All Questions
Tagged with reference-request nt.number-theory
1,409 questions
3
votes
0
answers
680
views
Birch/Swinnerton-Dyer "Notes on Elliptic Curves II"
I would like to know if any of you know if there is a more general treatment to what Birch and Swinnerton-Dyer did in "Notes on Elliptic Curves II" (http://www.ams.org/mathscinet-getitem?mr=179168).
...
2
votes
1
answer
543
views
main conjecture of Iwasawa theory implies Herbrand-Ribet
I need a reference/proof for "main conjecture of Iwasawa theory => refinement of Herbrand-Ribet ($v_p(B_{p-i}) = v_p(|A_i|)$, where $A_i$ denotes the $i$-th eigenspace of the Galois group acting on ...
3
votes
0
answers
293
views
multiplicity of automorphic representation of unitary similitude group
Let $G$ be a unitary similitude group over $\mathbb{Q}$ (as in the book of Harris-Taylor), $\pi$ an irreducible automorphic representation of $G(\mathbb{A})$. I'm looking for some results on its ...
8
votes
0
answers
335
views
Irreducibility of Galois representations attached to unitary groups
If $G$ is a unitary group in $n$ variables over $\mathbb Q$, attached to an hermitian form for an imaginary quadratic extension $E/\mathbb Q$ and if we suppose that the hermitian form is definite over ...
2
votes
1
answer
240
views
Congruence for the Apery Numbers
Is it true that $$A_n\equiv (-1)^n\;\;(\mathrm{mod}\;3)\;\;?$$
Here $A_n$ is the Apery number:
$$A_n=\sum\limits_{k=0}^n\binom{n}{k}^2\binom{n+k}{k}^2.$$
What is known about congruence properties ...
1
vote
0
answers
285
views
Davenport's proof that almost all integers are the sum of 4 cubes
Where can I find a pdf that describes Davenport's proof that almost all integers are the sum of $4$ cubes?
1
vote
0
answers
70
views
A non-surjective coboundary map induced by a central extension
Let $k$ be a number field and
$$ 1\to A \to B \to C \to 1$$ be a central extension of finite groups over $\mathcal{O}_k$ (the ring of integers of $k$), with $B$ non-commutative. Consider the induced ...
1
vote
0
answers
331
views
Where can I find the article of A. Borel: "Values of zeta-functions at integers, cohomology and polylogarithms"? [closed]
Where on the internet can I find this article?
I know that it is in this book: Current trends in mathematics and physics, Narosa, New Delhi, 1995.
7
votes
2
answers
522
views
How large (small) can be the measure of a set where a polynomial takes small values ?
A $n$-th degree polynomial has precisely $n$ roots. So it is natural to ask the question
how large ( and small) can be the measure of a set where a polynomial takes small values ?
This, and other ...
2
votes
0
answers
159
views
The dimension of the space of automorphic forms with multiplier system
Let $\Gamma$ be a discrete subgroup of $SL_{2}(\mathbb{Z})$ and $\vartheta$ a multiplier system of weight $k$ for $\Gamma$, by which we mean a function $\vartheta:\Gamma \rightarrow \mathbb{C}$ ...
1
vote
0
answers
122
views
Statements generalizing representability of integers by binary quadratic forms to $n$-variable higher homogeneous forms?
Representing integers through the theory of binary quadratic forms is a well studied topic. We know that given $a,b,c\in\Bbb N$, based on discrimant $b^2-4ac$, we can study the representability of ...
11
votes
3
answers
461
views
citation for first statement of the Re(s) = 6 conjecture on zeros of Ramanujan L function
Hi, for the bibliography of a paper I'm writing I seek a citation for the first statement of the conjecture that the nontrivial zeros of $F(s) = \sum_n\tau(n)n^{-s}$ all lie on the line Re(s) = 6. (...
2
votes
0
answers
179
views
Questions about transformation or integral transformation
I have asked several mathematicians about the following questions,but all of them think they are good questions,but can not give a complete answer.Now I have to come here to ask mathematicians all ...
2
votes
1
answer
164
views
A problem on the finiteness of solutions to a Diophantine equations
Given two positive integers $a,b$, and an odd prime $p$, I want to know whether the number of solutions to the following equation is finite:
$X^2=a+bp^{Y}$
where $X,Y$ are variables and are integers....
0
votes
1
answer
202
views
Reference request: on sums of the form $ax^m + by^n = h$
I know that equations of the form
$$\displaystyle ax^d + by^d = h$$
with $a,b,h \in \mathbb{Z}$ have been thoroughly investigated as a special (and interesting) case of the Thue-Mahler equation, for ...
1
vote
0
answers
165
views
Square-free sieve over number fields
I am trying to work on extending various methods to study square-free values of polynomials (or more generally, $k$-free values) over general rings of integers, and a literature review has yielded ...
3
votes
1
answer
528
views
Karolyi's theorem for finite groups and its extensions
Suppose that $\mathbb A = (A, +)$ is a (possibly non-commutative) group, and denote by $p(\mathbb A)$ the minimum of $|S|$ as $S$ ranges in the set of non-trivial subgroups of $\mathbb A$, with the ...
6
votes
0
answers
252
views
How big is the Fourier transform of the log of a polynomial over the p-adic numbers
Let $f(z_1,\dots,z_n)$ be a polynomial with $p$-adic coefficients, and let $g(z):=log\lvert f(z) \rvert$. If $\xi$ is a complex character of $\mathbb{Z}_p^n$ there exists a number $v=v(\xi)$ such that ...
2
votes
0
answers
114
views
Distribution of residue classes of totients of (univariate) polynomials
Let $\phi$ denote Euler's totient function and $f$ a non-constant univariate polynomial with integer coefficients such that $f(u) \in \mathbf N^+$ for all $u \in \mathbf N^+$ (assume $f$ is ...
8
votes
0
answers
595
views
A property of supersingular $j$-invariants (reference request)
Edit 2: For those who understandably don't want to read such a long post, I think Voloch's suggestion reduces the problem to asking whether $j$-invariants of supersingular curves are 3rd powers in $\...
7
votes
0
answers
695
views
modularity of elliptic curves with cm
I'd like to ask for references on the status of modularity results for elliptic curves with CM which are not necessarily defined over $\mathbb Q$. In the case of an elliptic curve with CM defined over ...
3
votes
0
answers
133
views
Reference for a lemma on the asymptotic upper density of special sets with large gaps and intervals
Update. Based on Anthony Quas' comment below, the proof can be made sensibly shorter and the lemma can be slightly generalized by weakening the old assumption (iii).
In a joint paper that I am ...
2
votes
0
answers
127
views
Nonnegative integers represented by $\prod_{i=1}^m \sum_{j=1}^n a_{i,j} x_j $, where the $a_{i,j}$ are positive integers
Fix $m, n \in \mathbf N^+$ with $m+n \ge 3$, and let $A = (a_{i,j})_{1 \le i \le m, 1 \le j \le n}$ be an $m$-by-$n$ matrix of positive integers. What is known about the asymptotic behavior of the ...
1
vote
1
answer
435
views
Elliptic subfields of a function field
Let $C$ be a curve and $K(C)$ be its function field of genus 2, where $K$ = $\mathbb{C}$.
The number of essential elliptic subfields of $K(C)$ is 0 or 2 or $\infty$.
Edit: I am looking for a proof. ...
0
votes
1
answer
510
views
Erdős-Straus with 4 terms
The Erdős-Straus conjecture states that any fraction of the form $\frac{4}{n}$ can be decomposed as an Egyptian fraction with just 3 terms. In related research, I've recently come across conditions on ...
2
votes
1
answer
569
views
Generalization of singular moduli
$j$-invariants of CM curves $E$ over (say) the complex numbers are known as singular moduli. As the theory of complex multiplication explains, singular moduli are algebraic integers of great ...
3
votes
3
answers
285
views
Limit connected with a periodic function
I am posting the following question from Math.Stackexchange:
Let $f$ be a $1$-periodic function, i.e., $f(x+1)=f(x)$, defined on the interval $(0, 1)$ by the formula
$$
f(x)=2x-1.
$$
For a real ...
0
votes
0
answers
121
views
Bound of Chebyshev function and zeros of zeta function
It is an elementary argument (such as in Multiplicative Number Theory, section 18) that, if the Chebyshev's function $f(x) = \sum_{n \le x} \Lambda(x) = x + O(x^\alpha)$ for some $\alpha < 1$, then ...
0
votes
1
answer
939
views
Asymptotic equivalence for functions with zeros
I am considering the relative asymptotic behavior of a pair of real functions on the positive real axis, say $f$ and $g$. There is no $x_0$ such that $f$ and $g$ are non-zero for all $x>x_0$.
...
18
votes
0
answers
718
views
Erdos-Kac for squarefree numbers
In its usual form, the Erdos-Kac Theorem states that if $f(n) : \mathbb{N} \rightarrow \mathbb{R}$ is a strongly additive function with $|f(p)| \le 1$ for all primes $p$, then
$$\frac{|\{n \le x : \...
2
votes
0
answers
227
views
Kloosterman-like sum with inverse to different moduli
In some recent work, the following strange-looking exponential sum arose:
$$
\sum_k \sum_r \sum_s e\bigg( \frac{r \bar s^{(r)} \bar k^{(r^2+s^2)}}{r^2+s^2} \bigg).
$$
Here $e(x) = e^{2\pi i x}$ as ...
3
votes
3
answers
670
views
Proof of infinitude of primes whose reversal in base 10 is also prime [closed]
Is there any proof of infinitude of A007500 primes?
If you want to generate them here is trivial and naive python program.
...
2
votes
2
answers
491
views
Summation of certain series
Suppose $f(n)$ is a periodic function with period $q$. Now from this paper we get that if $\displaystyle\sum_{n=1}^{q}f(n)=0$ then $\displaystyle\sum_{n=1}^{\infty}\frac{f(n)}{n}=-\frac{1}{q}\...
5
votes
0
answers
79
views
Some questions about the Lévy monoid of certain densities
Let $\bf H$ be a set, $f: \mathcal P({\bf H}) \rightharpoonup \bf R$ a partial function, and $\mathcal{D}$ the domain of $f$.
Next, denote by $\mathcal M(f)$ the set of all (total) functions $\theta: ...
1
vote
1
answer
1k
views
Lacunary sequence
Is there a standard definition for a lacunary sequence?
Suppose $0 < a_1 < a_2 < \cdots.$
I've read two papers using the term recently. One requires
$$
\liminf_n\frac{a_{n+1}}{a_n}>1
$$
...
4
votes
1
answer
928
views
Primes and Ackermann's function
If $A(m,n)$ is Ackermann's function, and $c$ is a fixed integer, are there any heuristics/conjectures/obvious things that can be said about primes of the form $A(m,n)+c$, $m \geq 4$,at all?
EDIT:
I ...
6
votes
2
answers
976
views
References for modular polynomials
I am teaching a graduate "classical" course on modular forms. I try to achieve the most elementary level for presenting modular polynomials. Serge Lang's "Elliptic functions" cover the topic quite ...
3
votes
0
answers
168
views
Invariant Theory over finite adeles
Classical invariant theory, among the other things, classifies polynomial functions over a vector space $V$ endowed with a quadratic form $Q$ which are invariant under the action of $SO(V,Q)$.
I am ...
3
votes
1
answer
151
views
Definability of orderings on a formally real number field
For vector basis $b_1,..,b_n$ on a finite extension $F$ of $\mathbb{Q}$, where $-1$ is not a sum of squares, each linear order on $F$ is determined by an order on the basis. This uses information ...
4
votes
0
answers
413
views
Maximal order of Hooley's Delta function?
There is a large literature on Hooley's
$$
\Delta(n)=\max_u\sum_{d|n,\ e^u\le d< e^{u+1}}1
$$
giving its normal and average order. What is known of its maximal order?
Clearly $\Delta(n)\le d(n)$ ...
1
vote
1
answer
446
views
Existence of Solutions to an Equation Involving the Sum-of-Divisors Function [Reference Request]
Let $\sigma(x) = \sigma_1(x)$ denote the sum of all the positive divisors of $x$.
If $n \in \mathbb{N}$ is odd and $\gcd(n, \sigma(n)) = 1$, then do there exist any solutions to the following ...
3
votes
0
answers
163
views
Oscillatory integral moments of $L(\frac{1}{2} + it, f \times f)$
Understanding moments and subconvexity bounds for $L$-functions is a big topic with a lot of activity. I'm currently looking at a related problem, bounding
$$
\int_0^T L\left(\tfrac{1}{2} + it, f \...
2
votes
1
answer
596
views
Principal term of the Dirichlet Divisor problem, from the work of A.F. Lavrik?
Ivic writes, at the beginning of chapter 13 of his The Riemann Zeta Function, about a method of expressing the principal terms of the Dirichlet Divisor Problem as polynomials of $log\\ n $ with ...
2
votes
0
answers
99
views
Does there exist $k\ge2$ s.t. $X \subseteq \mathbf N^+$ has positive upper Banach density if the counting function of $X$ is $\gg n/\log^{[k]}(n)$?
Does there exist an integer $k \ge 1$ such that ${\sf bd}^\ast(X) > 0$ whenever $X \subseteq \mathbf N^+$ and $\pi_X(n) \gg \frac{n}{\log^{[k]}(n)}$ as $n \to \infty$? Here, ${\sf bd}^\ast$ is the ...
3
votes
0
answers
102
views
Localized at $p$ integral representations of finite elementary $p$-groups
Let $C_p$ be a cyclic group of prime order $p$.
Let $F=C_p^n=C_p\times\dots\times C_p$ ($n$ times).
I would like to to classify finite dimensional representations of $F$ over ${\mathbb{Z}}$.
However, ...
4
votes
0
answers
117
views
Best constant for Maier's theorem?
Maier proved that, for fixed $\lambda>1,$
$$
\limsup_{x\to\infty}\frac{\pi(x+\log^\lambda x)-\pi(x)}{\log^{\lambda-1}x}>1
$$
and in particular
$$
\limsup_{x\to\infty}\frac{\pi(x+\log^\lambda x)-\...
2
votes
0
answers
207
views
n-ary quadratic forms with $S$-integer values
Let $Q(x_1,\ldots,x_n):=x_1^2+\cdots+x_n^2$ be an $n$-ary quadratic form.
Given a finite set of (rational) primes $S$ is there an algorithm or theorem that describes all solutions to $Q(x_1,\ldots,...
3
votes
0
answers
158
views
Conics over number fields
I am looking for a reference for the following fact.
Let $k$ be a number field and let $S$ be a finite set of places of $k$ of even cardinality. Then there exists a unique conic $C$ over $k$ such ...
1
vote
1
answer
1k
views
Good Minkowski Theory and Commutative Algebra Books
I am not so familiar with the theory of measures which Andre Weil uses to develope the Class Field Theory.
However, I am interested in learning algebraic number theory and I recently found that the ...
15
votes
0
answers
591
views
For how many primes does an elliptic curve over a totally imaginary field have supersingular reduction?
An elliptic curve over a finite field, $k$, of characteristic p is called supersingular if it has no $p$-torsion over $k^{\mathrm{alg}}$, or equivalently, if $\mathrm{End}(E)$ is an order in a ...