相关论文: Rankin-Selberg L-functions in cyclotomic towers, I

Following the prequel work \cite{VO3}, we prove a generalization of "Mazur's conjecture" for $L$-functions of elliptic curves in abelian extensions of imaginary quadratic fields, including the assertion that the Mordell-Weil rank of an…

Let $\pi$ be a cuspidal automorphic representation of $\operatorname{GL}_2$ over a totally real number field $F$. Let $K$ be a totally imaginary quadratic extension of $F$. We estimate central values of the $\operatorname{GL}_2 \times…

We show that a large number of elliptic curve L-functions do not vanish at the central point, conditionally on the generalized Riemann hypothesis and on a hypothesis on the regular distribution of the root number.

We obtain nonvanishing estimates for central values of certain self-dual Rankin-Selberg $L$-functions on $\operatorname{GL}_2({\bf{A}}_F) \times \operatorname{GL}_2({\bf{A}}_F)$, and more generally $\operatorname{GL}_r({\bf{A}}_F) \times…

We define new objects called 'horizontal $p$-adic $L$-functions' associated to $L$-values of twists of elliptic curves over $\mathbb{Q}$ by characters of $p$-power order and conductor prime to $p$. We study the fundamental properties of…

We unconditionally construct cyclotomic p-adic L-functions for Rankin-Selberg convolutions for GL(n+1) x GL(n) over arbitrary number fields, and show that they satisfy an expected functional equation.

We prove an algebraicity result for the central critical value of certain Rankin-Selberg L-functions for GL(n) x GL(n-1). This is a generalization and refinement of some results of Harder, Kazhdan-Mazur-Schmidt, Mahnkopf, and…

We prove an asymptotic expansion of the second moment of the central values of the $\mathrm{GL}(n)\times\mathrm{GL}(n)$ Rankin--Selberg $L$-functions $L(1/2,\pi\otimes\pi_0)$, for a fixed cuspidal automorphic representation $\pi_0$, over…

We formulate a general conjecture relating Chern classes of subbundles of Gauss-Manin bundles in Arakelov geometry to logarithmic derivatives of Artin L-functions of number fields. This conjecture may be viewed as a far-reaching…

We consider $L$-functions attached to representations of the Galois group of the function field of a curve over a finite field. Under mild tameness hypotheses, we prove non-vanishing results for twists of these $L$-functions by characters…

Deligne has formulated extremely influential conjectures about certain special values of the $L$-functions of (Grothendieck) motives over a number field $F$. Given the conjectural dictionary between motives and 'algebraic' automorphic…

We show how non-vanishing of p-adic L functions controls the dimensions of Selmer varieties associated to the complement of the origin in an elliptic curve with CM. As a corollary, one obtains a \pi_1-proof of the theorem of Siegel for such…

We derive new integral presentations for central derivative values of $L$-functions of elliptic curves defined over the rationals, basechanged to a real quadratic field $K$, twisted by ring class characters of $K$ in terms of sums along…

We prove new congruences between special values of Rankin-Selberg $L$-functions for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ over arbitrary number fields. This allows us to control the behavior of $p$-adic $L$-functions under Tate twists and…

We prove that all elliptic curves defined over the cyclotomic $\mathbb{Z}_p$-extension of a real quadratic field are modular under the assumption that the algebraic part of the central value of a twisted $L$-function is a $p$-adic unit. Our…

Notable results on the special values of $L$-functions of Siegel modular forms were obtained by J. Sturm in the case when the degree $n$ is even and the weight $k$ is an integer. In this paper we extend this method to half-integer weights…

In this article, we prove an explicit Waldspurger formula for the toric Hilbert modular forms. As an application, we construct a class of anticyclotomic p-adic Rankin-Selberg L-functions for Hilbert modular forms, generalizing the…

The conjectures of Deligne, Be\u\i linson, and Bloch-Kato assert that there should be relations between the arithmetic of algebro-geometric objects and the special values of their $L$-functions. We make a numerical study for symmetric power…

In this work we use the Rankin-Selberg method to obtain results on the analytic properties of the standard $L$-function attached to vector valued Siegel modular forms. In particular we provide a detailed description of its possible poles…

We prove non-vanishing theorems for the central values of $L$-series of quadratic twists of the Gross elliptic curve with complex multiplication by the imaginary quadratic field $\mathbb{Q}(\sqrt{-q})$, where $q$ is any prime congruent to…