${\it\lambda}$ be a Hecke character over $K$ such that the Rankin–Selberg convolution $f$ with the ${\it\theta}$-series associated with ${\it\lambda}$ is self-dual with root number 1. We consider the nonvanishing of the family of central-critical Rankin–Selberg $L$-values $L(\frac{1}{2},f\otimes {\it\lambda}{\it\chi})$, as ${\it\chi}$ varies over the class group characters of $K$. Our approach is geometric, relying on the Zariski density of CM points in self-products of a Hilbert modular Shimura variety. We show that the number of class group characters ${\it\chi}$ such that $L(\frac{1}{2},f\otimes {\it\lambda}{\it\chi})\neq 0$ increases with the absolute value of the discriminant of $K$. We crucially rely on the André–Oort conjecture for arbitrary self-product of the Hilbert modular Shimura variety. In view of the recent results of Tsimerman, Yuan–Zhang and Andreatta–Goren–Howard–Pera, the results are now unconditional. We also consider a quaternionic version. Our approach is geometric, relying on the general theory of Shimura varieties and the geometric definition of nearly holomorphic modular forms. In particular, the approach avoids any use of a subconvex bound for the Rankin–Selberg $L$-values. The Waldspurger formula plays an underlying role.

scholarly journals ON THE NON-TRIVIALITY OF THE -ADIC ABEL–JACOBI IMAGE OF GENERALISED HEEGNER CYCLES MODULO , II: SHIMURA CURVES

2015 ◽  
Vol 16 (1) ◽  
pp. 189-222 ◽  
Author(s):  
Ashay A. Burungale
Keyword(s):  
Quadratic Extension ◽  
Chow Group ◽  
Abelian Surface ◽  
Shimura Curves ◽  
Heegner Points ◽  
Shimura Curve ◽  
Heegner Cycles ◽  
Coniveau Filtration

Generalised Heegner cycles are associated to a pair of an elliptic newform and a Hecke character over an imaginary quadratic extension $K/\mathbf{Q}$. The cycles live in a middle-dimensional Chow group of a Kuga–Sato variety arising from an indefinite Shimura curve over the rationals and a self-product of a CM abelian surface. Let $p$ be an odd prime split in $K/\mathbf{Q}$. We prove the non-triviality of the $p$-adic Abel–Jacobi image of generalised Heegner cycles modulo $p$ over the $\mathbf{Z}_{p}$-anticyclotomic extension of $K$. The result implies the non-triviality of the generalised Heegner cycles in the top graded piece of the coniveau filtration on the Chow group, and proves a higher weight analogue of Mazur’s conjecture. In the case of weight 2, the result provides a refinement of the results of Cornut–Vatsal and Aflalo–Nekovář on the non-triviality of Heegner points over the $\mathbf{Z}_{p}$-anticyclotomic extension of $K$.

Derivative of the Analytic Kernel

2012 ◽  
Author(s):  
Xinyi Yuan ◽  
Shou-Wu Zhang ◽  
Wei Zhang
Keyword(s):  
Explicit Formula ◽  
General Formula ◽  
Kernel Function ◽  
Schwartz Function ◽  
Explicit Result ◽  
Analytic Kernel ◽  
Holomorphic Projection ◽  
Period Integral ◽  
Local Term

This chapter computes the derivative of the analytic kernel. It first decomposes the kernel function into a sum of infinitely many local terms indexed by places v of Fnonsplit in E. Each local term is a period integral of some kernel function. The chapter then considers the v-part for non-archimedean v. An explicit formula is given in the unramified case, and an approximation is presented in the ramified case assuming the Schwartz function is degenerate. An explicit result of the v-part for archimedean v is also introduced. The chapter proceeds by reviewing a general formula of holomorphic projection, and estimates the growth of the kernel function in order to apply the formula. It also computes the holomorphic projection of the analytic kernel function and concludes with a discussion of the holomorphic kernel function.

scholarly journals Singular Moduli of Shimura Curves

2011 ◽  
Vol 63 (4) ◽  
pp. 826-861 ◽  
Author(s):  
Eric Errthum
Keyword(s):  
Modular Forms ◽  
Complex Multiplication ◽  
Shimura Curves ◽  
Algebraic Integers ◽  
Quaternion Algebras ◽  
Shimura Curve ◽  
Modular Curve ◽  
Singular Moduli ◽  
Cm Points ◽  
Vector Valued

Abstract The j-function acts as a parametrization of the classical modular curve. Its values at complex multiplication (CM) points are called singular moduli and are algebraic integers. A Shimura curve is a generalization of the modular curve and, if the Shimura curve has genus 0, a rational parameterizing function exists and when evaluated at a CM point is again algebraic over Q. This paper shows that the coordinate maps given by N. Elkies for the Shimura curves associated to the quaternion algebras with discriminants 6 and 10 are Borcherds lifts of vector-valued modular forms. This property is then used to explicitly compute the rational norms of singular moduli on these curves. This method not only verifies conjectural values for the rational CM points, but also provides a way of algebraically calculating the norms of CM points with arbitrarily large negative discriminant.

Existence

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Structure Theorem ◽  
Quadratic Extension ◽  
Quadratic Space ◽  
Division Algebras ◽  
Separable Extension ◽  
Valued Fields ◽  
Quaternion Division Algebra

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.

Algorithm of target tracking based on Mean-Shift with adaptive bandwidth of kernel function

2009 ◽  
Vol 29 (6) ◽  
pp. 1680-1682
Author(s):  
Chang-tao CHEN ◽  
Qin ZHU ◽  
Sheng-yi ZHOU ◽  
Jia-ming ZHANG
Keyword(s):  
Target Tracking ◽  
Kernel Function ◽  
Mean Shift ◽  
Adaptive Bandwidth

Remaining Useful Life Prediction of Lithium-ion Batteries Using Multiple Kernel Extreme Learning Machine

2020 ◽  
Vol 13 ◽  
Author(s):  
Renxiong Liu
Keyword(s):  
Lithium Ion Batteries ◽  
Extreme Learning Machine ◽  
Kernel Function ◽  
Lithium Ion ◽  
Remaining Useful Life ◽  
Prediction Methods ◽  
Multiple Kernel ◽  
Kernel Extreme Learning Machine ◽  
Useful Life ◽  
Learning Machine

Objective: Lithium-ion batteries are important components used in electric automobiles (EVs), fuel cell EVs and other hybrid EVs. Therefore, it is greatly important to discover its remaining useful life (RUL). Methods: In this paper, a battery RUL prediction approach using multiple kernel extreme learning machine (MKELM) is presented. The MKELM’s kernel keeps diversified by consisting multiple kernel functions including Gaussian kernel function, Polynomial kernel function and Sigmoid kernel function, and every kernel function’s weight and parameter are optimized through differential evolution (DE) algorithm. Results : Battery capacity data measured from NASA Ames Prognostics Center are used to demonstrate the prediction procedure of the proposed approach, and the MKELM is compared with other commonly used prediction methods in terms of absolute error, relative accuracy and mean square error. Conclusion: The prediction results prove that the MKELM approach can accurately predict the battery RUL. Furthermore, a compare experiment is executed to validate that the MKELM method is better than other prediction methods in terms of prediction accuracy.

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