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

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16
votes
2answers
719 views

Special topics to include in course in algebraic number theory

I'll be teaching an introductory course in algebraic number theory this fall (stopping before class field theory). I'm looking for a good list of "special topics" I can include to illustrate the ...
2
votes
0answers
185 views

Quantum Mechanics derivation of Wallis' Formula?

Recently there was a proof of the Wallis Product using quantum mechanics on the arXiv. However, there are many proofs of the result, Wikipedia has 4. Fine Print the first proof has on Wikipedia, the ...
-1
votes
1answer
61 views

Equation with norms of cyclic extensions of coprime degrees

Let $\mathbb{K}$ be a quadratic extension of $\mathbb{Q}$ and $\mathbb{L}$ be a cyclic extension of $\mathbb{Q}$ of odd degree. Given a rational $r\neq 0$, does there always exist $k\in \mathbb{K}^*$ ...
1
vote
1answer
142 views

Local nontriviality of genus-one curves over extensions of degree dividing $6^n$

Suppose $p\geq 5$ is a prime, and $C$ a genus-one curve, defined over $\mathbf{Q}$. Is there always an extension $K/\mathbf{Q}_{p}$ whose degree divides a power of $6$, so that $C(K)$ is not empty? (I ...
11
votes
2answers
324 views

Can two rational rotations $F_2 = \langle A, B \rangle \to SO(3)$ efficiently approximate the $3 \times 3$ identity matrix?

Let $A,B$ be two rational rotations: $$ A = \left[\begin{array}{rcc} \frac{3}{5} & \frac{4}{5} & 0 \\ -\frac{4}{5} & \frac{3}{5} & 0 \\ 0 & 0 & 1 \end{array}\right] ...
-1
votes
1answer
124 views

Find the values of $n$ that satisfy this inequality involving a product over prime numbers [closed]

Inequality What values of $n$ satisfy the following inequality? $$2(n-2) < Ap_n\prod_{i=3}^n \left(\frac{p_i-2}{p_i}\right)$$ $p$ are prime numbers and the notation $p_i$ indicates the $i$th ...
0
votes
2answers
157 views

Reference : Partition of integer

In algebraic number theory we come across following formula: $n= e_1f_1+\cdots+e_rf_r$ where all $e_i$ and $f_i$ are positive integers. I am sure writing a positive integer n as above must be ...
4
votes
1answer
235 views

The values of $n$ which satisfy an inequality about prime numbers

For which values of $n$ does the following inequality hold for? $$2(n-2) < Ap_n\prod_{i=3}^n \left(\frac{p_i-1}{p_i}\right)$$ $p$ are prime numbers and the notation $p_i$ indicates the ...
7
votes
0answers
196 views

Does Stepanov's method extend to complete intersections?

Stepanov (circa 1970) created the polynomial method to limit the rational points of an algebraic curve over $\mathbb{F}_q$, leading to one of several alternative proofs of Weil's Riemann hypothesis ...
6
votes
2answers
219 views

What are the special parahoric subgroups in unitary groups?

Let $L$ be a $p$-adic field and let $L'/L$ be a quadratic extension. Let $U_{L'/L}(n)$ be a quasi-split unitary group of $n\times n$ matrices with entries in $L'$. I'm curious about what the special ...
0
votes
1answer
158 views

How to calculate $N_{L/k}$(roots of unity)?

Suppose that $L/k$ is a Galois extension of number fields and that $G$ is the corresponding Galois group. Further, for $\frak p$ a prime ideal of $\cal O$$_L$, let $K=L^{G(\frak p)}$, where [$L$ : ...
5
votes
2answers
142 views

Minimum length of a convex lattice polygon containing k lattice points?

Let $f(k)$ denote the minimum length of a convex lattice polygon containing exactly $k$ lattice points (including lattice points on the boundary). It is not too hard to show that $k = \frac{1}{4\pi} ...
3
votes
0answers
144 views

Computing algebraic properties of trace fields, as given by SnapPy

SnapPy can tell you the trace field of a hyperbolic $3$-manifold (which is awesome), but it specifies the field by outputting: the minimal polynomial of the field over $\mathbb{Q}$, and a decimal ...
2
votes
1answer
169 views

Reduction of Abelian Varieties with Complex Multiplication have Complex Multiplication

Let $A$ be an abelian variety of dimension $g$ over $C$ with complex multiplication by a CM field $K$ where $[K:Q] =2g$. By this I mean that End($A$) $\cong \mathcal{O}_K$. Then, $A$ has a model over ...
4
votes
0answers
166 views

$\text{PGL}_2(\mathbb{Q})$-equivalence versus $\text{PGL}_2(\mathbb{Z})$-equivalence

Let $V_{\mathbb{R}}$ be the space of binary quartic forms with real coefficients (so in particular $V_{\mathbb{R}}$ is a 5-dimensional vector space over $\mathbb{R}$), and define the twisted action of ...
2
votes
1answer
87 views

Set of triple-primes satisfying a certain equation

Is there a set of triple-primes satisfying the following equation? $p_1p_2+p_2p_3+p_3p_1+p_1+p_2+p_3=2^β,p_1p_2p_3=2^α-1,α>β.$ I have checked the first 11 numbers that no one satisfy the above ...
12
votes
1answer
207 views

A prime ideal $\mathfrak{p}$ decomposes in $\mathbb{Q}(\zeta_{24})/\mathbb{Q}(\sqrt{-6})$ iff it is generated by $\alpha\in1+2\Bbb{Z}[\sqrt{-6}]$

For a nonzero prime ideal $\mathfrak{p}$ of $\mathbb{Z}[\sqrt{-6}]$ which does not divide $2$, does $\mathfrak{p}$ decompose completely in the extension $\mathbb{Q}(\zeta_{24})/\mathbb{Q}(\sqrt{-6})$ ...
8
votes
1answer
246 views

Have the explicit Poisson-type formulas of Guinand and Meyer been observed before?

In a recent paper of Meyer Measures with locally finite support and spectrum PNAS vol. 113 no. 12:3152–3158 (behind a paywall, but see also these seminar notes) some new explicit Poisson-type formulas ...
7
votes
0answers
232 views

Has it been proved that odd perfect numbers cannot be triangular?

(Note: This question has been cross-posted from MSE.) Euclid and Euler proved that every even perfect number is of the form $m = \frac{{M_p}\left(M_p + 1\right)}{2}$ where $M_p = 2^p - 1$ is a prime ...
18
votes
1answer
298 views

On random divisor sums modulo $2^k$

Let $k,n,\ell$ be positive integers with $k,n\ge 2$ and $0\le \ell \le k-1$. For each integer $2\le j \le n$, choose a divisor $d_j$ of $j$, uniformly at random from the divisors of $j$. We denote by ...
2
votes
0answers
104 views

Extension to real number system [closed]

Suppose you have equation involving a number $s$ $s^2+ 1 = 0$, to solve it one needs to treat $s$ as complex number $s = \pm i$, and introduce $i$ as imaginary unit. Now suppose you have equation ...
7
votes
0answers
129 views

k-Almost Primes in short intervals

According to this question every interval $[x, x + x^{0.45}]$ contains a product of two primes, and this has been improved further slightly. Are there better results available for $k$-almost primes? ...
21
votes
2answers
499 views

CM $j$-invariants in $p$-adic fields

I'm trying to understand the $p$-adic distribution of $j$-invariants for elliptic curves with complex multiplication. Specifically, suppose $\sigma$ is some embedding $\sigma:\overline{\mathbb Q}\to ...
23
votes
1answer
470 views

Artin reciprocity $\implies $ Cubic reciprocity

I asked this on math.SE a few days ago with no reply, so I'm reposting it here. Hope this is not considered too elementary for MO (feel free to close if so). I'm trying to understand the proof of ...
2
votes
0answers
58 views

Property of a derivative in global field

Before posting I want to make it clear that I posted the same question in stack exchange awhile ago (http://math.stackexchange.com/questions/1533814/property-of-derivative-in-a-local-field) but didn't ...
6
votes
1answer
198 views

Can $b^4+1$ be a pseudoprime to base 2 (except for Fermat numbers)?

Cross-post: This very elementary question was first posted to Mathematics Stack Exchange but the response I got there (even after offering a bounty) was not useful. For the purpose of this question, ...
3
votes
0answers
72 views

Sign of bivariate polynomial evaluated over two algebraic numbers

I would like to compute the sign of a bivariate polynomial $f$ evaluated over two algebraic numbers $a$, $b$. The numbers are in "isolating interval representation" meaning that each one is defined by ...
12
votes
0answers
375 views

Elkies' supersingularity theorem in higher dimension (in terms of the associated Newton polygon)

Elkies' supersingularity theorem: Given an elliptic curve $E$ over $\mathbb{Q}$, there are infinitely many primes $p$ such that $E$ is supersingular over $\mathbb{F}_p$. I have seen another post on ...
6
votes
1answer
135 views

What is the cokernel of $O_S \to F_\infty/O_\infty$?

Let $k$ be a field of characteristic $\neq 2$ and consider the quadratic extension $F$ of $k(T)$ generated by $\sqrt{T^3 + 1}$. Let $X$ be the set of all places of $F$. Let $S = \{\infty\} \subset X$ ...
5
votes
1answer
195 views

Bounding $p$-adic characters and Jacquet-Langlands tranfert

I would like to bound uniformly in $\pi$ the $p$-adic Harisch-Chandra characters $\Theta_p$ for division quaternion algebras. By the Jacquet-Langlands correspondence, it is sufficient to bound it on ...
3
votes
1answer
299 views

Relation between the binary Goldbach problem and binary version of Mobius sum

What I want to ask is about the structure of the Goldbach function that defined by $$ R(x)=\#\{ p \mid x-p \in \mathbb{P} , \ p\leq x/2\}$$ for $x\in 2\mathbb{N}$, where $\mathbb{P}$ is the set of ...
2
votes
2answers
283 views

A class of quadratic equations

Let $f(x,y) = ax^2 + bxy + cy^2$ be an indefinite irreducible binary quadratic form with integer coefficients with non-zero discriminant. We can assume, without loss of generality, that $a \geq 1$. ...
35
votes
3answers
3k views

Why such an interest in studying prime gaps?

Prime gaps studies seems to be one of the most fertile topics in analytic number theory, for long and in lots of directions : lower bounds (recent works by Maynard, Tao et al. [1]) upper bounds ...
2
votes
0answers
189 views

On a sequence of L-functions having same zeros in critical strip and GRH

I had an idea on GRH involving a sequence of L-functions having same zeros, then at one step I need a bound on these function and I wonder if this bound is in fact not as hard as GRH itself ? Let's ...
2
votes
0answers
68 views

the least point on a variety over a finite field

Let $p$ be a large prime parameter and $V\subseteq \mathbb{P}^n_{\mathbb{F}_p}$ a variety defined over the finite field $\mathbb{F}_p$ with bounded degree and dimension (w.r.t. $p$). Assume that $V$ ...
0
votes
1answer
221 views

On elliptic curves, $\sqrt{x^2-101y^2} ,\sqrt{x^2+101y^2}$, and their ilk

I. Elliptic curves Given integers $a,b,m_k$. Let, $$x^2+a = m_1u_1^2\\x^2+b = m_1u_2^2\tag1$$ If there is a rational point $x_i$, then the pair (after a transformation) is birationally ...
0
votes
0answers
95 views

Abel summation formula versus Perron's formula to bound a partial sum

Taking $\chi$ a primitive character, with Abel summation it is easy to show that for $\epsilon >0$, there is a constant $M$ such that for all $x$ we have : $$|\sum_{n<x} ...
2
votes
0answers
91 views

A reference about a problem of the number of the rational points on a projective scheme

Let $X\hookrightarrow\mathbb{P}^n_{\mathbb{F}_q}$ be a pure dimensional projective scheme of dimension $d$. So we know a trivial estimate of the number of $X(\mathbb F_q)$ is that $\#X(\mathbb ...
0
votes
1answer
308 views

How do Modular Forms and the Geodesic Flow interact? [closed]

Textbooks talk at length of the modular properties of $\theta(z)$ or $\tau(z)$ and the prominent role of $SL(2,\mathbb{Z})$ or one of the congruence groups. In that case, aren't the basic objects ...
4
votes
3answers
386 views

On a theorem of Hensel about congruence of binomial coefficient

In the paper Binomial coefficients modulo prime powers, Andrew Granville stated the following theorem: Let $n, m$ and $r=n-m$ be three given positive integer and $p^k$ is the exact power of $p$ ...
3
votes
1answer
69 views

Finiteness of the $p$-primary subgroup of an elliptic curve over the cyclotomic $\mathbb{Z}_p$-extension

Let $E$ be an elliptic curve defined over a number field $F$ and $F_\infty$ be the cyclotomic $\mathbb{Z}_p$-extension of $F$. Is it true that the $p$-primary subgroup of $E$ over $F_\infty$ i.e. ...
8
votes
1answer
322 views

Is there any real quadratic ring for which the Euclidean algorithm is polynomial?

We know from Rolletschek's work that the Euclidean algorithm of $\mathbb{Z}[i]$ is polynomial. Indeed, let $n$ be the maximum number of steps in the Euclidean algorithm applied to $u,v ...
4
votes
0answers
119 views

extending $p$-adic character of the local intertia to the absolute Galois group

Suppose I have a number field $F$, and a finite place $v$ of $F$. Let $E$ be finite extension of $F_v$. I start with a continuous morphism $$ \chi \colon O_{F_v}^\times \to E^\times. $$ where ...
7
votes
0answers
184 views

Can primes be (almost) random sequence in von Mises sense?

Random models for primes (such as Cramer's model) have been extensively used for informal justification of various conjectures involving primes. It is crucial to understand in what sense sequence of ...
20
votes
3answers
736 views

Does X(13) have potentially good reduction at 13?

The complete level modular curve $X(p)$ does not have potentially good reduction at $p$ for any $p \neq 2,3,5,7,13$ because then there are cusp forms on $X_0(p)$ showing up in the cohomology of ...
0
votes
1answer
199 views

Is it true that singular fibers of elliptic fibrations that have the same Kodaira type are isomorphic schemes?

It is well known that Kodaira gave an essentially topological classification of the possible singular fibers of elliptic fibrations according to their type: ...
9
votes
2answers
1k views

Approximating any integer by multiples of 2 and 3

Given any integer $n$ sufficiently large, I want to prove (or disprove) that there exists another integer $m\ge n$ with the form $m=2^a3^b$ ($a,b$ are no negative integers) such that $m-n=o(n)$, i.e., ...
4
votes
1answer
182 views

Eisenstein Series on Siegel Space

I am looking for any reference dealing explicitly with Eisenstein series on Siegel space (the simplest case of $\rm{SP}_4$ is fine). Anything would be welcome, but in particular I'm interested in the ...
0
votes
1answer
133 views

An upper bound on $\sum_{n^{1/3}<p,q\leq n^{1/2}} \frac{n}{pq}-\lfloor \frac{n}{pq}\rfloor$

I would like to ask if there is a good upper bound on the difference $$D_2(n)=\sum_{n^{1/3}<p,q\leq n^{1/2}} \left(\frac{n}{pq}-\left\lfloor \frac{n}{pq}\right\rfloor\right)\quad (1) $$where $p$ ...
5
votes
1answer
165 views

Counting the number of permutations of $(1,\ldots,i,\ldots,j,\ldots,m)$, where $i < j$ and number of inversions is $k$

How can I prove the following: $d^{ij}(m,k) > d^{ji}(m,k)$ for all $k < \frac{1}{2}\binom{m}{2},$ where $d^{ij}(m,k)$ denotes the number of permutations of $(1,\ldots,i,\ldots,j,\ldots,m)$ ...