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Results tagged with nt.number-theory
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user 144623
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
2
votes
1
answer
164
views
Number of imaginary quadratic field with its ideal class group has $\Bbb{Z}/2\Bbb{Z}$ as 2 part
Let $K=\Bbb{Q}(\sqrt{D})$($D$ is a square free negative integer) be a quadratic number field.
Class number (order of ideal class group $Cl_K$ of $K$) is $1$ if only if $D=-2,-3,-7,-11,-19,-43,-67,-163 …
- 1,335
1
vote
1
answer
284
views
Does $17x^4+y^2=-1$ have solution in $\Bbb{Q}_2(\sqrt{-5})$?
This question raised when I tried to calculate $2$-Selmer group of elliptic curve $E:y^2=x^3+17x$ over $\Bbb{Q}(\sqrt{-5})$.
$17x^4+y^2=-1$ does not have solution in $\Bbb{Q}_2$
(https://math.stackexc …
- 1,335
6
votes
0
answers
437
views
Mistake in Silverman's book in proof of Neron-Ogg-Shafarevich criterion?
In Silverman's The arithmetic of elliptic curves, p. 201, theorem $7.1$ (Criterion of Neron-Ogg-Shafarevich),
he applies the theorem "When $K$ is complete with respect to it's discrete value, then, $[ …
- 1,335
1
vote
0
answers
83
views
Rayclass group and Hilbert class group, $\mathrm{Gal}(K(\mathfrak{a})/K)\cong(\mathcal{O}_K/...
Let $K$ be an imaginary quadratic field with class number $1$. Let $\mathfrak{a}$ be an ideal of $ \mathcal{O}_K$ and $K(\mathfrak{a})$ denote the ray class field of $K$ modulo $\mathfrak{a}$.
Why doe …
- 1,335
2
votes
1
answer
143
views
How can I calculate $\wp(αu), α\in \Bbb{C}$, $αL⊆L$
Let $\wp(u) = \frac{1}{u^2} + \sum\limits_{\omega \in L, \omega \neq 0} \left(\frac{1}{(u-\omega)^2} - \frac{1}{\omega^2}\right)$ be a Weierstrass pe function.
My question is, how can I calculate $\wp …
- 1,335
0
votes
0
answers
82
views
Relation between divisibility problem of Shafarevich group and group structure of $Ш(E/K)$
For abelian variety $A/K$,
divisibility problem (i.e. $\forall n≧1$, $Ш(A/K)⊂p^nH^1(G_K,A)$ holds for fixed prime $p$?) was asked by Cassels in 1962 and even now discussed.
On the other hand, once gro …
- 1,335
8
votes
1
answer
273
views
Is there a known elliptic curve, $E/\Bbb{Q}$, such that the rank of $E_D/\Bbb{Q}$ is bounded...
Let $E/\Bbb{Q}$ be an elliptic curve defined over $\Bbb{Q}$.
Let $E_D$ be a quadratic twist of $E$ also defined over $\Bbb{Q}$.
Is there a known elliptic curve, $E/\Bbb{Q}$, such that the rank of $E_D …
- 1,335
6
votes
1
answer
301
views
Cubic twist of elliptic curves and its rank
Let $E/\mathbb{Q}$ be an elliptic curve defined by $E: y^2 = x^3 + b$ (where $b \in \mathbb{Q}$).
Let $E_D$ be an elliptic curve defined by $E_D: y^2 = x^3 + bD^2$.
$E$ and $E_D$ are isomorphic over $ …
- 1,335
1
vote
0
answers
87
views
About the exact sequence $0\to $$Ш(E/\Bbb{Q})[\phi]\to Ш(E/\Bbb{Q})[2]\to Ш(E'/\Bbb{Q})[\hat...
This question is about Silverman's book,$7$-th line from the bottom of $350$ page of 'The arithmetic of elliptic curves' (http://www.pdmi.ras.ru/~lowdimma/BSD/Silverman-Arithmetic_of_EC.pdf) .
Let $E$ …
- 1,335
6
votes
1
answer
205
views
Ker of corestriction of Galois cohomology
Let $G$ be a Galois group and $H$ be its normal subgroup. Let $M$ be a $G$-module.
Consider the restriction map $res: H^1(G,M) \to H^1(H,M)$. Its kernel is given by $H^1(G/H,M^H)$.
On the other hand, …
- 1,335
1
vote
0
answers
71
views
Characterization of Selmer group in terms of two descent
This question is about p 337 of Silverman's book ''The arithmetic of elliptic curves'', p 337, http://www.pdmi.ras.ru/~lowdimma/BSD/Silverman-Arithmetic_of_EC.pdf.
Let $E:y^2=x^3+ax^2+bx$ and $E':Y^2= …
- 1,335
2
votes
0
answers
76
views
$K(S,2)=\{b \in K^{\times}/{K^{\times}}^2\mid v(b)≡0 \bmod2, \forall v \notin S\}$ and Selme...
This question is essentially related to the theory of elliptic curves (Selmer group), but this question itself is just a field theoretic one.
To calculate the Selmer group of given elliptic curve, we …
- 1,335
0
votes
0
answers
91
views
Identity component of $\mathrm{Ker}(E^n→E^m)$ in the advanced topics in the arithmetic of el...
$\DeclareMathOperator\Ker{Ker}$Silverman's "Advanced topics in the arithmetic of elliptic curves", p.115 reads $$0\to\mathfrak{a}^{-1}Λ\to\Bbb{C}\to\Ker(E^n\to E^m)\to Λ^n/A^tΛ^m\quad (1)$$ is exact i …
- 1,335
0
votes
0
answers
75
views
Local field such that the value group of $K^\text{perf}$ ( perfect closure of $K$) is $\bigc...
Let $K$ be a local field of positive characteristic.
I'm looking for a $K$ which satisfies the following condition.
Value group of $K^\text{perf}$ (perfect closure of $K$) is $\bigcup_{n\geqq1}(1/p^n …
- 1,335
-2
votes
1
answer
196
views
Special value of Hecke $L$ function
Let $E:y^2＝x^3-x/ \Bbb{Q}(i)$ be elliptic curve and $L(E,1)$ be a special value of $L$ function of $E$ at $1$.
Let $L(ψ,1)$ be value at $1$ of Hecke $L$ function with respect to Hecke character $ψ$, I …
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