Questions tagged [arithmetic-geometry]

A subject that lies at the intersection of algebraic geometry and number theory dealing with varieties, the Mordell conjecture, Arakelov theory, and elliptic curves.

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92 views

Finding formula that solves $w^4+x^4=y^4+z^4$ over the integers.

Several formulae that solve the diophantine equation $$w^4 + x^4 = y^4+z^4 \tag{1}$$ are presented in this collection. The simplest one bases on $$f_1 = a^7 + a^5 - 2 a^3 + 3 a^2 + a \tag{2}$$ and ...
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52 views

About the canonical map $R\rightarrow \prod_{i=1}^{r}R[g_{i}^{-1}]$

Let $R$ be a complete and separated adic ring with a finitely generated ideal of definition $I\subset R$. Then we say that $R$ does not have $I$-torsion, if the ideal $(I-\textrm{torsion})_{R}=\{r\in ...
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29 views

When is a horizontal divisor regular

Let $K$ be a number field and $\mathscr{X} \longrightarrow \mathcal{O}_K$ be an arithmetic surface (smooth and projective). Let $D$ be a horizontal divisor. We know $D=\overline{\{x\}}$ for some ...
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42 views

Generic point of projective space?

This may be a question with a trivial answer. Given a morphism of schemes $X \to \mathbf{P}_k^1$, where $k$ is a field and $\mathbf{P}_k^1$ the projective line over $k$, I am trying to understand what ...
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33 views

The reason why $E_1(K)$ does not depend on which minimal weierstrass equation we choose

Let $K$ be a local field and let $E$ be an elliptic curve on $K$. I would like to know why $E_1(K)$ does not depend on which minimal weierstrass equation we choose. A minimal weierstrass equation is ...
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47 views

Question about subgroups of elliptic curves

We discussed substructures in my Intro to Algebraic Structures class, and it got me thinking of structures that are defined by more than just its algebraic operations. Consider some elliptic curve $E$....
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Why is it better to define $ SO(q) $ using Clifford algebras?

http://math.stanford.edu/~conrad/papers/luminysga3.pdf, appendix C.2 defines $ \mathrm{SO}(q) $ as the kernel of the algebraic group morphism $ D_q: \mathrm{O}(q) \to (\mathbb Z / 2 \mathbb Z)_S $, ...
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3answers
237 views

Absolute irreducibility of a polynomial over a finite field

Let $k$ be a positive integer, $\mathbb{F}_q$ be the finite field with $q$ elements. Consider the polynomial $f(X, Y) = 1 + X^k + Y^k + X^{2k} + Y^{2k}$ over $\mathbb{F}_q$. How to verify whether it ...
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68 views

Restate Galois representation from category theory

Given an arbitrary category $C$, a representation of $G$ that is a category with a single object in $C$ is a functor from $G$ to $C$. A group representation is a representation of $G$ in the category ...
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79 views

Simplifying a proof of Langlands-Tunnnell Theorem

I read A SIMPLIFIED PROOF OF SERRE’S CONJECTURE recent paper by Luis Victor Dieulefait, Ariel Martín Pacetti to understand the proof of Serre's modularity theorem. According to this paper, the proof ...
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Why is $P^n(K)$ compact, where $K$ is a local field? [closed]

In Silverman's book AEC, question 7.6 asks to prove $E_0(K)$ has finite index in $E(K)$ for $K$ a local field. For part (a), I know the topology on $P^n(K)$ is the quotient topology on $K^{n+1}$, and ...
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1answer
80 views

Criterion for a group scheme to be smooth

Let $G$ be an algebraic group over a field $k$ and fix a $k$-point $x \in G$ (let's just take the unit $ e \in G(k) $). $G \to \mathrm{Spec}(k)$ is smooth iff it is smooth in $e$ (see proposition 1.22 ...
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27 views

What is the closure of a field with respect to a valuation?

I have a problem when reading Berkovich's Spectral Theory and Analytic Geometry Over non-Archimedean Fields. I was confused about "the closure of a quotient field with respect to the valuation&...
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58 views

How to construct the following exact sequence?

Let $A$ be an adic ring and $\mathfrak{a}$ be its ideal of definition. Let $\textrm{Spf }A$ denote the set of all open prime ideals $\mathfrak{p}\subset A$, then $\textrm{Spf }A\cong \textrm{Spec }A/\...
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1answer
96 views

Why are the Weil Conjectures stated via zeta functions?

Let $C$ be a smooth curve over $\mathbf{F}_p$, and let $\zeta$ be the zeta function of $C$. The Weil Conjectures for $C$ are usually stated something like this: The zeta function $\zeta(s)$ is a ...
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Reference request- Galois Coverings over genus 1 curves

Would anyone have any good recommendations on learning about Galois coverings of genus 1 curves/torsors over genus 1 curves? More specifically learning about $H^{1}_{ét}(E,\mathbb{Q}/\mathbb{Z})$ ...
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1answer
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Can $P$ in $Q=[(E,P)] \in X_1(N)(K)$ always be defined over $K$?

Let's say we have an algebraic number field $K$ and a point $Q\in X_1(N)$ that is not a cusp. Now $Q$ can be represented as $Q=[(E,P)]$, where $E$ is an elliptic curve and $P$ is a pont on $E$ of ...
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On well-definedness of Shimura curve.

Suppose that we have a quaternion algebra $D$ over a totally real number field $K$ such that $[K \colon {\Bbb Q}] = {\mathrm{odd}}$. We assume that $D$ splits everywhere at finite places of $K$ and at ...
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1answer
31 views

Computing cusps of the Drinfeld modular curve $X_1(t^2)$

Let $K$ be the global function field $\mathbb{F}_3(t)$ and set $$ \Gamma_1(t^2)=\left\{\begin{pmatrix} a & b\\ c & d \end{pmatrix}\in \text{GL}_2(\mathbb{F}_3[t]) : a\equiv 1 \pmod{t^2} \text{ ...
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1answer
48 views

Reduction types of $y^{2} = x^{3}+a_{6}$ over wildly ramified field extensions

In J. Silverman's Advanced Topics in the Arithmetic of Elliptic Curves, he proposed this question in exercise 4.51(b). Take $K/\mathbb{Q}_{3}$ a 3-adic field, and let $L/K$ be a wildly ramified ...
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The textbook of Abelian varieties included Grothendieck’s semi-stable reduction Theorem

I would like to learn abelian varieties, semi-stable abelian varieties, and the proof of Grothendieck’s semi-stable reduction Theorem. I have briefly learned abstract algebra, category theory, ...
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1answer
35 views

Are Weil differentials on a Function Field same as the differentials on the corresponding Riemann surface?

I was reading "Number Theory on Function Fields" by Michael Rosen, and there is a notion of "Weil differentials" on a Function Field. It intuitively seems to me that a function ...
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3answers
118 views

Genus Degree Formula for Curves over Arbitrary Fields and a Reference Request

There exists the genus-degree formula for plane, projective, nonsingular curves that relates the (arithmetic) genus of a curve $C_F$ with the degree of the polynomial $F$ by the following relation: $$...
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38 views

How should I think of the nearby cycle $\mathrm R\Psi\overline{\mathbb Q_{\ell}}$ with its Galois action on a smooth projective scheme?

Let $F$ be a $p$-adic field with ring of integers $\mathfrak o$ and residue field $k$. Let $X$ be a smooth equidimensional projective scheme over $\mathrm{Spec}(\mathfrak o)$ of dimension $d$. Denote ...
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39 views

Difference between modular curves $X_0(N)$ and $X_1(N)$

I'm having trouble understanding how these 2 curves differ, i.e. how the points on these curves differ. As I understand, for an algebraic number field $K$, a point on $X_1(N)(K)$ parameterizes a pair $...
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53 views

The orders of points in the function field of elliptic curves

Let $K$ be the function field of an elliptic curve $C$ over a finite field $\mathbb F_q$, and $|C|$ be the set of closed points of $C$, i.e. the set of places of $K$. Let $A$ be any element of ...
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57 views

In which field $K$, elliptic curve is connected?

In which field $K$, elliptic curve is connected? If $K$=$\Bbb C $, elliptic curve over $\Bbb C $ is just homeo to torus, so connected. But in other field? If $K$=$\Bbb R $, $y^2=x^3-x$ seems ...
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42 views

In which field, minimal Weierstrass equation of elliptic curve can be defined?

For which fields can a, minimal Weierstrass equation of elliptic curve be defined? If $K$ is a local field, valuation takes discrete value, so we can define a minimal Weierstrass equation. But I heard ...
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1answer
80 views

Is it possible to find a circle with exactly 5 rational points?

Can we find a circle in $\mathbb{R}^2$ with exactly 5 points with rational coordinates? What is obvious is that a circle with a rational center and a rational radius has infinitely many rational ...
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1answer
49 views

Intuition behind supersingular reduction of elliptic curves

I would like to know if there's an obvious (or almost obvious) reason why we should expect elliptic curves over $\mathbb{Q}$ to have supersingular reduction at infinitely many primes. Thank you for ...
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1answer
93 views

Isogenies of Elliptic curves with complex multiplication

This is a slight continuation of a previous question of mine. Given an elliptic curve $E$ over $\mathbb{Q}$ which has complex multiplication. How would one find each $p$ such that $E$ admits a $p$-...
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1answer
25 views

Number of points over a curve over finite fields

Artin's conjecture is that for $f$ a separable polynomial irreducible modulo $p$ of degree 3, the equation $y^2 \equiv f(x) \mod p$ has a number of solutions $n$ satisfying $$n = p + O(\sqrt{p}).$$ I ...
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60 views

Historical references for Birational Geometry.

I'm interested in learning more about the mathematical history of birational geometry. I'm writing my thesis about certain obstructions to various types of rationality (ie. intermediate jacobians, ...
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1answer
43 views

$H^1(E(C), Z_\ell)$ is isomorphic to $T_\ell(E(C))$ and Lefschetz principle

I know $H^1(E(C), Z_\ell)$ is isomorphic to $T_\ell(E(C))$ as group. I heard there are principle known as the lefschetz principle. The legschetz principle says, roughly, that algebraic geometry over ...
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1answer
74 views

Show that $\mathcal C : Y^2=X(X^2-pX+p^2)$ has rank $0$, where $p$ is a prime $\equiv$ 2 (mod 3) and 5 (mod 8)

The question is from a masters' level elliptic curves exam (Oxford, 2017): Let $p$ be a prime congruent to 2 (mod 3) and 5 (mod 8). Show that the elliptic curve $\mathcal C : Y^2=X\left(X^2-pX+p^2\...
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37 views

To prove $Φ$:$\Bbb C\to E(\Bbb C)$ : $t\mapsto (\wp(t),\wp'(t))$ is a group hom

This is a question from Silverman's 'the arithmetic of elliptic curves', p171. If $\Lambda$ is a lattice in $\Bbb C$ the map $$z\mapsto (\wp(z),\wp'(z))$$ is a parametrisation of the complex points of ...
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2answers
52 views

Definitions of CM abelian varieties

I was reading through a presentation by Oort (https://www2.math.upenn.edu/~chai/UPenn2013-beamer.pdf) and noticed something which disturbed me: he defines (slide 38) a simple CM abelian variety to be ...
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2answers
120 views

Why $\mathbb{Z}_{p}[p^{1/p^{\infty}}]/p$ $\cong$ $\mathbb{F}_{p}[t^{1/p^{\infty}}]/t$?

I'm reading Peter Scholze's Perfectoid Space, and I'm confused with the isomorphism $\mathbb{Z}_{p}[p^{1/p^{\infty}}]/p$ $\cong$ $\mathbb{F}_{p}[t^{1/p^{\infty}}]/t$. What is the meaning of $\mathbb{Z}...
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How to understand $\mathbb{Q}_{p}(p^{1/p^{\infty}})$?

It is known that $\mathbb{Q}_{p}(p^{1/p^{\infty}})$ is defined to be $\bigcup_{n>0} \mathbb{Q}_{p}(p^{1/p^{n}})$, which means adjoining all $p$-power roots of $p$ to the mixed characteristic field $...
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27 views

Group action on ringed space

I am trying to understand the action of a finite group on a ringed space. let $G$ be a finite group acting on $(X, O_X)$. I know that $g \in G $ induces a morphism $g: X\longrightarrow X$ and for the ...
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1answer
120 views

Computing the endomorphism ring of an elliptic curve over a finite field (using SAGE)

$ \newcommand{\End}{\mathrm{End}} \newcommand{\Gal}{\mathrm{Gal}} \newcommand{\kb}{\overline{k}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\F}{\mathbb{F}} \newcommand{\Q}{\mathbb{Q}} $ I would like to ...
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63 views

The Segre embedding and addition on elliptic curves

Let $$E:Y^2Z=X^3+AXZ^2+BZ^3$$ be an elliptic curve with $A,B\in \mathbb{Z}$. I would like to understand the map \begin{align} G:E\times E &\to E\times E\\ (P,Q)&\mapsto (P+Q,P-Q). \end{align} ...
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72 views

Finite type scheme over a ring of $S$-integers is separated?

I have a question in the following setting: Let $k$ be a number field with ring of integers $O_k$. For a finite set of places $S$ of $k$ we can form the ring of $S$-integers $O_{k,S}$. Let $\mathbb{A}...
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0answers
63 views

How to calculate which primes have good reduction?

I'm trying to find for which primes the elliptic curve over $\mathbb{Q}$ defined by $y^2z+yz^2=x^3-xz^2$, for example, has good reduction. I think the way to go would be to find the minimal ...
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0answers
25 views

Transcendence degree of intersection of Field extensions

So my question goes as follows: Suppose we have the function field $\mathbb{C}(x)$ (no particular reason for choosing $\mathbb{C}$ - in principle, choosing $\mathbb{Q}$ might be a better idea), where $...
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49 views

A question on the Chow-Kunneth decomposition of a smooth hypersurface.

Let $K$ be an algebraically closed field. There is for any non-singular hypersurface $Y \subseteq \mathbb{P}^n_K$ a "Chow-Kunneth decomposition" (see page 38 in the link L1 below). Given any ...
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39 views

$\Gamma$-equivalence definition of zero-cycles

Let $\Gamma$ is a smooth projective curve, $x_1, x_2$ two distinc fixed points on $\Gamma$, and $X$ a smooth projective variety. In definition-Lemma 1 Roitman paper "On $\Gamma$-equivalence of $...
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54 views

In the process of counting the number of supersinguar elliptic curve

Let $φ:E→E$ be $q$-th Frobenius map and $φ'$ be it's dual isogeny. And we define $a$ as $a=1-deg(1-φ)+deg(φ)$(what we call trace of frobenius). Then, My question is , Why $[a]=φ+φ'$ ? These questions ...
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1answer
29 views

$E$ is supersingular if and only if dual of $p^r$ -Frobenius map $F:E\to E^{(p^r)}$ is inseparable?

Using the definitions that an elliptic curve $E$ over a finite field $K$ of characteristics $p$ is supersingular if dual of $p^r$-Frobenius map $F:E\to E^{(p^r)}$ is purely inseparable for all $r≧1$, ...
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2answers
38 views

supersingular if and only if $E[p]\cong 0$?

Using the definitions that an elliptic curve $E$ over a finite field $K$ of characteristics $p$ is supersingular if $E[p^r]=0$ for all $r≧1$, how can I show that $E$ is supersingular if and only if $E[...

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