# Questions tagged [algebraic-number-theory]

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

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### Central division algebras over $\mathbb{Q}$

Quaternions over $\mathbb{Q}$ are an example of a Central Division algebra over $\mathbb{Q}$ for which the basis elements $\{ i,j,ij \}$ other than $1$ are represented by skew-symmetric matrices ...
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### Torsion points on $E/\mathbb{Q}$ with large coordinates

Let $E/\mathbb{Q}$ be an elliptic curve with finitely many rational points. What are some examples where at least one rational point has large coordinates (compared to the height of $E$)?
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### How to compute the intersection of an ideal with the maximal order of a subfield?

I asked this earlier on math.stackexchange but I think this is a better place for this question. Computing the intersection of ideals belonging to the same maximal order of a number field $K$ can be ...
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### Elliptic curve over $\mathbb{Q}(T)$ receiving only constant maps from modular curves

Is there an elliptic curve $E/\mathbb{Q}(T)$ such that any map $X_1(N)\to E$ for any $N>0$ is constant?
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### Are class numbers of number fields with prime degree often $1$?

I have taken a look at the class number statistics of the L-functions and Modular Forms Database: https://www.lmfdb.org/NumberField/stats, table "Distribution by class number". It appears ...
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### Commutative algebra details on patching when proving $R = \mathbb{T}$ theorem (Calegari-Geraghty Paper)

I have originally posted this on math.SE and been suggested to post this here. I'm merely an undergraduate student and it is the first time for me to ask questions here. I'm sincerely sorry if these ...
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### Computing the genus of certain ternary indefinite lattices

For $k$ a positive integer, let us consider the rank 3 ternary indefinite lattice $L=L_k$ with quadratic form $$6kx^2-2(y^2+yz+z^2).$$ Its discriminant group has length $2$. Question. Is this lattice ...
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### Why are these sets divisible by n?

Suppose we have a polynomial $z \to f_c(z)$ defined over $\mathbb Z$ with a free parameter $c$, for instance $z \to z^2 + c$ and we consider the iterates $z \to f_c^{(n)}(z)$ and define the ...
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### Using algebraic geometry to understand class field theory

In Algebraic Number Theory, S. Lang says "[a geometrical approach] allows one to have a much clearer insight into the whole class field theory, since the existence theorem and the reciprocity law ...
### Why is the regulator of the degree 11 extension of $\mathbb{Q}$ so large?
Let $K_\infty/\mathbb{Q}$ denote the $\hat{\mathbb{Z}}$-extension of $\mathbb{Q}$. Then for each $n\geq1$, $K_\infty$ has a unique subfield $K_n$ of degree $n$ over $\mathbb{Q}$. The fields $K_n$ are ...