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Results tagged with algebraic-number-theory
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user 70751
Algebraic number fields, Algebraic integers, Arithmetic Geometry, Elliptic Curves, Function fields, Local fields, Arithmetic groups, Automorphic forms, zeta functions, $L$-functions, Quadratic forms, Quaternion algebras, Homogenous forms, Class groups, Units, Galois theory, Group cohomology, Étale cohomology, Motives, Class field theory, Iwasawa theory, Modular curves, Shimura varieties, Jacobian varieties, Moduli spaces
3
votes
1
answer
318
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 …
- 1,805
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.
- 1,805
5
votes
2
answers
611
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 …
- 1,805
7
votes
1
answer
387
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 …
- 1,805
2
votes
0
answers
134
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 …
- 1,805
4
votes
0
answers
248
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 …
- 1,805
7
votes
0
answers
151
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 …
- 1,805