# Questions tagged [nt.number-theory]

Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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### Special values of Artin L-functions

This question might be naive and might carry the heuristic that we are living in the best possible world a little too far. If so, I appreciate being told so. Background: Stark's conjecture interprets ...
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### Finite version special case Jacobi triple product formula

In this paper, Shanks uses the following formula: $$\sum_{s=0}^{n-1}q^{s(2n+1)} \times \left( \prod_{k=s+1}^{n} \dfrac{1-q^{2k}}{1-q^{2k-1}}\right) = \sum_{s=1}^{2n} q^{\frac{s(s-1)}{2}}$$ to get a ...
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### Failure of local Fontaine Mazur

This question unfortunately has a very similar name to this one, but I what want to ask here is different. Let $K$ be a finite extension of $\mathbb{Q}_p$. It seems to be well known that the local ...
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### If $ab^2$ is a sum of three squares, then so is $a$. How to see it quickly?

Here $a, b$ are positive integers, and the squares are the squares of integers. This follows from Legendre's three squares theorem, but is there a direct way?
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### Explicit example of elliptic curve of the kind needed for IUTT

At the nLab, we are currently trying to illustrate the definition of initial theta-data in Mochizuki's first IUTT paper by means of an explicit example. The exposition should end up at the following ...
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### Moments of derivatives of $L$-functions

I'd like to know why it is important to know the moments of the derivatives of $L$-functions. The moments of $L$-functions are related to the Lindelöf Hypothesis, but what about the moments of the ...
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### Status of the “anabelian dream” ($\mathrm{dim} \leq 1$)

The anabelian conjectures for small dimensions have been known for quite some time. In full generality the results are: Dimension 0. Finitely generated fields are anabelian (Pop) Dimension 1. ...
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### Counting abelian varieties over finite fields in a given isogeny class

Let $f(x) \in \mathbb Z[x]$ be a monic polynomial of degree $g$ with all roots having absolute value $\sqrt{q}$. How many principally polarized abelian varieties over $\mathbb F_q$ have $f(x)$ as the ...
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### Is the absolute Galois group of the rationals Hopfian?

Is every continuous epimorphism from the absolute Galois group of $\mathbb{Q}$ to itself injective?
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### Can there be arbitrarily many cubic fields unramified outside $\{p,\infty\}$?

Observe, trivially, that since quadratic fields correspond to rational integers modulo squares (viz. discriminants), there are (roughly about, but certainly at most) $2^{|S|+1}$ quadratic fields ...
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### On quintic roots $x_1^{1/5}+x_2^{1/5}+\dots+x_5^{1/5}$

(Major revision to incorporate new results in this MO cubic version.) Note: All coefficients are in the rationals. I. Cubic In the linked post, it was shown that given a general cubic (via its ...
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### What's the dimension of the space of CM cusp forms?

I would guess that the following is very well known, but I don't know the answer and I couldn't find anything with some googling. Let $\Gamma \subset \mathrm{SL}(2,\mathbf Z)$ be a congruence ...
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### Positive binary quadratic form plus univariate monic cubic (giving Hilbert class field)

We have the Lucas numbers, $$L_1 = 1, \; L_2 = 3, \; L_3 =4, \; L_4 = 7, L_5 = 11, \; L_{n+2} = L_{n+1}+ L_n \; .$$ Question: is it the case that $$f(x,y,z) = 4 x^2 + 3 x y + 9 y^2 + z^3 + 3 z$$...
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### Is there an infinite field of characteristic 2 whose multiplicative group is torsion free and (direct-sum) indecomposable?

Let $F$ be a infinite field of characteristic 2 whose multiplicative group $F^*$ is torsion free. I would like to conclude that $F^*$ is decomposable or find an example where $F^*$ is indecomposable. ...
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### Structure of the algebra of mod $p$ modular forms

Let me first define the algebra $M$ I am talking about: let us fix a prime $p$, an integer $N$ not divisible by $p$. For $k$ an integer, let me call $N_k$ the $\mathbb{Z}$-module of modular forms of ...
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### Cohomological interpretations of quadratic form invariants over rings?

The standard approach to classifying of quadratic forms over $\Bbb Q$ is to use the Hasse (local-global) principle together with a system of standard invariants of quadratic forms over the local ...
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### Any references on zeta-function like sums of inverse determinants over lattices of matrices?

I'm sorry for the title, it was little difficult to phrase.. Let us consider a matrix lattice $L\subset M_n(\mathbb{C})$. By this I mean a discrete additive group in $M_n(\mathbb{C})$. Let us ...
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### Galois theory: Generalization of Abel's Theorem?

Let $L$ stand for the field obtained by adjoining to ${\Bbb Q}$ all roots of all polynomials of the form $x^n+ax+b$, $a,b\in {\Bbb Q}$. What polynomials $p$ don't split over $L$? In particular, ...
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### Geometry of algebraic curve determined by point counts over all number fields?

Let $C$ be a smooth irreducible projective curve of genus $g>1$ over a number field $K$. The Mordell conjecture (first proved by Faltings) says that for any finite field extension $L/K$ (say, ...
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### When does the product equal the sum?

Let $R$ be a commutative ring with identity and $R^n$ be the direct sum of $R$. Find all $a_1, a_2, \cdots, a_n \in R$ such that $$a_1 + a_2 + \cdots + a_n = a_1a_2\cdots a_n,$$ or, in other words, if ...
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### Hensel lemma and rational points in complete noetherian local ring

Let $A$ be a complete noetherian local ring and $\mathfrak{m}$ be its maximal ideal. If we have several polynomials $f_i \in A[X_1, \dots, X_m]$ which have a common zero $x_n$ in $A/\mathfrak{m}^n$ ...
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### Is the set of numbers $\{ [n^{3/2}] \mid n\text{ an integer}\}$ a basis of order 3?

A set of integers is called a basis of order h if every nonnegative integer can be written as the sum of h not necessarily distinct elements of the set.(Nathanson, Additive Number Theory, page 7) ...