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1.
We consider the Weil elliptic curve E/ℚ and let
be its canonical L-series. Admitting the Birch-Swinnerton-Dyer conjecture and fixing the curve E, a criterion is given for
the finiteness of the group ED(ℚ) for twisted elliptic curves ED, defined by the condition
相似文献
2.
A. A. Panchishkin 《Mathematical Notes》1975,17(2):148-153
Let τ(n) be Ramanujan's function, $$x\prod _{m = 1}^\infty (1 - x^m )^{24} = \sum\nolimits_{n = 1}^\infty {\tau (n)x^n .} $$ In this paper it is shown that the Ramanujan congruence τ(n)=σd/nd11 mod 691 cannot be improved mod 6912. The following result is proved: for arbitrary r, s mod 691 the set of primes such that p ≡ r mod 691,τ (p) ≡ p11+1+691 · s mod 6912 has positive density. 相似文献
3.
4.
A. A. Panchishkin 《Journal of Mathematical Sciences》2008,149(3):1246-1254
We discuss modular forms as objects of computer algebra and as elements of certain p-adic Banach modules. We discuss a problem-solving approach in number theory, which is based on the use of generating functions
and their connection with modular forms. In particular, the critical values of various L-functions of modular forms produce nontrivial but computable solutions of arithmetical problems. Namely, for a prime number
we consider three classical cusp eigenforms
5.
Alexei Panchishkin 《Israel Journal of Mathematics》2011,185(1):343-368
For a prime p and a positive integer g, by making use of certain lifting procedures, we study some constructions of p-adic families of Siegel modular forms of genus g and associated p-adic L-functions. Describing L-functions attached to Siegel modular forms and their analytic properties from the point of view of motivic L-functions studied by Deligne and Yoshida, we discuss critical values of the L-functions and p-adic interpolation problems. In particular, we formulate a general conjecture on the existence of the modularity lifting
from GSp
r
× GSp2m
to GSp
r+2m
for some positive integers r and m. 相似文献
6.
7.
A. A. Panchishkin 《Journal of Mathematical Sciences》1983,23(6):2707-2736
In this survey there are included results of recent years, concerning the theory of modular forms and representations connected with them of adele groups and Galois groups. There is discussed the hypothetical principle of functoriality of automorphic forms and other conjectures of Langlands concerning automorphic forms and the L-functions connected with them. 相似文献
8.
A. A. Panchishkin 《Israel Journal of Mathematics》2000,120(1):467-509
An Eisenstein measure on the symplectic group over rational number field is constructed which interpolatesp-adically the Fourier expansion of Siegel-Eisenstein series. The proof is based on explicit computation of the Fourier expansions
by Siegel, Shimura and Feit. As an application of this result ap-adic family of Siegel modular forms is given which interpolates Klingen-Eisenstein series of degree two using Boecherer’s
integral representation for the Klingen-Eisenstein series in terms of the Siegel-Eisenstein series. 相似文献
9.
Alexei Panchishkin 《Journal of Mathematical Sciences》2012,180(5):626-640
Let p be a prime, and let
G = \textS\textpg( \mathbbZ ) \Gamma = {\text{S}}{{\text{p}}_g}\left( \mathbb{Z} \right) be the Siegel modular group of genus g. This paper is concerned with p-adic families of zeta functions and L-functions of Siegel modular forms; the latter are described in terms of motivic L-functions attached to Sp
g
; their analytic properties are given. Critical values for the spinor L-functions are discussed in relation to p-adic constructions. Rankin’s lemma of higher genus is established. A general conjecture on a lifting of modular forms from
GSp2m
× GSp2m
to GSp4m
(of genus g = 4 m) is formulated. Constructions of p-adic families of Siegel modular forms are given using Ikeda–Miyawaki constructions. 相似文献
10.
A. A. Panchishkin 《Mathematical Notes》2010,88(3-4):544-551
For a prime p and a positive integer n, using certain lifting procedures, we study some constructions of p-adic families of Siegel modular forms of genus n. Describing L-functions attached to Siegel modular forms and their analytic properties, we formulate two conjectures on the existence of the modularity liftings from GSp r × GSp2m to GSp r+2m for some positive integers r and m. 相似文献
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