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# 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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### Monogenic cubic rings and elliptic curves

By an elliptic curve over $\mathbb{Q}$, we mean a genus 1 curve with a $\mathbb{Q}$-point. By a monogenic cubic order, we mean a unital cubic ring $R$ isomorphic to $\mathbb{Z}^3$ as a $\mathbb{Z}$-...
120 views

### Newer versions of Mahler's Lemma

I'm trying to find a way to numerically ensure that two constructible numbers are equal (this would be done by a computer). The idea is to find a polynomial $p(x)$ that contains both numbers as roots ...
377 views

### On the product $\prod_{k=1}^{(p-1)/2}(x-e^{2\pi i k^2/p})$ with $x$ a root of unity

Let $p$ be an odd prime. Dirichlet's class number formula for quadratic fields essentially determines the value of the product $\prod_{k=1}^{(p-1)/2}(1-e^{2\pi ik^2/p})$. I think it is interesting to ...
107 views

### $L$-series and the $\zeta$-function of ideal classes modulo $f$

Let $K$ be an imaginary quadratic field and let $\mathcal O_f$ be an order in $K$ with conductor $f$. Let $\chi$ be a proper character of $\operatorname{Cl}(\mathcal O_f)$ (non-principal if $f=1$). ...
317 views

### The sign of an interesting sum involving a Dirichlet character

Let $\chi_{q}$ be a primitive Dirichlet character with modulus $q$ (see definition at wikipedia ). For example for $q=5$ we have \begin{aligned} \chi_{5,1}&=(1, 1, 1, 1, 0),\\ ...
316 views

130 views

### Rank mod $p$ of a non-singular matrix with given determinant

Let $A$ be a non-singular $n$-by-$n$ matrix with integer entries. Assume that $p^r\nmid \det(A)$. Does it follow that $A$ has an $(n-r+1)$-by-$(n-r+1)$ minor that is non-singular modulo $p$? If the ...
36 views

### Specialization of an irreducible polynomial and monogeneous ring of integers

Let $P\in\mathbb{Q}[X_1,\ldots,X_n][T]$ be an irreducible polynomial, which is monic in $T$ of degree $d\geq 1$. For $x=(x_1,\ldots,x_n)\in \mathbb{Q}^n$, let $P_x=P(x_1,\ldots,x_n,T)$. It is well-...
239 views

### Is there any conditions on a finite abelian group so that it cannot be class group of any number field?

The Cohen-Lenstra paper says the probability that the odd part of a class group being cyclic is close to 0.98. So I was thinking: can we find any conditions on a finite abelian group so that it cannot ...

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