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Results tagged with nt.number-theory
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user 70751
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
2
votes
1
answer
160
views
Local nontriviality of genus-one curves over extensions of degree dividing $6^n$
Suppose $p\geq 5$ is a prime, and $C$ a genus-one curve, defined over $\mathbf{Q}$. Is there always an extension $K/\mathbf{Q}_{p}$ whose degree divides a power of $6$, so that
$C(K)$ is not empty?
(I …
- 1,727
3
votes
1
answer
807
views
Cohomology of elliptic curves
Assume $K$ is an imaginary quadratic extension of $\mathbb{Q}$, and $E$ an elliptic curve defined over $\mathbb{Q}$.
Let $p\neq l$ be primes in $\mathbb{Q}$ where $E$ has good reduction. Assume $p$ sp …
- 1,727
12
votes
0
answers
759
views
Is the number $\sum_{p\text{ prime}}p^{-2}$ known to be irrational?
Is the number $$\sum_{p\text{ prime}}p^{-2}$$ known to be irrational?
The limit exists, since $$\sum_{p\text{ prime}}p^{-2}<\sum_{i=1}^{\infty}i^{-2}=\frac{\pi^{2}}{6}$$.
- 1,727
1
vote
0
answers
97
views
Is $(pE(L))^{\operatorname{Gal}(L/K)}/pE(K)=0$ for almost all $p\geq 5$ if $rank(E)\geq 1$, ...
Let $K$ be a number field (possibly of infinite degree over $\mathbb{Q}$) and $E$ an elliptic curve without complex multiplication.
Let $L:= K(E_{5^{\infty}7^{\infty}11^{\infty}...})$ be the field ob …
- 1,727
1
vote
0
answers
81
views
Reference request: Structure of $H^1({{\mathbf{Q}}_{q}},{{{{E}}_{{p}^{\infty}}}})$
I need reference on the structure of ${H}^{1}({{\mathbf{Q}}_{q}},{{E}_{{{p}^{\infty}}}})$, in particular when:
(1.) $q=p$
and/or
(2.) $E$ has multiplicative reduction at $q$.
Here, $E$ is an ellip …
- 1,727
2
votes
1
answer
137
views
Points $\alpha_n$ of $A$ over the $m_n$-th layer in a $\mathbb{Z}_p$-ext. of $K$, where $A$ ...
I have the following setting:
1.) A Galois extension of number fields $K\hookrightarrow L$, with $\operatorname{Gal}(L/K)=\mathbb{Z}_{p}$. In my terminology, number field does not imply finiteness ov …
- 1,727
3
votes
1
answer
295
views
Is there an infinite family of primes $q_{1},q_{2},...$ so that the rank of $E(\mathbb{Q}(\s...
It is known that the group of $K$-rational points of an elliptic curve $E$ is finitely generated if $K$ is a number field of finite degree over $\mathbb{Q}$.
Much less is known if $K$ is infinite-dime …
- 1,727
3
votes
Accepted
Is there an infinite family of primes $q_{1},q_{2},...$ so that the rank of $E(\mathbb{Q}(\s...
Pasten has answered the question: Murty (MS1106677, Corollary to Theorem 2) has shown that the quadratic twist of $E$ by a prime $q$ has rank zero for infinitely many primes $q$, if GRH holds.
- 1,727
5
votes
1
answer
213
views
Is there $t\in\operatorname{Gal}(\overline{K}/K)$ s.t. $\operatorname{rank}_{\mathbf{Z}_p}((...
Let $E$ be an elliptic curve defined over $\mathbb{Q}$, let $$K:=\varinjlim_{k\in\mathbb{Q}[\mu_{p^\infty}]} \mathbb{Q}\left[\mu_{p^\infty},k^{1/p^\infty}\right]$$
and $G:=\operatorname{Gal}(\overline …
- 1,727
-3
votes
1
answer
232
views
Is $\sharp E((\mathbb{F}_{p^{2}})/E(\mathbb{F}_{p}))=1$ for almost all primes $p$? [closed]
Let $E$ be an elliptic curve over $\mathbb{Q}.$
Is $\sharp E((\mathbb{F}_{p^{2}})/E(\mathbb{F}_{p}))=1$ for almost all primes $p$?
- 1,727
8
votes
2
answers
518
views
What do we know about the structure of $J_{0}(N)$ over $\mathbb{Q}[{\mu}_{{p}^{\infty}},{{k}...
What is known about the structure of $J_{0}(N)$ over $\mathbb{Q}[\mu_{p^{\infty}}]$?
More generally, what do we know about $J_{0}(N)$ over
$\mathbb{Q}[\mu_{p^{\infty}},k^{1/p^{n}}]$, where $k\in\mat …
- 1,727
4
votes
2
answers
561
views
Mordell-Weil rank of an elliptic curve over $\mathbb{Q}(\sqrt{-1},\sqrt{2},\sqrt{3},\sqrt{5}...
It is known that the group of $K$-rational points of an elliptic curve $E$ is finitely generated if $K$ is a number field of finite degree over $\mathbb{Q}$.
The picture is less clear if $K$ is infini …
- 1,727
6
votes
1
answer
496
views
Good references for K-theory of modular curves?
The title says it. I am looking for a good exposition on the K-theory of the curves $X_{i}(N)$, $Y_{i}(N)$, where $i\in\{0,1\}$.
I have some background in $K$-theory and also some background in modula …
- 1,727
1
vote
0
answers
145
views
Existence of infinite or bad places that ramify in $K(p^{-1}E(K))/K$ where $p$ is a prime of...
Let $E$ be an elliptic curve, $K$ a number field so that $\operatorname{rank}_{K}(E)\geq 1,$ $p$ a prime of $\mathbb{Q}$ at which $E$ reduces well.
We know (see for instance Silverman, The Arithmetic …
- 1,727
2
votes
0
answers
82
views
Rank growth in ray class fields of primes that are inert in an imaginary quadratic extension
If $K=\mathbb{Q}[\sqrt{-d}]$, $E$ is an elliptic curve and $\operatorname{rank}(E(K))=\operatorname{rank}(E(\mathbb{Q}))$, are there infinitely many primes $l$ of $\mathbb{Q}$ so that:
(1.) $\operato …
- 1,727
5
votes
0
answers
303
views
Existence of infinitely many Heegner points that are divisible by $p^{n}$ in $K_{\lambda}$
Let $E$ be an elliptic curve over $\mathbb{Q}$, $p>2$ a prime and $n,s\in\mathbb{N}$.
For $j\in\{1,...,s\}$ let $n_{j}\in\mathbb{N}$ be a natural number which may or may not be coprime to $p$.
Let $K/ …
- 1,727
3
votes
0
answers
317
views
Lifting a real quadratic twist of an Elliptic Curve to the modular curve
Let $E$ be an elliptic curve of conductor $N\cdot p^2$ over $\mathbb{Q}$, defined by the equation
$$y^2=x^3+p^2b\cdot x + p^3\cdot c$$
and parametrized by a map
$$X_{0}(N\cdot {p}^{2})\rightarrow E$$
…
- 1,727
7
votes
1
answer
371
views
Existence of imaginary quadratic fields of class numbers coprime to $p$ with prescribed spli...
Let $x\in\{\text{totally ramified, inert, totally split}\}.$
If $p\geq 5$ is a prime, are there infinitely many imaginary quadratic fields $K=\mathbb{Q}(\sqrt{-d})$ of class number coprime to $p$ so …
- 1,727
8
votes
0
answers
149
views
Looking for an elliptic curve E st ${\large Ш}(\mathbb Q,E)$ cont. an element of order $p^2$...
I am looking for an elliptic curve $E$ with Weierstraß coefficients in $\mathbb{Q} $ so that for some prime $p$ the following conditions are satisfied:
(1) ${\large Ш}_{p^{\infty}}(\mathbb{Q},E)$ con …
- 1,727
3
votes
1
answer
394
views
Selmer and free rank of Elliptic Curves
If I am not mistaken, the equality of the $p$-Selmer rank and the free rank of an elliptic curve are conjectured to be equal.
This is one of the many implications of the Birch and Swinnerton-Dyer conj …
- 1,727
4
votes
0
answers
245
views
Height pairings of Heegner points of nontrivial conductor
I am studying Gross's and Zagier's proof of the BSD conjecture for elliptic curves of rank $\leq 1.$ Their calculation essentially boils down to the following ingredients:
(1.) Finding a suitable imag …
- 1,727
6
votes
1
answer
771
views
Relationship between Tate-Shafarevich group and the BSD conjecture
The finiteness of the Tate-Shafarevich group is known to be equivalent to BSD for elliptic curves over function fields over $\mathbb{F}_{q},$ this result is due to Kato and Trihan if I am not mistaken …
- 1,727
1
vote
1
answer
224
views
Is an Isomorphism from an Abelian variety to a Shimura variety always defined over a solvabl...
Are there counterexamples to the following:
Given two varieties $A$, $\tilde{A}$, both defined over $\mathbb{Q}$, one of which, say $A$, is a Shimura variety. Then, every isomorphism defined over $\ov …
- 1,727
2
votes
0
answers
118
views
Differential equation of Van Gorder type for zeta of global fields, or: Does the zeta functi...
Let
\begin{equation*}
\zeta(s):=\prod_{p\text{ prime}}\frac{1}{1-p^{-s}}
\end{equation*}
be the Riemann zeta function. Van Gorder has shown that $\zeta$ satisfies a differential equation
\begin{equati …
- 1,727
2
votes
0
answers
461
views
Confusion regarding Proposition 1.1 in Wiles's Fermat paper
This is from p. 459 of Wiles's Fermat paper.
Theorem: If $D_{p}$ is a decomposition group at $p$, $A$ is an Artinian local ring with maximal ideal $\mathfrak{m}$ and finite residue field $k$ of charac …
- 1,727
7
votes
0
answers
113
views
Relation between the additive Haar measure on $(K,+)$ and the multiplicative Haar measure on...
The following question comes from my studying of Alain Connes's paper Trace Formula in Noncommutative Geometry and the Zeros of the Riemann Zeta Function. In it, on p. 11, Connes notes that if $K$ is …
- 1,727