共查询到20条相似文献,搜索用时 15 毫秒
1.
Laura Paladino 《Annali di Matematica Pura ed Applicata》2010,189(1):17-23
Let ${\mathcal{E}}Let E{\mathcal{E}} be an elliptic curve defined over
\mathbbQ{\mathbb{Q}} . Let
P ? E(\mathbb Q){P\in {\mathcal{E}}(\mathbb {Q})} and let q be a positive integer. Assume that for almost all valuations
v ? \mathbbQ{v\in \mathbb{Q}} , there exist points
Dv ? E(\mathbb Qv){D_v\in {\mathcal{E}}(\mathbb {Q}_v)} such that P = qD
v
. Is it possible to conclude that there exists a point
D ? E(\mathbb Q){D\in {\mathcal{E}}(\mathbb {Q})} such that P = qD? A full answer to this question is known when q is a power of almost all primes
p ? \mathbbN{p\in \mathbb{N}} , but some cases remain open when p ? S={2,3,5,7,11,13,17,19,37,43,67,163}{p\in S=\{2,3,5,7,11,13,17,19,37,43,67,163\}} . We now give a complete answer in the case when q = 4. 相似文献
2.
Artūras Dubickas 《Archiv der Mathematik》2010,95(2):151-160
Let α be a complex number of modulus strictly greater than 1, and let ξ ≠ 0 and ν be two complex numbers. We investigate the distribution of the sequence ξ α n + ν, n = 0, 1, 2, . . . , modulo ${\mathbb{Z}[i],}Let α be a complex number of modulus strictly greater than 1, and let ξ ≠ 0 and ν be two complex numbers. We investigate the distribution of the sequence ξ
α
n
+ ν, n = 0, 1, 2, . . . , modulo
\mathbbZ[i],{\mathbb{Z}[i],} where i=?{-1}{i=\sqrt{-1}} and
\mathbbZ[i]=\mathbbZ+i\mathbbZ{\mathbb{Z}[i]=\mathbb{Z}+i\mathbb{Z}} is the ring of Gaussian integers. For any
z ? \mathbbC,{z\in \mathbb{C},} one may naturally call the quantity z modulo
\mathbbZ[i]{\mathbb{Z}[i]}
the fractional part of z and write {z} for this, in general, complex number lying in the unit square
S:={z ? \mathbbC:0 £ \mathfrakR(z),\mathfrakJ(z) < 1 }{S:=\{z\in\mathbb{C}:0\leq \mathfrak{R}(z),\mathfrak{J}(z) <1 \}}. We first show that if α is a complex non-real number which is algebraic over
\mathbbQ{\mathbb{Q}} and satisfies |α| > 1 then there are two limit points of the sequence {ξ
α
n
+ν}, n = 0, 1, 2, . . . , which are ‘far’ from each other (in terms of α only), except when α is an algebraic integer whose conjugates over
\mathbbQ(i){\mathbb{Q}(i)} all lie in the unit disc |z| ≤ 1 and
x ? \mathbbQ(a,i).{\xi\in\mathbb{Q}(\alpha,i).} Then we prove a result in the opposite direction which implies that, for any fixed
a ? \mathbbC{\alpha\in\mathbb{C}} of modulus greater than 1 and any sequence
zn ? \mathbbC,n=0,1,2,...,{z_n\in\mathbb{C},n=0,1,2,\dots,} there exists
x ? \mathbbC{\xi \in \mathbb{C}} such that the numbers ξ
α
n
−z
n
, n = 0, 1, 2, . . . , all lie ‘far’ from the lattice
\mathbbZ[i]{\mathbb{Z}[i]}. In particular, they all can be covered by a union of small discs with centers at
(1+i)/2+\mathbbZ[i]{(1+i)/2+\mathbb{Z}[i]} if |α| is large. 相似文献
3.
Paul D. Nelson 《The Ramanujan Journal》2012,27(2):235-284
Let
\mathbbF\mathbb{F} be a totally real number field, and let f traverse a sequence of non-dihedral holomorphic eigencuspforms on
\operatornameGL2/\mathbbF\operatorname{GL}_{2}/\mathbb{F} of weight
(k1,?,k[\mathbbF:\mathbbQ])(k_{1},\ldots,k_{[\mathbb{F}:\mathbb{Q}]}), trivial central character and full level. We show that the mass of f equidistributes on the Hilbert modular variety as
max(k1,?,k[\mathbbF:\mathbbQ]) ? ¥\max(k_{1},\ldots,k_{[\mathbb{F}:\mathbb{Q}]}) \rightarrow \infty. 相似文献
4.
Amol Sasane 《Complex Analysis and Operator Theory》2012,6(2):465-475
Let
\mathbb Dn:={z=(z1,?, zn) ? \mathbb Cn:|zj| < 1, j=1,?, n}{\mathbb {D}^n:=\{z=(z_1,\ldots, z_n)\in \mathbb {C}^n:|z_j| < 1, \;j=1,\ldots, n\}}, and let
[`(\mathbbD)]n{\overline{\mathbb{D}}^n} denote its closure in
\mathbb Cn{\mathbb {C}^n}. Consider the ring
Cr([`(\mathbbD)]n;\mathbb C) = {f:[`(\mathbbD)]n? \mathbb C:f is continuous and f(z)=[`(f([`(z)]))] (z ? [`(\mathbbD)]n)}C_{\rm r}(\overline{\mathbb{D}}^n;\mathbb {C}) =\left\{f: \overline{\mathbb{D}}^n\rightarrow \mathbb {C}:f \,\, {\rm is \,\, continuous \,\, and}\,\, f(z)=\overline{f(\overline{z})} \;(z\in \overline{\mathbb{D}}^n)\right\} 相似文献
5.
Let E be an elliptic curve defined over
\mathbbQ{\mathbb{Q}}. Let Γ be a subgroup of rank r of the group of rational points
E(\mathbbQ){E(\mathbb{Q})} of E. For any prime p of good reduction, let [`(G)]{\bar{\Gamma}} be the reduction of Γ modulo p. Under certain standard assumptions, we prove that for almost all primes p (i.e. for a set of primes of density one), we have
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