All Questions
38 questions
1
vote
0
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87
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Equidistribution of Frobenius Classes
Let $G$ be a reductive group over $\mathbb{Q}$. Let $K$ be a maximal compact subgroup of $G(\mathbb{C})$. Let $S$ be a finite set of primes. For each prime $p$ not in $S$, let $Frob_p$ be a conjugacy ...
- 11
7
votes
0
answers
124
views
Projections of closed geodesics under the modular function
In the answers to this question it was shown that for closed geodesics on $\mathbb{H}^2/\Gamma(2)$, the projection under the modular function $\lambda$ is an immersed topological component of a real ...
- 68.9k
6
votes
0
answers
338
views
Deeper meaning behind the occurrence of the factor $\frac{\log q}{i}$ in Deninger's results
In two papers Deninger proved the following:
If $q=p^{n}$ and $p$ is a finite prime of $\mathbb{Z}$, $B=\mathbb{C}[\mathbb{C}]$ is generated by symbols of the form $e^{\alpha}$, $\alpha\in\mathbb{C}$,...
- 1,805
0
votes
0
answers
107
views
Cubic monic polynomial over z_p
Let
$$
f_{a}(x)=x^3+(u-2-a)x^2+ax+1,
$$
where $u\in\mathbb{Z}_p^*$ is fixed. Let $S$ be the set consisting of all $a\in\mathbb{Z}_p$ such that $f_{a}(x)$ factor linearly. Then what is the cardinality ...
user126352
3
votes
1
answer
230
views
Cubic polynomial over $\mathbb{Z}_p$
Let
$$
f_{a,b}(x)=x^3+(u-1-a-b)x^2+ax+b,
$$
where $u\in\mathbb{Z}_p^*$ is fixed. Let $S$ be the set consisting of all pairs $(a,b)\in\mathbb{Z}_p^2$ such that $f_{a,b}(x)$ factor linearly. Then what ...
user126352
2
votes
0
answers
226
views
Number of roots of a multivariate polynomial
What could be the best known asymptotic for the number of solution of the following polynomial in $(F_p)^s$:
$$
(1-x_1)(1-x_2)\cdots(1-x_s)(1-x_1x_2...x_s)=ux_1x_2...x_s
$$
where $u$ is a non-zero ...
user126352
1
vote
0
answers
98
views
Hardness of solving $0=\sum_{i=1}^k \operatorname{linear}_i(x_1,\ldots,x_n)^D$ over the rationals
This is related to cryptography and this question
and another question.
In short, we are asking about decomposing multivariate polynomial
as sum of perfect powers of linear polynomials.
Working over $\...
- 25.4k
5
votes
0
answers
192
views
On the elementary proof of Dirichlet theorem on arithmetic progressions
In [Cassels, JWS, Rational quadratic forms, p. 333], the autor says: "In fact the elementary proof of Dirichlet's theorem [Selberg (1949)] makes essential use of the existence of genera".
In ...
4
votes
0
answers
211
views
Rational solutions to Catalan's equation
Famous Catalan's conjecture, now a theorem proved by Mihăilescu, states that the only solution in the natural numbers of the equation
$$
x^{a}-y^{b}=1.
$$
for $a, b > 1$ and $x, y > 0$ is $x = 3,...
- 7,182
3
votes
1
answer
261
views
Examples of non-singular hypersurfaces exhibiting Hasse principle failures
Suppose that $f\in \mathbb{Z}[x_1,\dots,x_n]$ and $f$ is a homogenous polynomial of degree $d$. Can we always construct $f$ such that the hypersurface $S_f=\{x \in \mathbb{Z}^n:f(x)=0\}$ exhibits the ...
- 107
1
vote
0
answers
172
views
Does Lemma 5.4 in Deligne's Ramanujan paper generalize to Shimura varieties of PEL type?
It is generally not known if a smooth variety over a perfect field embeds into a smooth proper variety.
Lemma 5.4 in Formes modulaires et représentations $\ell$-adiques provides such an embedding for ...
- 31
2
votes
1
answer
215
views
The growth of the number of Fano complete intersection families
I recently calculated the number (possible multidegrees) of Fano complete intersections of dimension $n$ , because I wanted to make the remark that it grows "very rapidly" as $n \rightarrow \...
- 6,995
4
votes
0
answers
204
views
$\DeclareMathOperator\sym{sym}$Does $L(s, \sym^2 f \times \sym^2 g)$ have a pole at $s=1$?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\sym{sym}$I encountered a question on the poles of $\GL_3\times \GL_3$ $L$-function, which needs the knowledge of the experts ...
- 563
3
votes
0
answers
186
views
Maximum value of newform from Galois representation
One can attach $\ell$-adic Galois representations to holomorphic cuspidal newforms of weight $2$ on the upper half-plane.
If a newform is $L^2$-normalized, can one extract its maximum value from the ...
- 39
1
vote
0
answers
62
views
Polynomial sized arithmetic map from circle to ellipse preserving integral points
Let $n$ be a square free integer and a product of $O(m/\log m)$ number of primes $1\bmod 4$ where $m$ is $\log_2n$.
Take the circle around origin of radius $n^2$. It has ${\exp}(m/\log m)$ number of ...
- 13.9k
-1
votes
1
answer
342
views
Finite or polynomial number integral points clarification on Coppersmith's theorems (possibility of ellipse counter example?)
Coppersmith states if $f(x,y)$ is an irreducible bivariate with total degree $\delta$ then if he can list all roots $(X,Y)$ of the polynomial in $\mathsf{poly}(\log D,\delta)$ time if the roots ...
- 13.9k
9
votes
1
answer
860
views
Complex manifold defined over $\mathbb{Q}$
If we consider complex projective varieties, to be defined over $\mathbb{Q}$ means that there is a projective embedding whose image is the vanishing locus of a polynomial system with coefficients in $\...
user164740
4
votes
1
answer
282
views
Mean square estimate for the Kloosterman sums
For $m,n\in \mathbb{N}$, denote the Kloosterman sum
$$S(m,n;c)=\sum_{a\bmod c}e\left( \frac{ma+n \overline{a}}{c}\right),$$where $\overline{a}$
denotes the multiplicative inverse of $a\bmod c$.
Does ...
- 353
1
vote
0
answers
102
views
Maximum number of bounded primitive integer points in a zero-dimensional system
Given a set of $n$ many degree $2$ algebraically independent and thus zero-dimensional system of homogeneous polynomials in $\mathbb Z[x_1,\dots,x_n]$ with absolute value of coefficients bound by $2^{...
- 1,826
3
votes
0
answers
148
views
Maximum number of integral roots in degree $d$ polynomial?
Given $f(x_1,\dots,x_n)\in\mathbb Z[x_1,\dots,x_n]$ such that
Each coefficient is bound in absolute value by $B$
Degree of each variable in any monomial is bound by $d$
Total degree is $d'$
$f(x_1,\...
- 13.9k
10
votes
2
answers
1k
views
Are the ideles literally a Picard group?
I understand that in the number field / function field analogy, the ideles $\mathbb I_K$ of a number field $K$ are supposed to be analogous to the Picard group of a function field.
Question: Is this ...
- 63.9k
35
votes
0
answers
1k
views
Is there a rigid analytic geometry proof of the functional equation for the Riemann zeta function?
The adèles $\mathbb A$ arise naturally when considering the Berkovich space $\mathcal M(\mathbb Z)$ of the integers. Namely, they are the stalk $\mathbb A = (j_\ast j^{-1} \mathcal O_\mathbb Z)_p$ ...
- 63.9k
0
votes
1
answer
125
views
Polyhedral conditions for quadratic inequalities in fixed dimension
Denote $\mathcal T$ be set of $(T_1,T_2,T_3,T_4)\in\mathbb Z^4$ that satisfy
$$0<T_1,T_2,T_3,T_4$$
conditions?
Define the level set $$M_{\gamma}(Q,\mathcal T)=\{(T_1,T_2,T_3,T_4)\in\mathcal T:Q(...
- 13.9k
18
votes
0
answers
740
views
Infinite extensions such that every elliptic curve has finite rank
The comments to this answer seem to make the following claim.
Claim. Let $K$ be the maximal abelian extension of $\mathbf Q$ that is unramified away from $p$ (more generally, away from a finite set $S$...
- 28.7k
2
votes
0
answers
249
views
An exponential sum like the Kloosterman sums
I encounter a tricky sum like the Kloosterman sum
$${\sum_{x \mod P}}^\ast e\left(\frac{ax+\overline{x}}{P}+\frac{lx^2}{P^2}\right),$$
where $l$ is a positive integer co-prime with $P$ and here $P$ ...
- 353
5
votes
0
answers
280
views
Proving that a certain function (related to a volume of a region) has a bounded derivative
Let $F$ be a homogeneous form in $n$ variables with integer coefficients.
Let $D$ be a closed box in $\mathbb{R}^n$ (product of closed and bounded intervals). Assume that the partial $\partial F/\...
- 3,625
7
votes
3
answers
768
views
Sato-Tate and the angles of split primes
I was reading this blog by Evan Chen about complex multiplication. He's discussing Sato-Tate Conjecture. We can have elliptic curve $E/\mathbb{Q}$ and solve it over finite fields.
\begin{eqnarray*}
...
- 22.8k
14
votes
3
answers
666
views
Patterns in solutions to $a^2 + b^2 + c^2 = n$
I have plotted solutions $(a,b,c)$ to $a^2 + b^2 + c^2 = n$ for $12000 \leq n < 12100$, rescaled to $S^2$ and projected onto the first two coordinates. (these are read from the lower left, across ...
- 22.8k
5
votes
1
answer
1k
views
Reference to "bounds of Weil and Deligne"
In the this paper by Friedlander and Iwaniec, it is said that they are "able to avoid much of the high-powered technology frequently used in modern analytic number theory such as the bounds of Weil ...
- 1,890
3
votes
1
answer
664
views
Colmez conjecture and endomorphism rings
It is given by the Colmez conjecture that if $A$ has a CM-type $(k, \phi)$ and $\text{End}(A) = \mathcal{O_k}$ $$\displaystyle h_{\text{fal}}(A) = \sum_{\text{irr} \hspace{1 mm}\rho: G(k^{\text{nor}}/\...
- 201
12
votes
2
answers
902
views
Infinitely many irreducible polynomials of the form f(X^2) + X mod 3?
Are there infinitely many polynomials $f \in \mathbb{F}_3[X]$ for which $f(X^2) + X$ is irreducible?
- 11.3k
2
votes
0
answers
146
views
Odds of projections of a point not on the hyperplane
Let $\mathcal{L}=\{\Bbb x\in\Bbb R^n:x_1+x_2+\dots+x_n=0\}$ be a specific hyperplane.
Let a projection of $c\in\Bbb R^n$ be $p(c)=[p_1,p_2,\dots,p_n]$ where $p_i\neq c_i\implies p_i=0$.
Let $\...
- 13.9k
17
votes
1
answer
874
views
How fast can we numerically calculate Kloosterman sums?
Define the usual Kloosterman sum by $$S(m,n;c) = \sum_{\substack{x \pmod{c} \\ (x,c) = 1}} e\Big(\frac{mx + n\overline{x}}{c}\Big),$$
where $x \overline{x} \equiv 1 \pmod{c}$, and $e(x) = e^{2 \pi i x}...
- 4,671
17
votes
1
answer
1k
views
On the Hasse-Weil L-function of $P^n$
So let us start with the "simplest" scheme over $Spec(\mathbf{Z})$ namely $X_0=Spec(\mathbf{Z})$. Then the (reciprocal) Weil zeta function of $X_0$ at a prime $p$ is given by $Z_p(T)=1-T$ (a ...
- 7,609
2
votes
2
answers
606
views
Axioms for zeta functions
The Selberg class is an axiomatization of arithmetically significant zeta functions (a.k.a. L-functions) by a few analytic properties (functional equation etc.) However there do exist other zeta ...
- 333
4
votes
2
answers
1k
views
on the Zeroes of Hasse -weil L-function
my question is that
already we know that the Birch and Swinnerton Dyer conjecture ,formally conjectures that the Hasse-weil L-function should have a zero at $s=1$ when curves have infinitely many ...
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6
votes
1
answer
3k
views
Intuition behind the Tamagawa numbers
i have read many books concerning the definition of tamagawa numbers ,but none of the books explained an intuition behind the concept ,
i mean what could be the intuitive definition of tamagawa number ...
Trust God
1
vote
3
answers
2k
views
Where Can i find the lecture Videos of BSD 2011
i recently heard that there was a conference on Birch and Swinnerton dyer conjecture Held at Cambridge on May 4 until May 6,
the main theme is "The conference marks the 50th anniversary of the Birch-...
Trust God