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2 votes
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Rational points on a special class of surfaces

Consider a smooth surface of the following form $$ S = \{f(x,y,t) = p_0(t)x^2+p_1(t)xy+p_2(t)x+p_3(t)y^2+p_4(t)y+p_5(t) = 0\}\subset\mathbb{A}^3 $$ over $\mathbb{Q}$, and set $$ U_S = \{t' \in \mathbb{...
Puzzled's user avatar
  • 8,998
4 votes
1 answer
394 views

Reductive subgroups of $\mathrm{GL}_2$ over an algebraically closed field of characteristic zero

I am reading a very nice paper of Newton and Thorne, Symmetric power functoriality for holomorphic modular forms, and there is an argument concerning the (Zariski-closure of) image of certain $p$-adic ...
babu_babu's user avatar
  • 241
4 votes
0 answers
244 views

Torsionness of the kernel of the pullback map of Picard groups of a normalization map

Let $X$ be a (irreducible) projective variety over a number field $k$, $\pi: \tilde X \to X$ be its normalization, and $\pi^{*}: \mathrm{Pic}(X) \to \mathrm{Pic}(\tilde X)$ be the corresponding map of ...
penseur_32's user avatar
12 votes
1 answer
1k views

An omission in K. Conrad's notes on the conductor ideal

I am referring to the very useful K. Conrad's notes on the conductor ideal of an order in a Dedekind domain: https://kconrad.math.uconn.edu/blurbs/gradnumthy/conductor.pdf $\DeclareMathOperator\Cl{Cl}$...
Hair80's user avatar
  • 675
9 votes
1 answer
472 views

Why is the category of motives generated by varieties?

I'm reading Ayoub's paper Motifs des varietes analytiques rigides, but I'm not quite familiar with motives. In this paper, he defines the category of motives to be $\mathbf{RigDM}^{\rm eff}_{\rm Nis}(...
Chen Zekun's user avatar
2 votes
0 answers
149 views

Singular fiber of a family of elliptic curves with Kodaira's symbol mI_V

From https://en.wikipedia.org/wiki/Elliptic_surface, one classifies all fibers of a minimal elliptic fibration via Kodaira's symbols: $I_v$, $mI_v$, $II$, $III$, $IV$, $I_v^*$, $II^*$, $III^*$, $IV^*$ ...
Yang's user avatar
  • 429
2 votes
0 answers
124 views

About finite dimensionality of Chow groups of zero cycles

Let $S$ be a connected smooth complex projective surface. Let $Sym^{d}(S)$, $d\in \mathbb{Z}^+_0$, be the $d$-th symmetric product of $S$ parametrizing $0$-cycles of degree $d$. Let $Sym^{d,d}(S)=...
Roxana's user avatar
  • 519
-1 votes
1 answer
186 views

Public key cryptography based on non-invertible matrices, part II

Closely related to this question and extending comment of R. van Dobben de Bruyn. Working over $\mathbb{F}_p$ and all matrices of square $n \times n$. Alice chooses invertible $X_A$ and non-...
joro's user avatar
  • 25.4k
6 votes
1 answer
566 views

Public key cryptography based on non-invertible matrices?

Added Wed 13 Apr 2022 I have written a short note with experimental data, which shows not all pseudo keys are good keys. Public key cryptography based on non-invertible matrices We got public key ...
joro's user avatar
  • 25.4k
6 votes
1 answer
543 views

Chebotarev density theorem and pure weight local systems

How do we deduce the following statement from the Chebotarev density theorem? The statement is from Ngo's Fundamental Lemma paper. Let $U$ be a scheme of finite type over $\mathbb{F}_q$. Let $\...
userabc's user avatar
  • 677
2 votes
0 answers
214 views

Number of lines on a weak del Pezzo surface

By a line I mean a (-1)-curve. Given a weak del Pezzo surface $X$ of degree $d$, how many lines would $X$ contain?
H U's user avatar
  • 481
5 votes
1 answer
371 views

Number of points on schemes modulo $p^k$

Let $X$ be a finite type scheme over $\mathbb{Z}_p$ for some prime $p$. Assume that $X_{\mathbb{Q}_p}$ is smooth of dimension $n$, but not necessarily irreducible. Then is $$X(\mathbb{Z}/p^k\mathbb{Z})...
Daniel Loughran's user avatar
2 votes
0 answers
242 views

Solving $x^k+y^k+z^k=w^k$ non-trivially in strictly positive integers

Consider the equation $x^k+y^k+z^k=w^k$ in $x$, $y$, $z$ and $w$ with $k\in\mathbb{N}_{\geq2}$. If we look for solutions that are strictly positive and non-trivial i.e. $x\neq-y$, $x\neq w$ etc... ...
Ivan Meir's user avatar
  • 4,862
1 vote
1 answer
180 views

On integral points of $f(x,y)=z g(x,y)$

Let $f(x,y),g(x,y)$ be polynomials with integer coefficients. Consider the surface $$ f(x,y)=z g(x,y) \qquad (1)$$ (1) has parametrization over the rationals given by $z=\frac{f(x,y)}{g(x,y)}$. Q1 ...
joro's user avatar
  • 25.4k
9 votes
0 answers
274 views

$y^3=x^4+x+1$, and rational points on rank 2 Picard curves

What are (a) integer, (b) rational solutions to the equation $$ y^3 = x^4 + x + 1 ? $$ There are obvious solutions $(x,y)=(-1,1)$ and $(0,1)$, are they the only ones? Context: There are a lot of ...
Bogdan Grechuk's user avatar
2 votes
1 answer
167 views

Existence of reduced norms for CSAs using fpqc descent

Let $k$ be a field and $A$ be a central simple algebra over $k$. It's known that $A$ has a splitting field (i.e. a field $K/k$ such that $A_K\cong M_n(K)$ for some $n$) which is finite and Galois. ...
Gabriel's user avatar
  • 711
8 votes
0 answers
516 views

Galois rigidity for ℙ¹ with infinitely many punctures

A well-known result (due first to Nakamura I think) is that given a number field $K$, and a variety $U = \mathbb P^1 \setminus (\text{finitely many points})$ over $K$, the étale fundamental group of $...
Mark OSS's user avatar
  • 159
6 votes
0 answers
196 views

Weyl group and Galois action on cubic surfaces

Let $X$ be a smooth cubic surface over a field $k$. Denote by $\bar{k}$ the separable closure of $k$ and $\bar{X}:=X\times_{k}\bar{k}$. Then it is well know that there exists a homomorphism $$ \phi:\...
H U's user avatar
  • 481
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 ...
user859588's user avatar
7 votes
0 answers
451 views

Is it unconditionally known that abc conjecture can't fail on a variety?

Background: this question gives the identity: $$(x+z)^5+(y-z)^5 = (-3 x + 4 y)^2 (x + y)^3 + (x+y) f(x,y,z)$$ The curve $C : f(x,y,z)=0$ is genus 1, have infinitely many rational and integral points ...
joro's user avatar
  • 25.4k
10 votes
1 answer
462 views

Homomorphisms between Oort–Tate group schemes

Let $R$ be a complete local $\mathbf{Z}_p$-algebra, for some prime $p$. In the 1970 paper Group schemes of prime order by Oort and Tate, they write down an explicit finite flat group scheme $G_R(a, b)$...
David Loeffler's user avatar
7 votes
1 answer
547 views

Étale fundamental group of rigid analytification

Let $X$ be a quasi-projective variety over a $p$-adic field. Denote by $X^{an}$ its rigid analytification. Does $\pi_1^{et}(X)=\pi_1^{et}(X^{an})$?
Yang's user avatar
  • 429
2 votes
1 answer
326 views

$2$-isogenous to a curve in the Tate normal form

It is well-known that an elliptic curve $E$ that has a point of order $2$ and is represented as $E=[0,a,0,b,0]$ has a $2$-isogenous curve $E^\prime=[0,-2a,0,a^2-4b,0]$, see e.g. p. 507 in A. Dujella, ...
Maksym Voznyy's user avatar
5 votes
0 answers
206 views

$\operatorname{GL}_2$-type abelian varieties with full level-$p$ structure

For a prime $p$, does there exist a $\operatorname{GL}_2$-type abelian variety $A$ over $\mathbb{Q}$ having full level-$p$ structure? That is, $A[p]=(\mathbb{Z}/p)^d \oplus \mu_{p}^{d}$ as a $G_{\...
mtc's user avatar
  • 51
4 votes
1 answer
366 views

Splitting the Witt vectors of $\overline{\mathbb{F}_p}$

Let $\overline{\mathbb{F}_p}$ be the algebraic closure of $\mathbb{F}_p$. Let $W({-})$ denote the functor of taking $p$-typical Witt vectors. Then the extension $\mathbb{F}_p\rightarrow \overline{\...
kiran's user avatar
  • 2,052
87 votes
12 answers
12k views

Why do we make such big deal about the 'unsolvability' of the quintic?

The unsolvability of a general quintic equation in terms of the basic arithmetic operations and $n$th roots (i.e. the Abel–Ruffini theorem) is considered a major result in the mathematical canon. I ...
Arthur's user avatar
  • 1,389
7 votes
2 answers
605 views

ℤ/18ℤ elliptic curves over cubic fields

I am working on $\mathbb{Z}/18\mathbb{Z}$ elliptic curves over cubic fields. The curves are created using the formulas on p. 584 of D. Jeon, C. H. Kim, Y. Lee, Families of elliptic curves over cubic ...
Maksym Voznyy's user avatar
4 votes
1 answer
315 views

Criteria for Zariski density of subgroups of reductive groups

Let $G$ be a reductive group over a number field $K$. Let $\Gamma\subset G(K)$ be a subgroup. My extremely naive question is - When can you deduce that $\Gamma$ is Zariski-dense? I'm looking for ...
stupid_question_bot's user avatar
3 votes
1 answer
256 views

Rationalizing and minimizing elliptic curve coefficients

I am working on elliptic curves with torsion group $\mathbb{Z}/14\mathbb{Z}$ over quadratic fields. The curves are constructed using the model $E_1=[0,a,0,b,0]$ following the formulas on p. 13 of L. ...
Maksym Voznyy's user avatar
0 votes
1 answer
370 views

Systems of equations for elliptic curves without $3$-torsion

In his YouTube video New rank records for elliptic curves having rational torsion, Noam Elkies uses systems of equations at 6:16 and 8:38 to present $\mathbb{Z}/3\mathbb{Z}$ curves of rank 14 and rank ...
Maksym Voznyy's user avatar
4 votes
1 answer
204 views

Groups suitable for algebraic group factorizations of integers

Quoting Wikipedia on Algebraic-group factorisation algorithm Algebraic-group factorisation algorithms are algorithms for factoring an integer N by working in an algebraic group defined modulo N whose ...
joro's user avatar
  • 25.4k
3 votes
1 answer
272 views

Product of Abelian varieties with complex multiplication

We take Abelian varieties $A_1, A_2,\dotsc,A_n$ over a number field. If $A_1, A_2,\dotsc,A_n$ have complex multiplication, then does the product $A_1\times A_2 \times \dotsb \times A_n$ have complex ...
OOOOOO's user avatar
  • 349
7 votes
1 answer
502 views

When must a set of sections which is Zariski dense in the generic fiber also be dense in some special fiber?

Let $f : X\rightarrow S$ be a flat finite type morphism of schemes with $S$ integral and Noetherian. Let $\eta\in S$ be the generic point. Let $\{\sigma_i\}$ be a collection of sections of $f$ (...
stupid_question_bot's user avatar
5 votes
0 answers
278 views

Torsion points of an elliptic curve over number fields. Another proof of Silverman AEC theorem

I am studying the following theorem from Silverman's AEC: I am wondering whether there exists another proof that doesn't make use of formal groups and is still valid for a number field $K$. Could you ...
cartesio's user avatar
  • 233
4 votes
0 answers
219 views

Generate periods only by smooth varieties

Like explained in this passage that a period is a complex number whose real and imaginary parts are integrations of rational functions over $\mathbb{Q}$ on some $\mathbb{Q}$-semi-algebra set in $\...
CO2's user avatar
  • 275
2 votes
0 answers
161 views

Nearby cycle is tamely ramified?

Let $S$ be a henselization of a closed point $s$ in a smooth algebraic curve $C$ over some finite field $\mathbb{F}_q$. Then we can consider nearby cycles over $S$. Let $s$ be the closed point of $S$ ...
Yang's user avatar
  • 71
4 votes
1 answer
344 views

Tannakian fundamental group of automorphic representations

Let $\mathcal{C}_{\mathrm{aut}}(G, F)$ be the category of automorphic representations of a connected reductive group $G$ over a number field $F$. If this is a Tannakian category, it has an associated ...
Sam's user avatar
  • 87
2 votes
0 answers
278 views

Rational points on surfaces

Let $k$ be a field of characteristic zero. In the affine space $\mathbb{A}_{x,y,t}^3$ consider a surface $S$ of the form $$ S = \{a_0(t)x^2+a_1(t)xy+a_2(t)x+a_3(t)y^2+a_4(t)y+a_5(t) = 0\} $$ where $...
user avatar
6 votes
1 answer
438 views

$y^3 = x^4 + x + 2$, and existence of rational points on rank 0 Picard curves

Do there exists rational numbers $x$ and $y$ such that $$ y^3 = x^4 + x + 2 ? $$ Context: There are a lot of publications about computing rational points on elliptic and hyperelliptic curves, and ...
Bogdan Grechuk's user avatar
1 vote
0 answers
261 views

Integer points on genus 1 curves using CAS

How can I practically find integer points on genus 1 curves with small coefficients using computer algebra systems (CAS), like Mathematica, Maple, SageMath, Magma, etc.? As a specific example, do ...
Bogdan Grechuk's user avatar
2 votes
1 answer
273 views

Does $L$-functions of elliptic curves over $\mathbb{Q}$ being meromorphic obviously imply modularity?

If I somehow know that for each elliptic curve over $\mathbb{Q}$ the $L$-function has a meromorphic continuation to the whole plane can I easily deduce modularity from that? If not is there a way to ...
novler's user avatar
  • 441
3 votes
0 answers
188 views

Does the construction of arithmetic toroidal compactification of $A_{g}$ depend on semistable reduction theorem?

If there is a good theory of arithmetic toroidal compactification over $\mathbb{Z}_{p}$ of the Siegel modular variety with deep enough level structure, then it seems like semistable reduction theorem ...
GTA's user avatar
  • 1,024
4 votes
0 answers
205 views

Grothendieck group of admissible $p$-adic representations

Let $K$ be a $p$-adic local field; $G = \mathop{\mathrm{Gal}}(\overline K | K)$; $\tau \in \{\text{HT}, \text{dR}, \text{crys}\}$, $B_\tau$ the corresponding period ring; $\mathop{\mathrm{Rep}}_{\...
Aoi Koshigaya's user avatar
7 votes
0 answers
307 views

Number of rational points over finite fields mod $q$ is birational invariant

I heard that if $\mathbf F_q$ is a finite field, $X, Y$ are birational smooth proper variety over $\mathbf F_q$, then $\#(X(\mathbf F_q)) \equiv \#(Y(\mathbf F_q)) \pmod q$, and I heard that the proof ...
Aoi Koshigaya's user avatar
6 votes
0 answers
198 views

$\mathbb{Z}$-points in a given $\widehat{\mathbb{Z}}$-isomorphism class

Given a finite type $\mathbb{Z}$-scheme $X$ with $X(\widehat{\mathbb{Z}})\neq\emptyset$ can we find a finite type $\mathbb{Z}$-scheme $Y$ with $X\times \widehat{\mathbb{Z}}\cong Y\times\widehat{\...
user avatar
3 votes
0 answers
377 views

Meaning of "the" general fiber in the paper "La conjecture de Weil : I"

In section 4.1, chapter 4 of Pierre Deligne's paper La conjecture de Weil : I (french version, translation to English) he states: Let $X$ be a non singular analytic space and purely of dimension $n+1$....
Roxana's user avatar
  • 519
4 votes
1 answer
363 views

The numbers of isomorphism classes of abelian variety over finite fields

It is known that there are only finitely many isomorphism classes of abelian variety over a finite field. I am curious about the exact number of these isomorphism classes. Explicitly, fix $g$, let $\...
Yuan Yang's user avatar
  • 547
14 votes
2 answers
567 views

Does every non-isotrivial 1-parameter family of elliptic curves have a positive rank specialization?

Let $\mathcal{E}/\mathbb{Q}(t)$ be given by $$y^2=x^3+A(T)x+B(T)$$ for some $A(T),B(T)\in\mathbb{Q}[T]$ and assume $\mathcal{E}$ is non-isotrivial (the $j$-invariant $\frac{6912 A(T)^3}{4A(T)^3 + 27B(...
Jonathan Love's user avatar
5 votes
1 answer
402 views

Extending rational to integral points

Let $p: \mathcal{X} \rightarrow \text{Spec } \mathcal{O}_K$ be a normal proper Artin stack with finite diagonal. A $K$-rational point is by definition a section $x: \text{Spec}(K) \rightarrow \mathcal{...
P. Koymans's user avatar
6 votes
1 answer
247 views

Sets of $\mathbb{F}_p$-points of closed subvarieties of $\mathbb{A}^n$

Let $p$ be a prime and let $n\geq 2$ be an integer. The set $\mathbb{A}^n(\mathbb{F}_p)$ has $p^n$ elements so it has $2^{p^n}$ subsets. How many of those subsets are of the form $V(\mathbb{F}_p)$ ...
Qasim's user avatar
  • 103

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