# 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

2,960
questions with no upvoted or accepted answers

**11**

votes

**0**answers

477 views

### Between Fermat's primes and the twin primes

Let me start with a curiosity. The integers $11,13,17,19$ are prime numbers, and $101,103,107,109$ are prime as well. One might wonder whether there is another occurrence where $10^m+1,10^m+3,10^m+7$ ...

**11**

votes

**1**answer

981 views

### Automorphic factorization of Dedekind zeta functions

It is well known that for abelian number fields, the factorization of its Dedekind zeta function goes like this:
$$\zeta_K(s)=\zeta(s)\prod_{\chi \neq 1} L(s,\chi)$$
with the Dirichlet characters ...

**11**

votes

**0**answers

162 views

### Hyperelliptic curves over $\mathbb{Q}$ with a $\mu_p^2$ subgroup in their Jacobian

Given a prime number $p>2$, I'm looking for a smooth projective hyperelliptic curve $C$ defined over $\mathbb{Q}$ whose Jacobian $J(C)$ has a subgroup isomorphic to $\mu_p^2$ as a $Gal(\overline{\...

**11**

votes

**0**answers

469 views

### Can an abelian variety/Q have no non-trivial points over Q_sol?

Let $A/\mathbb{Q}$ be an abelian variety. Must there be a finite solvable
extension $K/\mathbb{Q}$ such that $A(K)$ is nontrivial?
This follows from the conjecture that the maximal (pro-)solvable ...

**11**

votes

**0**answers

632 views

### What is miraculous about the mirabolic subgroup?

I recently asked this question about Euler subgroups and generalizing the automorphic theory of $\mathrm{GL}_n$ to a more general setting. My question here is more specific.
As mentioned there, the ...

**11**

votes

**0**answers

407 views

### Euler Subgroups and Automorphic L-functions

Recently, I have read about the Whittaker expansion for $\mathrm{GL}_n$ and was struck by the utility of the mirabolic subgroup, $\mathrm{P}_n\subset \mathrm{GL}_n$ of matrices with bottom row $(0\; 0 ...

**11**

votes

**0**answers

1k views

### The zeta function and classical mechanics

In this paper, Guilherme França and André LeClair show that $$\gamma_{y}\sim 2 \pi \left(y-11/8\right)/W\left((y-11/8)e^{-1}\right)$$ where $W$ is the Lambert W function, and $\gamma_{y}$ is the ...

**11**

votes

**0**answers

390 views

### How tight is the Weil bound for this exponential sum?

Let $f$ be a nonconstant polynomial of degree $d$. Let $\psi$ be a character of the additive group of $\mathbb F_p$. By Weil, we have:
$$ \left| \sum_{x\in \mathbb F_p} \psi ( f(x)) \right| \leq (d-1)...

**11**

votes

**0**answers

770 views

### Erdos multiplication problem revisited

The well-known problem is acquiring a cardinality of the set of distinct numbers in the multiplication table n x m.
The very problem has been discussed in-depth and, as such, I require no further ...

**11**

votes

**0**answers

252 views

### What is the lowest-weight non-cyclotomic Galois representation in $\overline{\mathcal M}_{g,n}$?

I want to know about low-weight Galois representations in $H^i(\overline{\mathcal M}_{g,n}, \overline{\mathbb Q}_\ell)$ that aren't cyclotomic. This should be equivalent to finding $p,q$ such that $H^{...

**11**

votes

**0**answers

408 views

### Power series defined by Witt vectors / Teichmüller representatives of p-adics

Let $K$ be $\mathbb{Q}_p$ for some prime $p$ (or more generally an unramified extension $W(\mathbb{F}_q)$ of $\mathbb{Q}_p$). If $\xi \in K$, we can write it in a unique way in the form $\sum a_i p^i$...

**11**

votes

**0**answers

933 views

### Are the twin primes the only positive double zeros of this real function?

Agno's answer
was extremely helpful.
For $x \in \mathbb{R}, x \ge 1$ define
$$ f(x) = \sin\frac{\pi(\Gamma(x)+1)}{\lfloor x \rfloor}$$
By Wilson's theorem the positive integer zeros of $f(x)$ are ...

**11**

votes

**0**answers

362 views

### Average ranks of abelian surfaces

Most people nowadays believe that over a fixed global field, $50$% of the elliptic curves have $0$ rank, $50$% have rank $1$, and $0$% have higher rank. A significant advance in this direction has ...

**11**

votes

**0**answers

719 views

### What are the limits of the Erdős-Rankin method for covering intervals by arithmetic progressions?

To construct gaps between primes which are marginally larger than average, Erdős and Rankin covered an interval $[1,y]$ with arithmetic progressions with prime differences. A nice short exposition is ...

**11**

votes

**0**answers

731 views

### What numbers are integrally represented by $4 x^2 + 2 x y + 7 y^2 - z^3$

This is related to my first MO question and Kevin Buzzard's conjecture at
Integers not represented by $ 2 x^2 + x y + 3 y^2 + z^3 - z $
In December 2010 my question appeared in the M.A.A. Monthly, ...

**11**

votes

**0**answers

350 views

### Galois invariants in a ring of fractional power series over a finite field

Let $\mathbf{F}_q$ be a finite field, and let $A=\mathbf{F}_q [[ x^{1/q^\infty} ]]$ be the completion of $\mathbf{F}_q[x^{1/q^\infty}]$ with respect to the $x$-adic topology. Then the $q$th power ...

**11**

votes

**0**answers

414 views

### Linear eta product identities - how many are there?

For the Dedekind eta function, defined as usual by $\eta(q) = q^{\frac1{24}} \prod\limits_{n=1}^{\infty} (1-q^{n}) $, let for brevity $e_k:=\eta(q^k)$. With this notation, a blog entry of Michael ...

**11**

votes

**0**answers

890 views

### Isogenies between supersingular elliptic curves

Suppose we are given two non-isomorphic supersingular elliptic curves $C$ and $C'$ (in characteristic $p$). Is there an isogeny $C\to C'$ of a given degree (say, power of a prime $l$ different from ...

**11**

votes

**0**answers

542 views

### Sums of four squares and the modular invariant

The question accounts my curiosity only, and may not be as deep as I think.
One of recent talks at our local seminar was devoted to the proof of the
classical formula
$$
F(q)=\sum_{n=0}^\infty r_4(n)...

**11**

votes

**0**answers

939 views

### Linear algebra of elliptic curves over p-adic fields

Unfortunately, the following question is somewhat ill-posed. However, I hope to make what I'm looking for sufficiently clear.
Given two elliptic curves (or Abelian varieties) over $\mathbb{C}$, one ...

**11**

votes

**0**answers

736 views

### Points of bounded height in a number field

Let $K$ be a number field of absolute degree $d$, let $B$ be a positive real number, and write $S(K, B) = \{x \in K : H(x) \leq B\}$. Here $H$ is the absolute multiplicative height of an algebraic ...

**11**

votes

**0**answers

344 views

### What are the endomorphisms of Drinfeld's “special formal O_D-modules”?

Let $F$ be a nonarchimedean local field, and let $D/F$ be the central division algebra of invariant $1/d$. Let $k$ be the algebraic closure of the residue field of $F$ and let $\pi$ be a uniformizer.
...

**11**

votes

**0**answers

729 views

### Why are Goldbach laggards biased towards $2 \mod 6$?

For even $n$,let $g(n)$ be the number of ways to write $n$ as a sum of two primes $n=p+q$ with $p \le q$. Define $a_k$ to be the largest $n$ with $g(n)=k$. I would bet money that no-one will disprove (...

**11**

votes

**0**answers

555 views

### What's known about the mod 2 reduction of the level l Jacobi modular equation?

Motivation:
Let $\ell$ be an odd prime. Let $A$ in ${\mathbb Z}/2[[x]]$ be $x+x^9+x^{25}+x^{49}+...$, and $B=A(x^\ell)$. One can use the level $\ell$ Jacobi modular equation to get a polynomial ...

**11**

votes

**0**answers

843 views

### On the relation of special values of motivic L functions and partial zetas

Let $K$ be a number field, $L$ a finite abelian extension and $\chi \in \widehat{Gal(L/K)}$ a (non-trivial) character. If we multiply out the associated Artin L-function $L(\chi,s)$ we can write this ...

**11**

votes

**0**answers

1k views

### Do the Standard Conjectures imply parts of the “Weil II” Riemann Hypothesis?

It is known that Grothendieck's Standard Conjectures on algebraic cycles imply the Riemann Hypothesis of the original Weil Conjectures. However, do they also say something about the version of the ...

**11**

votes

**0**answers

487 views

### Adeles of Holomorphic Functions

In number theory, an adele is a restricted product of elements of the completion at each prime. For function fields, we take (a kind of) product of the completion at each point, and at non-singular ...

**11**

votes

**0**answers

648 views

### A number theoretic identity

Let $n$ be a positive integer such that $2n+1$ is prime.
The elements of the factor group $G = \mathbb{F}^\times_{2n+1}/\{\pm 1\}$ can be represented by the integers $1,2,\ldots,n$. For every $x \in \...

**11**

votes

**0**answers

546 views

### Dihedral extensions and the Ankeny - Artin - Chowla conjecture

Jensen and Yui (Polynomials with Dp as Galois group
J. Number Theory 15, 347-375 (1982)) proved that if p = 4n+1
is a regular prime, then there is no normal extension of the
rationals with Galois ...

**11**

votes

**0**answers

719 views

### Lower bounds for linear forms of logarithms (a la Baker)?

Let $\lambda_1$, $\lambda_2$, and $a$ be three fixed complex algebraic
numbers.
For a given integer $n$, write
$\Theta(n) = \arg(a \lambda_1^n + \lambda_2^n)$.
Assuming $\Theta(n)$ is not zero, I am ...

**10**

votes

**0**answers

576 views

### Two questions around the $abc$-conjecture

Let $d(a,b) = 1-\frac{2 \gcd(a,b)}{a+b}$, $d_{ABC}(a,b) = 1-\frac{2\gcd(a,b)^3}{ab(a+b)}$ be two metrics on natural numbers.
The abc-conjecture can be formulated using these two metrics as:
For ...

**10**

votes

**0**answers

389 views

### Bounding $1/\zeta(s)$ given RH

Let $T\geq 0$. Assume RH(T+100), that is, assume that all non-trivial zeros $\rho$ of the Riemann zeta function with $|\Im(\rho)|\leq T+100$ satisfy $\Re(\rho)=1/2$. Can we then give a good upper ...

**10**

votes

**0**answers

171 views

### On the infinity of $\{p\in \mathbb {N}:\exists n\in\mathbb{N}~p| \left \lfloor{r^n}\right \rfloor\}$

I've already asked this same question on MSE here, but didn't get much help, so I will try on this site as well.
For which $r\in\mathbb{R}$ is the set $\mathscr{P}_r=\{p \in \mathbb{P}:\ (\exists n\...

**10**

votes

**0**answers

420 views

### Reverse Mathematics of Euclid's theorem

Euclid's theorem that there are infinitely many prime numbers has multiple proofs, ranging from Euclid's original theorem that constructs a new prime from a finite list of such, to Euler's proof that ...

**10**

votes

**0**answers

253 views

### Easy cases of Herbrand's theorem

$\def\QQ{\mathbb{Q}}\def\ZZ{\mathbb{Z}}$ I recall Herbrand's theorem about class groups of cyclotomic fields: Let $p$ be an odd prime, let $\zeta$ be a primitive $p$-th root of $1$ and let $K = \QQ(\...

**10**

votes

**0**answers

897 views

### Euler's totient function and Riemann hypothesis

I am looking for an upper-bound of the Euler's totient function $\varphi$ which would be equivalent to the Riemann hypothesis (RH). There is the following Nicolas' criterion about primorial numbers $...

**10**

votes

**0**answers

226 views

### Published reference on the automorphism group of modular curves $X_1(N)$?

I wish to cite that the automorphism groups of $X_1(N)$ have already been completely calculated, and what they are, but I am having difficulty finding this calculation in the literature.
I have ...

**10**

votes

**0**answers

223 views

### Recognizing the Galois group from the field discriminant

Along the lines of the general question "How much does the discriminant of a number field reveal about the field?", I was wondering how often it happens that the discriminant of some number field ...

**10**

votes

**0**answers

269 views

### The trace formula over function fields

There are many examples in number theory where an "arithmetic" problem (i.e. for number fields) has an easier analogue for function fields over finite fields. This is also true for questions ...

**10**

votes

**0**answers

357 views

### Is every integer a difference of two powers?

True or false? (I don't know.) Every positive integer is the difference of two powers. Examples:
$ 1 = 3^2 - 2^3 $
$ 2 = 3^3 - 5^2 $
$ 3 = 2^7-5^3 $
$ 4 = 2^3-2^2 = 5^3-11^2 $
$ 5 = 2^5 - 3^3 $
...

**10**

votes

**0**answers

258 views

### Non-vanishing modular forms

Prompted by this MO question, I have the following question about modular forms which do not vanish on the upper-half plane.
Q1. Let $N \geq 1$ be an integer and let $\Gamma(N)$ be the principal ...

**10**

votes

**0**answers

185 views

### Product of four consecutive primes plus $1$ equals square

Some days ago, I noticed that $3\cdot 5\cdot 7\cdot 11 +1=34^2$.
I am almost sure that if we denote four consecutive primes by $p, q, r, s$ then the equation
$$p\cdot q\cdot r\cdot s+1=x^2 \quad ...

**10**

votes

**0**answers

787 views

### Where stands functoriality in 2017?

In 2002, R. Langlands put forward a new strategy to prove the general functoriality conjecture in the Beyond endoscopy paper. The main purpose of this strategy is to detecting the automorphic ...

**10**

votes

**0**answers

160 views

### Explicit L-factor for supercuspidals

I am interested in L-functions for the quasi-split unitary group $U$ in three variables, following the construction by zeta integrals of Gelbart-Piatetski-Shapiro. My aim is to understand the ...

**10**

votes

**0**answers

798 views

### Nick Katz observation: “the rationality of the zeta function!”

In the proceedings "Algebraic Geometry - Arcata 1974" edited by R. Hartshorne there is an article by Nick Katz called "$p$-adic $L$-functions via moduli of elliptic curves". He starts by recalling $p$-...

**10**

votes

**0**answers

203 views

### Sums of areas of unit pythagorean triangles

Can the sum of the area of two rational unit (as in length of hypotenuse) pythagorean triangles be the area of another rational unit pythagorean triangle? It is known that the pythagorean triangles ...

**10**

votes

**0**answers

265 views

### Curious properties of 79 and 859 in sums of two Fibonacci numbers

Let $\mathrm{Fib}_2$ be the set of sums of two Fibonacci numbers:
$$\mathrm{Fib}_2 = \{ 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 18, 21, 22, 23, 24, 26, \dots \}. $$
The elements in $\mathrm{...

**10**

votes

**0**answers

195 views

### Galois action on units of totally real cyclic numberfields

Given a cyclic totally real number field $K/\mathbb{Q}$ of degree n with unit group isomorphic to $\mathbb{Z}/2 \times \mathbb{Z}^{n-1}$, how much is known about the action of Gal$(K/\mathbb{Q})$ on ...

**10**

votes

**0**answers

359 views

### horocycle flow and the prime number theorem

Looking at Zagier's Eisenstein Series and the Riemann Zeta Function, we get a proof of the prime number theorem using horocycles. I would really love it if there were a geometric proof like this.
...

**10**

votes

**0**answers

254 views

### The non-abelian Gauss-Manin connection; non-abelian M_dR; a Grothendieck lemma for cyrstals

I'm interested in understanding the non-abelian Gauss-Manin connection on Carlos Simpson's relative de Rham Moduli space $M_{dR}(X/S,n)$ for a smooth projective morphism of schemes $X/S$. The scheme $...