Fix a number field k with its adele ring A. Let G=O(n+3) be an orthogonal group of k-rank 1 and H=O(n+2) a k-anisotropic subgroup. We have previously [arXiv:0908.3521] described how to factor the global period of a spherical Eisenstein series of G against a cuspform F of H into an Euler product. Here, we describe how to evaluate the factors at even primes. When the local field is unramified, we carry out the computation in all cases. We show also concrete examples of the complete period when k=Q. The results are consistent with the Gross-Prasad conjecture.
On the Ramification of Hecke Algebras at Eisenstein Primes
利用Wiles的模技巧，研究了权为2、素级为N的Hecke代数在Eisenstein素数上的局部化.在此基础上，我们从形变理论的角度恢复了Mazur的一些结果（“模曲线和Eisenstein理想”）。结合Merel的一个定理，我们得到了Q（N^（1/p））的p-部分的新信息，其中p和N是素数，N = 1 mod p.
Let $E/\mathbb{Q}$ be an elliptic curve, and $p$ a prime where $E$ has good reduction, and assume that $E$ admits a rational $p$-isogeny. In this paper, we study the anticyclotomic Iwasawa theory of $E$ over an imaginary quadratic field in which $p$ splits, which we relate to the anticyclotomic Iwasawa theory of characters following the method of Greenberg--Vatsal. As a result of our study, we obtain a proof, under mild hypotheses, of Perrin-Riou's Heegner point main conjecture, as well as a $p$-converse to the theorem of Gross--Zagier and Kolyvagin and the $p$-part of the Birch--Swinnerton-Dy…
On Greenberg's conjecture for rational elliptic curves at Eisenstein primes
University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；Morningside Center of Mathematics（中国科学院数学与系统科学研究院）；Academy of Mathematics and Systems Sciences, Chinese Academy of Science
We prove for primes $p\ge 5$ a conjecture of Coleman on the analytic continuation of the family of modular functions $\frac{E^\ast_κ}{V(E^\ast_κ)}$ derived from the family of Eisenstein series $E^\ast_κ$. The precise, quantitative formulation of the conjecture involved a certain on $p$ depending constant. We show by an example that the conjecture with the constant that Coleman conjectured cannot hold in general for all primes. On the other hand, the constant that we give is also shown not to be optimal in all cases. The conjecture is motivated by its connection to certain central statements in…
For any rational prime $p$, we define a certain $p$-stabilization of holomorphic Siegel Eisenstein series for the symplectic group $\text{Sp}(2n)_{/\mathbb{Q}}$ of an arbitrary genus $n \ge 1$. In addition, we derive an explicit formula for the Fourier coefficients and conclude their $p$-adic interpolation problems. Consequently, for any odd prime $p$, we deduce the existence of a $Λ$-adic form (in the sense of A. Wiles, H. Hida and R.L. Taylor) such that after taking a suitable constant multiple, it interpolates $p$-adic analytic families of the above-mentioned $p$-stabilized Siegel Eisenstei…
For an even positive integer $n$, we study rank-one Eisenstein cohomology of the split orthogonal group ${\rm O}(2n+2)$ over a totally real number field $F.$ This is used to prove a rationality result for the ratios of successive critical values of degree-$2n$ Langlands $L$-functions associated to the group ${\rm GL}_1 \times {\rm O}(2n)$ over $F$. The case $n=2$ specializes to classical results of Shimura on the special values of Rankin - Selberg $L$-functions attached to a pair of Hilbert modular forms.
Let $p$ be a prime $\ge 5$. We establish explicit rates of overconvergence for members of the "Eisenstein family", notably for the $p$-adic modular function $V(E_{(1,0)}^{\ast})/E_{(1,0)}^{\ast}$ ($V$ the $p$-adic Frobenius operator) that plays a pi\-votal role in Coleman's theory of $p$-adic families of modular forms. The proof goes via an in-depth analysis of rates of overconvergence of $p$-adic modular functions of form $V(E_k)/E_k$ where $E_k$ is the classical Eisenstein series of level $1$ and weight $k$ divisible by $p-1$. Under certain conditions, we extend the latter result to a vast g…
We study nearly holomorphic Siegel Eisenstein series of general levels and characters on $\mathbb{H}_{2n}$, the Siegel upper half space of degree $2n$. We prove that the Fourier coefficients of these Eisenstein series (once suitably normalized) lie in the ring of integers of $\mathbb{Q}_p$ for all sufficiently large primes $p$. We also prove that the pullbacks of these Eisenstein series to $\mathbb{H}_n \times \mathbb{H}_n$ are cuspidal under certain assumptions.
