All Questions
Tagged with reference-request nt.number-theory
1,408 questions
6
votes
1
answer
497
views
Half integral weight Hecke operators
I would like to find a source giving the exact formula for the product of two Hecke operators $T_{\kappa}(n^2)$ and $T_\kappa(m^2)$ of half integral weight. That is, $\kappa \in \frac 12 \mathbb{Z} - \...
5
votes
2
answers
1k
views
Request: Kato's article "Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions." Part II
The question (similar to MO.96531) is about the article by Professor Kazuya Kato in this book.
In this article, Professor Kato indicates the contents of the second part.
MathSciNet does not list it, ...
- 3,867
10
votes
0
answers
323
views
The mod 3 reduction of some powers of delta
Let f in Z/3[[x]] be the mod 3 reduction of the Fourier expansion of the normalized weight 12 cusp form delta for the full modular group. The exponents appearing in f are all 1 mod 3. Fix
k>0 and ...
- 5,422
4
votes
2
answers
655
views
Intersection of Hilbert class fields of imaginary quadratic fields
In this question Hilbert class field of Quadratic fields it is mentioned that if $d\equiv 1 \mod 4$ then the Hilbert class field of $\mathbb{Q}(\sqrt{-d})$ contains $\mathbb{Q}(i,\sqrt{d})$.
Could ...
- 1,905
2
votes
2
answers
1k
views
Place stabilizers for the absolute Galois Group
Fix an algebraic closure, $\overline{\mathbb{Q}}$ for the rationals and consider the set, $B_p$, of all places of $\overline{\mathbb{Q}}$ over a fixed (possibly infinite) prime, $p$, of $\mathbb{Q}$. ...
- 1,049
5
votes
1
answer
455
views
Large gaps between P2s
Gaps between consecutive primes are $O(n^{\theta+\varepsilon})$ for $\theta=0.525$ and any $\varepsilon>0.$ I was wondering if a better result is known for gaps between numbers with at most two ...
- 9,114
11
votes
1
answer
2k
views
The Class Number One Problem for Real Quadratic Fields
An approach to the Gauß class number one problem for imaginary quadratic fields is to determine the integral points on the modular curve $Y_{nonsplit}(n)$ for a suitable $n$. Here follows a quick ...
10
votes
3
answers
638
views
Last term of repeating continued fraction expansion
Once again, working with stable vector bundles on $\mathbb{P}^2$ I have run into a question that is really out of my area. (Thanks to everybody who helped out with my last question!)
Let $D>9$ be ...
- 5,857
6
votes
1
answer
1k
views
Must the $j$-invariant of an elliptic curve with an isogeny be integral?
Let $K$ be a quadratic field, and $E/K$ a non-CM elliptic curve with a $K$-rational $p$-isogeny, for $p$ a prime. I would like to say the following:
For large enough $p$, the $j$-invariant $j(E)$ ...
- 2,827
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 ...
- 2,235
11
votes
1
answer
1k
views
Extending an assignment property from Q to R (or C)
Property of any odd number of nonnegative integers:
Given $x_1 \leq \cdots \leq x_{2n + 1}$ with each $x_i \in \mathbb{Z}_{\geq 0}$, suppose that for any $x_i$ we remove, the remaining numbers can be ...
- 7,831
4
votes
2
answers
506
views
Empty lattice simplex or White's theorem
White has proved (White, G. K. Lattice tetrahedra -- Canad. J. Math. 16 1964 389–396.) the following theorem:
If $T$ is a closed tetrahedron and $\Lambda$ is a lattice which contains the vertices of $...
- 12.3k
4
votes
1
answer
505
views
Number of divisors of a sum / ABC conjecture equivalent statement
I have two related questions that I can't seem to find any literature on:
1) What can be said about $\tau(a+b)$ knowing $\tau(a)$ and $\tau(b)$ (where $\tau(n)$ is the number of positive divisors of $...
- 2,235
1
vote
0
answers
162
views
Construction of RM abelian variety from eigenform
Let $f$ be a normalized eigenform of weight $2$ level $N$. If the Fourier coefficients of $f$ generate a totally real field $F$, then we associate to $f$ a system of $\ell$-adic Galois representations ...
- 15.4k
6
votes
2
answers
381
views
Lattice-cube minimal blocking sets
Let $C_d(n)$ be the lattice cube consisting of the $n^d$ points with
each of its $d$ coorindates in $\lbrace 1,2,\ldots,n \rbrace$.
Define a blocking set for a lattice cube to be a set of points
in ...
- 151k
9
votes
1
answer
1k
views
Modern Proof of the Theorem of the Base
I am looking for a modern proof of the so-called "Theorem of the Base"--that the Neron-Severi rank of a smooth projective variety is finite. One can prove this for varieties over $\mathbb{C}$ easily ...
- 23k
9
votes
2
answers
584
views
Number Fields Arising from Newforms
It is well-known that, given a normalized eigenform $f=\sum a_n q^n$, its coefficients $a_n$ generate a number field $K_f$.
In their 1995 paper "Fermat's Last Theorem", Darmon, Diamond, and Taylor ...
- 1,422
6
votes
3
answers
966
views
congruences for Fourier coefficients of modular forms
Are there other good articles on congruences for Fourier coefficients of modular forms beside Swinnerton-Dyer's article in "Modular Functions of One Variable III"?
I am looking for generalisations ...
user19475
7
votes
1
answer
313
views
Prescribed values for the uniform density
Strauch & Tóth [1] Georges Grekos [3][4] showed that for any choice of upper and lower density, there is some subset of $\mathbb{N}$ with the chosen densities, provided the lower is no more than ...
- 9,114
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
$$
\liminf_n\frac{a_{n+1}}{a_n}>1
$$
...
- 9,114
4
votes
1
answer
288
views
Reference request for an identity for tangent numbers
The tangent numbers $(T_{2n+1})=(1,2,16,272,7936,...)$ (cf. OEIS: A000182) satisfy many recurrences. I would be interested to find references for the following which I think must be very old:
$T_3 -...
- 5,622
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 ...
- 217
7
votes
2
answers
1k
views
questions on Néron-Tate canonical height
I have three questions regarding height pairings:
In [Serre, Lectures on the Mordell-Weil theorem], p. 85 f., it is stated that the following function is a local height function:
"Let $V/R$ be a ...
user19475
15
votes
1
answer
954
views
Funktorialität in der Theorie der automorphen Formen
In 2010 Langlands wrote an article with the title Funktorialität in der Theorie der automorphen Formen: Ihre Entdeckung und ihre Ziele. On the IAS website, he says that
This note ... was written ...
- 18.7k
5
votes
3
answers
3k
views
Effective way of finding generators on the curve and the rank conjecture
Hello everyone,
I have never heard of a polynomial time running algorithm that finds the generators of elliptic curves efficiently. I do know that Nagell-Lutz theorem is useful in computing the ...
-2
votes
1
answer
1k
views
Why should I believe in the Siegel's and Hasse's rationale ?
Hello everyone,
I was deeply attracted by the Hasse and Siegel's theorems while studying $p$-adic analysis. While reading a paper B.J. Birch and H.P.F. Swinnerton-Dyer - Notes on elliptic curves. I, ...
5
votes
1
answer
374
views
Where can I read about exponential sums corresponding to Jones Polynomial?
I remember reading that a number theoretic analogue of Witten's path integral formula for the Jones polynomial:
$$\text{Jones}_K(e^{2\pi i/(k+2)})=\int_{\text{$SU(2)$ connections on $\mathbb S^3$}/\...
- 18.7k
14
votes
4
answers
3k
views
Fourier decay rate of Cantor measures
For $0<\theta<\frac{1}{2}$, denote by $C_\theta$ the Cantor set with dissection ratio $\theta$, i.e. the Cantor set obtained from dissection parttern $(\theta, 1-2\theta,\theta)$. It is known ...
- 981
6
votes
2
answers
861
views
Number of integers coprime to l
A long time ago I've seen a paper considering, given $\ell$ fixed, estimates for
$$
\sum_{n \leq x, (n, \ell) = 1} 1
$$
Of course, this is easy to estimate with a trivial error term of $O(\varphi(l))...
- 293
4
votes
1
answer
338
views
Reference for Rank Distribution Conjecture.
I am currently writing my master's thesis and I was wondering if the rank distribution conjecture was ever formally written down. Recall that it says that:
Half of all elliptic curves have rank $0$, ...
- 1,458
22
votes
3
answers
7k
views
A recommended roadmap to Fermat's Last Theorem
I was inspired to undertake math as a career after watching a documentary on the proof of Fermat's Last Theorem. As such it's been a small goal of mine to understand Wiles et al's proof.
In a ...
- 1,458
3
votes
1
answer
318
views
Minor Arc Estimates for an Exponential Sum for a Quadratic Polynomial Over the Primes
Let $f$ be a quadratic polynomial with leading coefficient $\alpha$, and suppose $\alpha$ is in a "minor arc" in the sense that $\alpha$ is not within $\frac{(\log N)^A}{q N^2}$ of any rational number ...
- 33
4
votes
5
answers
688
views
Nonlinear equations in integers
Linear patterns in subset of the integers (for example, primes) such as arithmetical progressions is a hot topic in mathematics. Recently, much progress has been made in this area. For example,
the ...
- 71
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$.
...
- 2,480
2
votes
3
answers
339
views
Gauss sums over multiplicative subgroups
Hello,
Is anyone here aware of a well-motivated exposition of the Bourgain-Glibichuk-Konyagin estimate for exponential sums (or Gauss sums) over multiplicative subgroups? If any of you has a write-up ...
- 8,842
13
votes
1
answer
1k
views
Reference for: CM Hilbert Modular forms arise from Hecke characters
For classical modular forms, the correspondence between the form having CM by an imaginary quadratic field $K$ and it being induced from a Hecke character on $K$ is well-known. (Ribet's paper is a ...
- 659
34
votes
2
answers
3k
views
Shimura-Taniyama-Weil VS Grothendieck's dessins
When listening to the beautiful lectures by Gilles Schaeffer at
the SLC68, the following (perhaps crazy) question occurred to me:
did anyone attempt (succeed?) to combinatorially prove modularity of ...
7
votes
2
answers
2k
views
Tamagawa Number of Elliptic Curves over $\mathbb{Q}$
I am currently reading a paper by De Weger and one theorem in it proves a bound for the Tamagawa number of any elliptic curve defined over $\mathbb{Q}$.
I was wondering if anyone has any good ...
- 1,458
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. ...
- 471
11
votes
1
answer
1k
views
Finiteness of Tate-Shafarevich
Does anyone happen to know who conjectured the finiteness of the Tate-Shafarevich group?
We recall the conjecture. Let $E/K$ be an elliptic curve where $K$ is a number field. Then $Ш(E/K)$ is finite.
- 1,458
7
votes
3
answers
510
views
Proto-Euclidean algorithm
Consider the Euclidean algorithm (EA) as a way to measure the relative length $b/a$ of a shorter stick $b$ compared to a longer one $a$ by recursively determining
$$q_i = \left\lfloor \frac{r_i}{r_{...
- 9,720
26
votes
3
answers
5k
views
Questions about the Bernstein center of a $p$-adic reductive group
Dear all,
The "Bernstein center" of a $p$-adic reductive group appears frequently in the literature of automorphic forms, often without a precise definition. For example, in page 233 of Moeglin-...
- 809
5
votes
3
answers
881
views
A question about partial Euler products
Let $K/{\mathbb Q}$ be an extension of degree $d$. Let $S$ be the set of primes $p$ which split completely in $K$. What can one say about the analytic properties of
$$
\zeta_{K, S}(s) : = \prod_{p \...
- 1,362
6
votes
3
answers
555
views
Source for embedding multiplicative group of an algebraic closure of a finite field?
It's easy to embed the (cyclic) multiplicative group of a finite field into the multiplicative group of $\mathbb{C}$ (or other algebraically closed field of characteristic 0): assign to a generator of ...
- 52.9k
5
votes
1
answer
655
views
A theorem of Stickelberger on the number of prime ideals in a decomposition
Suppose that $p$ is unramified in a number field $K$ of degree $n$. Apparently, Stickelberger proved that $\big( \frac{Disc(K)}{p}\big) = (-1)^{n - g}$, where $g$ is the number of prime ideal factors ...
- 7,347
8
votes
1
answer
1k
views
Generalization of Hilbert 94 and capitulation
Let $L/K$ be a finite, cyclic extension of number fields, say with $\mathrm{Gal}(L/K)=G$. In my context $G$ is actually of order $p$, an odd prime number, but let me state my question for every cyclic ...
- 5,670
7
votes
2
answers
785
views
Inverse map for partition transform
Let $(a_n)$, $n\in\mathbb{N}$, be a sequence of complex numbers, then formally one has
(1)
$$\prod_{1}^{\infty}\left(1-a_nx^n\right)^{-1}=1+\sum_{1}^{\infty}\left(\sum_{j_1+2j_2+\cdots +nj_n=n}a_1^{...
- 2,480
7
votes
2
answers
570
views
For an approach to the Hadamard-matrix-problem: is there a proof, that the iterative plane-wise orthogonal rotations (Quartimax/Varimax) converge to global maximum?
I've asked this question at stat-exchange and at the "Semnet"-mailing list of professionals in statistics. The reference to some articles in Psychometrica (for instance ten Berge 1995, Jennrich 2001) ...
- 5,342
5
votes
2
answers
471
views
Kronecker theorems on linear forms.
Dickson's History of the Theory of Numbers, vol II p. 94 refers to some theorems of Kronecker on linear forms:
...find integers $w$ and $w^\prime$
such that $aw+a^\prime w^\prime$ takes
a value ...
- 11.1k
6
votes
1
answer
1k
views
Reference request: Dickman, On the frequency of numbers containing prime factors
I've been trying without success to find the paper
Dickman, Karl, "On the frequency of numbers containing prime factors of a certain relative magnitude." Ark Mal., Astronomi och Physik, 22A (10), ...
- 1,077