All Questions
Tagged with reference-request nt.number-theory
1,408 questions
1
vote
1
answer
250
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Link between integral points on varieties and solutions to Diophantine equations
Let $k$ be a number field, $S$ a finite set of places of $k$ including the infinite ones and $F(X_1,\dots,X_n)$ a polynomial in $k[X_1,\dots,X_n]$.
I am looking for notes, books or surveys detailing ...
2
votes
0
answers
245
views
Help for reference of moduli stack of fake elliptic curves
I see everywhere the following:
Let $B$ be an indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $D$, $\mathcal{O}_B$ be a maximal order, $N$ be an positive integer coprime to $D$.
...
- 21
7
votes
1
answer
644
views
(Finite) Models of two subtheories of Peano Arithmetic
Consider first-order theory (with identity) of Peano Artithmetic built in the language $\{S,+,\times,0\}$ and with the following set of axioms:
\begin{align}
\neg Sx&=0\tag{1}\\\
Sx=Sy&\...
- 1,112
5
votes
1
answer
455
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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
5
votes
0
answers
359
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Examples of Rankin-Selberg L-functions from Eisenstein series
I've been digging for awhile to not much success, so I figure I would try here:
I am looking for some references which compute explicitly examples of Rankin-Selberg L-functions from the constant ...
- 3,799
9
votes
4
answers
1k
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cohomology of moduli spaces
Does anyone know if there's any reference on the $\ell$-adic cohomology of some simple moduli spaces/Shimura varieties, like Siegel moduli varieties $A_{g,N}$ of genus $g$ and level $N,$ for small $g$ ...
- 4,265
6
votes
2
answers
861
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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
6
votes
1
answer
192
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Monte-Carlo computation of the Smith normal form
Quite some time ago I saw an article where a Monte-Carlo algorithm for computing the Smith normal form of an integer matrix was described. In this article the following problem was posed:
Suppose $P, ...
6
votes
0
answers
332
views
Criteria for irreducibility using the location of complex roots
I would like to see criteria for the irreducibility of a polynomial over $\mathbb{Z}$ based (mainly) on the location of the roots of the polynomial in the complex plane. An example of such a criterion ...
- 11.3k
7
votes
2
answers
2k
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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
2
votes
0
answers
104
views
A question on exponential increasing sequences of natural numbers
Let $(a_i)$ be an exponential increasing sequences of natural numbers; there are constants $a\in(1,2)$, $b>0$ such that $|a_{i}-ba^i|$ is exponentially decreasing. Let $(s_{k})$ be a sequences in $\...
- 1,648
1
vote
1
answer
231
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If $q^k n^2$ is an odd perfect number with Euler prime $q$, are the following statements known to hold in general? [closed]
(Note: This has been cross-posted to MSE. However, I feel that it is more likely to receive a good answer here, because I believe that it is a research-level question. For the mathematicians who ...
13
votes
2
answers
880
views
Arithmetic progressions modulo $p$ under the squaring map
I feel that the following problem should be known, but I'm not sure where to look for it.
Fix a real constant $\frac{1}{2} \ge \epsilon > 0$. For varying primes $p$, Let $A_p$ denote the set of ...
user631
16
votes
0
answers
11k
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Deligne's letter to Jean-Pierre Serre
I'm looking for another letter of Pierre Deligne, this time to Jean-Pierre Serre (from around 1974 I think), in which he proves that the Galois representation associated to a certain Hecke eigenform ...
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} - \...
2
votes
0
answers
487
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On Descartes / spoof odd perfect numbers
Descartes, Frenicle, and subsequently Sorli, conjectured that $k = 1$, if $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\...
17
votes
1
answer
1k
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Serre and Tate's conjectures on étale cohomology
In the appendix of Serre and Tate "Good Reduction of Abelian Varieties" [Annals of Mathematics 88 (1968), 492-517], the authors make the following conjectures.
Suppose that $X$ is a smooth proper ...
- 1,832
2
votes
0
answers
564
views
Sets of coprime numbers
Consider the set $\{0, 3, 7, 15\}$ of four integers. If you add each of these numbers to a fixed power of 2, then the resulting four numbers are pairwise coprime. For example, $\{4, 7, 11, 19\}$ are ...
- 595
7
votes
1
answer
313
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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
2
votes
1
answer
364
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A question on the bounds of the $n$-th composite $c_n$
While trying to prove the inequality $$c_{p_n-m}+c_{m-n}>p_n+2$$ I tried the bounds of $c_n$ (denotes the $n$-th composite number) given in this paper to prove that the sum $c_{p_n-m}+c_{m-n}$ ...
user57432
3
votes
0
answers
97
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A sieve result with two parameters
I proved the following sieve result and - since the proof is quite long and I need to use it in a work - I am looking for a reference to it (or at least something from which it could be proved quickly)...
user40023
2
votes
0
answers
149
views
$f(x)$-th largest number of prime factors
Given a sufficiently well-behaved function $1\le f(x)\le x$ and a multiset $S=\{\omega(n): 1\le n\le x\}$, what can be said about the asymptotics of the $f(x)$-th largest member of $S$? In other words,...
- 9,114
5
votes
2
answers
1k
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Survey of Algebraic K-Theory Since 1980?
I just came across Charles Weibel's Development of Algebraic K-Theory until 1980, and found it really helpful. Is there been anything analogous which surveys the developments in the last 30 years? I'...
- 1,138
2
votes
2
answers
334
views
What should I read if I want to learn about integral structures on classical algebraic groups?
I'm looking to learn about integral structures (or models?) on classical algebraic groups.
To begin with I have been learning about algebraic groups, quadratic forms and lattices. And also looking at ...
- 787
0
votes
1
answer
1k
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Name of a conjecture on difference of prime numbers? [closed]
Hello Dear
there is a conjecture for which I do not know how it is called. The conjecture is:
Every even number can be always written as the difference between two prime numbers.
Could you please ...
- 3
3
votes
0
answers
220
views
Almost primes in short intervals
Define an integer $n$ to be a $k$-almost prime if it has at most $k$ distinct primes factors. A detecting function for the set of such numbers is the generalized von Mangoldt function given by $\...
- 141
2
votes
1
answer
223
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Level-Lowering in Weight 2
Let $N$ and $p$ be relatively prime integers with $p$ a prime. Suppose $f$ is a weight $k=2$ (normalized, cuspidal, etc) newform of level $\Gamma_1(N) \cap \Gamma_0(p)$. I seem to recall the existence ...
- 1,422
2
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1
answer
208
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Enumerating positive fractions (reference missing)
I remember that the recursion
$r(0)=0, \ \
r(n+1)=\frac{1}{2 [r(n)]+1-r(n)}$
produces a sequence of rational values $ 0 \mapsto 1 \mapsto 1/2 \mapsto 2 \mapsto 1/3 \mapsto ... $ which exausts the ...
- 373
4
votes
1
answer
590
views
To which automorphic forms/rep's over a function field can we associate a Galois representation?
As far as I understand it, by the work of Lafforgue (cf. Laumon, "Cohom. of Drinfeld ... II", Thm 12.4.1) there is a Galois representation associated to an irreducible cuspidal automorphic ...
- 43
12
votes
3
answers
881
views
What does the computer suggest about the parity of p(n), for n in a fixed arithmetic progression?
Let p(n) be the number of partitions of n. A famous theorem of Euler allows one to compute
the parity of p(n) quickly for quite large n. In:
On the distribution of parity in the partition function, ...
- 5,422
3
votes
1
answer
246
views
Numbers with balanced diophantine approximations
This is a follow-up to Question 146635, namely Expected symmetry in the diophantine approximations of an irrational number, which I will refer to for notation and terminology used here without ...
- 10.5k
1
vote
0
answers
87
views
Diophantine equation $z=(ax+by+c)/(dxy)$, references? [closed]
I am looking for some sources (books or papers) which discuss the Diophantine equation
$$
z=\frac{ax+by+c}{dxy}
$$
where $a,b,c,d$ are given positive integers. Could anyone give some references?
...
- 841
4
votes
3
answers
1k
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Companion forms
What is the best known result concerning the existence of companion forms for classical modular forms? Gross' tameness criterion paper is always mentioned with a "unchecked compatibility" caveat? ...
- 173
16
votes
1
answer
2k
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Reference for the `standard' Tate curve argument.
I'd like a reference (e.g. something published somewhere that I can cite in a paper) for the proof of the following:
Let $E$ be an elliptic curve over $\mathbb Q$ with minimal discriminant $\Delta$...
- 10.5k
-1
votes
1
answer
376
views
Collatz property implying infinite "fall below" trajectories, is it known?
(this was discovered analyzing Collatz empirically.)
a key aspect of resolving Collatz involves looking at the number of iterations for trajectories to "fall below" the initial value.
consider a ...
- 529
-2
votes
1
answer
1k
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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, ...
3
votes
1
answer
303
views
Higher dimensional analogue of Thue's equation
The classical Thue equation is
$$\displaystyle F(x,y) = h,$$
for a binary form $F(x,y) \in \mathbb{Z}[x,y]$. Recall that a binary form is a polynomial in two variables which is homogeneous, and $h$ ...
- 26.9k
2
votes
3
answers
912
views
Reference on generators of subgroups of symplectic groups
We should start with the definition of the symplectic group for an arbitrary ring $R$.
The symplectic group $Sp(g,R)$ is the subgroup of $SL(2g,R)$ such that all elements satisfy $M=J_g^t M J_g$ with $...
- 85
11
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0
answers
282
views
Reference request: a combinatoric result [closed]
When I tried to construct a counterexample in my research, I encountered the following result, which should be true.
Let $m=m(n)$ be a function that grows faster than $\sqrt n$, so $m(n) = \omega(\...
- 211
7
votes
1
answer
288
views
Expected symmetry in the diophantine approximations of an irrational number
Given $x \in \mathbb{R}$ we will write $\{x\}$ for the fractional part of $x$ and $\|x\|$ for the distance of $x$ from the nearest integer, in such a way that $\{x\} = x - \lfloor x \rfloor$ and $\|x\|...
- 10.5k
3
votes
1
answer
562
views
Volume of PGL(2,F) \ PGL(2, A)
Let $F$ be a global field. What is the measure of $PGL_2(F) \backslash PGL_2(\mathbb{A})$?
This depends of course on the normalizations of the Haar measures on $PGL_2(F)$ and $PGL_2(\mathbb{A})$. ...
- 11.2k
1
vote
1
answer
122
views
Source for equations involving congruences of Fibonacci and Lucas numbers
In a paper of Cohn (see here), he uses some formulae involving congruences of Lucas- and Fibonacci-numbers (equations 11,12,13 in the preliminaries section). Does anyone know a source for these (and ...
- 1,101
5
votes
0
answers
143
views
Reference request: "effective'' semistable reduction
I am looking for the origin of the following idea: suppose $m$ and $n$ are relatively prime integers $\geq 3$. Let $E$ be an elliptic curve over a number field $K$. Let $L/K$ be a finite extension ...
- 51
5
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1
answer
2k
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Galois representation associated to a modular form is crystalline iff...
I am looking for the reference for the following fact (used, for example, in the proof of theorem 4.4. in Breuil's expose about local-global compatibility at Bourbaki):
For $f$ a modular cuspidal ...
6
votes
0
answers
380
views
Large sets not containing arithmetic progressions of length 3 in intervals
Given a large enough natural number $N$, let $\Delta_N=\{A \subseteq [N, 2N]: A$ contains no arithmetic progressions of length $3 \},$ where for natural numbers $N<M$ we have $[N, M]=\{N, N+1, ..., ...
- 32.2k
3
votes
0
answers
181
views
Factorization of linear recurrences
For each (commutative unitary) ring $R$, let $\mathfrak{R}(R)$ be the set of all linear recurrences over $R$, that is, the set of all sequences $(a(n))_{n \geq 0}$ in $R$ such that
$$a(n+k) = r_1 a(n+...
user40023
2
votes
1
answer
264
views
The number of different lattice triangles
Two convex lattice polygons are equivalent if there is a lattice-preserving affine transformation mapping one of them to the other. Equivalent polygons have the same area. Let $H(A)$ denote the number ...
- 12.3k
7
votes
1
answer
653
views
Closed form for derivatives $\zeta^{(n)}(1/2)$
According to mathworld
41,42. "Derivatives $\zeta^{(n)}(1/2)$ can also be given in closed form"
with example for the first derivative.
What is the closed form? References?
The motivation is that ...
- 25.4k
5
votes
0
answers
522
views
Moduli interpretation of Hecke operators on Shimura curves
In his book on Automorphic Forms, Shimura gives (chapter 9) definitions of the the Hecke operators for Shimura curves.
One can give definitions of the Hecke and Atkin-Lehner operators in terms of the ...
- 3,555
1
vote
0
answers
262
views
$\mathfrak{q}$-ideal class bound
Let $K$ be a number field, $\mathcal{O}_K$ be its ring of integers. Let $\mathfrak{q}$ be a nonzero ideal in $\mathcal{O}_K$.
The $\mathfrak{q}$-ideal class group consists of equivalence classes of ...
- 3,320