In this paper we study the eigenvalue problem for integro-differential operators on a lasso graph.The trace formula of the operator is established by applying the residual technique in complex analysis.
Let f be a fixed Maass form for SL_3(Z)with Fourier coefficients A_(f)(m,n).Let g be a Maass cusp form for SL_2(G)with Laplace eigenvalue(1/4)+k^(2) and Fourier coefficientλ_(g)(n),or a holomorphic cusp form of even ...Let f be a fixed Maass form for SL_3(Z)with Fourier coefficients A_(f)(m,n).Let g be a Maass cusp form for SL_2(G)with Laplace eigenvalue(1/4)+k^(2) and Fourier coefficientλ_(g)(n),or a holomorphic cusp form of even weight k.Denote by S_(X)(f×g,α,β)a smoothly weighted sum of A_(f)(1,n)λ_(g)(n)e(αn~β)for X 0 are fixed real numbers.The subject matter of the present paper is to prove non-trivial bounds for a sum of S_(X)(f×g,α,β)over g as k tends to∞with X.These bounds for average provide insight for the corresponding resonance barriers toward the Hypothesis S as proposed by Iwaniec,Luo,and Sarnak.展开更多
A spectral interpretation for the poles and zeros of the L-function of algebraic number fields is given by Meyer. As Meyer works with Schwartz spaces which are not Hilbert spaces, the information on the location of ze...A spectral interpretation for the poles and zeros of the L-function of algebraic number fields is given by Meyer. As Meyer works with Schwartz spaces which are not Hilbert spaces, the information on the location of zeros of the L-function is lost. In 1999, A. Connes gave a spectral interpretation for the critical zeros the Riemann zeta function. He works with Hilbert spaces. In this paper, we show that a variant of Connes’ trace formula is essentially equal to the explicit formula of A. Weil.展开更多
In this paper, we consider the eigenvalue problem for integro-differential operators with separated boundary conditions on the finite interval and find a trace formula for the integro-differential operator.
1 Introduction Jacobi forms are the generalization of Jacobi theta series and the coefficients of the Fourier-Jacobi expansion of a Siegel modular form. The theory develops systematically in recent years and has many ...1 Introduction Jacobi forms are the generalization of Jacobi theta series and the coefficients of the Fourier-Jacobi expansion of a Siegel modular form. The theory develops systematically in recent years and has many interesting applications in theory of modular forms and number theory.展开更多
We show that the cuspidal component of the stable trace formula of a split special odd orthogonal group over a number field, satisfies a weak form of beyond endoscopic decomposition. We also study the r-stable trace f...We show that the cuspidal component of the stable trace formula of a split special odd orthogonal group over a number field, satisfies a weak form of beyond endoscopic decomposition. We also study the r-stable trace formula, when r is the standard or the second fundamental representation of the dual group, and show that they satisfy a similar kind of beyond endoscopic decomposition. The results are consequences of Arthur's works(2013) on endoscopic classification of automorphic representations, together with known results concerning a class of Langlands L-functions for special odd orthogonal groups.展开更多
In this paper, we define the localization operator associated with the spherical mean operator, and show that the localization operator is not only bounded, but also in Schatten-Von Neumann class. We also give a trace...In this paper, we define the localization operator associated with the spherical mean operator, and show that the localization operator is not only bounded, but also in Schatten-Von Neumann class. We also give a trace formula when the symbol function is a nonnegative function.展开更多
The purposes of this article are to discuss the symplectic transformation laws on theta series and to give some explicit formulas for the trace of the symplectic operator.
This work studies the asymptotic formulas for the solutions of the Sturm-Liouville equation with the polynomial dependence in the spectral parameter. Using these asymptotic formulas it is proved some trace formulas fo...This work studies the asymptotic formulas for the solutions of the Sturm-Liouville equation with the polynomial dependence in the spectral parameter. Using these asymptotic formulas it is proved some trace formulas for the eigenvalues of a simple boundary problem generated in a finite interval by the considered Sturm-Liouville equation.展开更多
Langlands program is the central concern of modern number theory. In general, we think that the main tool for studying this program is the trace formula. Arthur-Selberg's trace formula is successful in solving the...Langlands program is the central concern of modern number theory. In general, we think that the main tool for studying this program is the trace formula. Arthur-Selberg's trace formula is successful in solving the endoscopic case of Langlands program. Arthur stabilized the geometric side of the local trace formula in the general case, but did not stabilize the general spectral. This paper aims to solve the general case by different method in the Archimedean case, by directly stabilizing the spectral side of the local trace formula, and obtains the stable local trace formula.展开更多
基金Supported by the National Natural Science Foundation of China(11871031)the National Natural Science Foundation of Jiang Su(BK20201303).
文摘In this paper we study the eigenvalue problem for integro-differential operators on a lasso graph.The trace formula of the operator is established by applying the residual technique in complex analysis.
文摘Let f be a fixed Maass form for SL_3(Z)with Fourier coefficients A_(f)(m,n).Let g be a Maass cusp form for SL_2(G)with Laplace eigenvalue(1/4)+k^(2) and Fourier coefficientλ_(g)(n),or a holomorphic cusp form of even weight k.Denote by S_(X)(f×g,α,β)a smoothly weighted sum of A_(f)(1,n)λ_(g)(n)e(αn~β)for X 0 are fixed real numbers.The subject matter of the present paper is to prove non-trivial bounds for a sum of S_(X)(f×g,α,β)over g as k tends to∞with X.These bounds for average provide insight for the corresponding resonance barriers toward the Hypothesis S as proposed by Iwaniec,Luo,and Sarnak.
文摘A spectral interpretation for the poles and zeros of the L-function of algebraic number fields is given by Meyer. As Meyer works with Schwartz spaces which are not Hilbert spaces, the information on the location of zeros of the L-function is lost. In 1999, A. Connes gave a spectral interpretation for the critical zeros the Riemann zeta function. He works with Hilbert spaces. In this paper, we show that a variant of Connes’ trace formula is essentially equal to the explicit formula of A. Weil.
基金Supported by the National Natural Science Foundation of China(No.11171152)the Natural Science Foundation of Jiangsu(No.BK 2010489)Scientific Research Project Unit of the Firat University(No.1881)
文摘In this paper, we consider the eigenvalue problem for integro-differential operators with separated boundary conditions on the finite interval and find a trace formula for the integro-differential operator.
文摘1 Introduction Jacobi forms are the generalization of Jacobi theta series and the coefficients of the Fourier-Jacobi expansion of a Siegel modular form. The theory develops systematically in recent years and has many interesting applications in theory of modular forms and number theory.
文摘We show that the cuspidal component of the stable trace formula of a split special odd orthogonal group over a number field, satisfies a weak form of beyond endoscopic decomposition. We also study the r-stable trace formula, when r is the standard or the second fundamental representation of the dual group, and show that they satisfy a similar kind of beyond endoscopic decomposition. The results are consequences of Arthur's works(2013) on endoscopic classification of automorphic representations, together with known results concerning a class of Langlands L-functions for special odd orthogonal groups.
文摘In this paper, we define the localization operator associated with the spherical mean operator, and show that the localization operator is not only bounded, but also in Schatten-Von Neumann class. We also give a trace formula when the symbol function is a nonnegative function.
文摘The purposes of this article are to discuss the symplectic transformation laws on theta series and to give some explicit formulas for the trace of the symplectic operator.
文摘This work studies the asymptotic formulas for the solutions of the Sturm-Liouville equation with the polynomial dependence in the spectral parameter. Using these asymptotic formulas it is proved some trace formulas for the eigenvalues of a simple boundary problem generated in a finite interval by the considered Sturm-Liouville equation.
基金supported by National Natural Science Foundation of China(Grant No.11471154)
文摘Langlands program is the central concern of modern number theory. In general, we think that the main tool for studying this program is the trace formula. Arthur-Selberg's trace formula is successful in solving the endoscopic case of Langlands program. Arthur stabilized the geometric side of the local trace formula in the general case, but did not stabilize the general spectral. This paper aims to solve the general case by different method in the Archimedean case, by directly stabilizing the spectral side of the local trace formula, and obtains the stable local trace formula.
