All Questions
Tagged with examples nt.number-theory
15 questions
84
votes
31
answers
70k
views
Applications of the Chinese remainder theorem
As the title suggests I am interested in CRT applications. Wikipedia article on CRT lists some of the well known applications (e.g. used in the RSA algorithm, used to construct an elegant Gödel ...
71
votes
2
answers
8k
views
Galoisian sets of prime numbers
The question is about characterising the sets $S(K)$ of primes which split completely in a given galoisian extension $K|\mathbb{Q}$. Do recent results such as Serre's modularity conjecture (as proved ...
- 18.7k
33
votes
3
answers
3k
views
Arithmetic geometry examples
(This is inspired by Algebraic geometry examples.)
I want to collect here (counter)examples in arithmetic geometry.
Curves violating the Hasse principle: The Selmer curve $3X^3 + 4Y^3 + 5Z^3 = 0$. ...
29
votes
12
answers
6k
views
When does 'positive' imply 'sum of squares'?
Does anyone have examples of when an object is positive, then it has (or does not have) a square root? Or more generally, can be written as a sum of squares?
Example. A positive integer does not ...
25
votes
5
answers
5k
views
Proof of no rational point on Selmer's Curve $3x^3+4y^3+5z^3=0$
The projective curve $3x^3+4y^3+5z^3=0$ is often cited as an example (given by Selmer) of a failure of the Hasse Principle: the equation has solutions in any completion of the rationals $\mathbb Q$, ...
- 6,734
25
votes
2
answers
2k
views
Examples where the analogy between number theory and geometry fails
The analogy between $O_K$ ($K$ a number field) and affine curves over a field has been very fruitful. It also knows many variations: the field over which the curve is defined may have positive or zero ...
18
votes
3
answers
2k
views
Eta-products and modular elliptic curves
Recently the elliptic curve $E:y^2+y=x^3-x^2$ of conductor $11$ (which appears in my answer) became my favourite elliptic over $\bf Q$ because the associated modular form
$$
F=q\prod_{n>0}(1-q^n)^2(...
- 18.7k
17
votes
2
answers
3k
views
Consequences of the Birch and Swinnerton-Dyer Conjecture?
Before asking my short question I had made some research. Unfortunately I did not find a good reference with some examples. My question is the following
What are the consequences of the Birch and ...
13
votes
2
answers
4k
views
Example of connected-etale sequence for group schemes over a Henselian field?
Can someone give a really concrete example of such a sequence? I am looking at several notes related with such things, but haven't seen any well-calculated example. And I'm really confused at this ...
- 1,503
12
votes
1
answer
532
views
Finite Galois module whose Ш¹ is nonzero?
In algebraic number theory, we constantly make use of the nine-term Poitou-Tate sequence: Let $K$ be a number field and $M$ a finite $K$-Galois module. Then we have the nine-term exact sequence
$$
H^0(...
- 980
9
votes
6
answers
5k
views
Examples of naturally occurring Quadratic forms or quadrics.
I am always fascinated when a quadratic form (or a quadric) arises naturally. I have
some elementary examples, but most of all, I want to learn more examples. I hope this question isn't considered too ...
6
votes
1
answer
668
views
Examples of "nice" properties of algebraic extensions of $\mathbb{Q}$
I am writing a short survey of some "nice'' properties of algebraic extensions of $\mathbb{Q}$. Let's say a property (P) is nice if
every finite extension of $\mathbb{Q}$ satisfies (P), and
if $K \...
4
votes
1
answer
399
views
Examples when one can use the the symmetric power $L$-functions to study topics related to the number theory
"The symmetric power $L$-functions are a powerful tool for studying algebraic or geometric objects through analytic methods." I read this sentence in the introduction of a Master thesis. I want to ...
- 1,257
2
votes
0
answers
105
views
Trivial fine Selmer group in the cyclotomic extension
In explicit examples that I have seen worked out, it appears that when the fine Selmer group is finite in the cyclotomic extension it is in fact trivial.
Is there any reason to expect that this ...
- 1,283
0
votes
0
answers
102
views
On simple examples of unimodularity
$w=z=x+ 1 =y−1$ provides $wz−xy=w^2−(w−1)(w+ 1) = 1$. Hence if $x,y$ are odd then $w,z$ are even and all four integers are close.
Is there elementary example where only $w$ is even and all four ...
- 13.9k