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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What are some topics of advanced number theory every young geometers should know? (soft question)

By "advanced number theory", I mean topics like arithmetic/Diophantine geometry, modular/automorphic forms and Shimura varieties. I'm interested in derived/non-commutative algebraic geometry, some ...
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Tate curve and cusps

I know this is a naive question, but what is the relation between the Tate curve and cusps on a modular curve? Naive googling seems to suggest that level structures on the Tate curve (up to ...
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458 views

A Guide to the Proof of Fermat's Last Theorem from the Modularity Theorem

A few years ago, a friend of mine told me that he had taken an advanced undergraduate number theory class (so something that assumed only a knowledge of algebra and mathematical maturity) which ended ...
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Contraction of curves on a surface and stalks

Let $$f:X \rightarrow S$$ be a fibered surface over a Dedekind scheme of dimension $1.$ Let $$s_1, \ldots, s_n$$ be closed points of S and $\{E_{ij}\}$ irreducible vertical divisors of $X$ with $E_{ij}...
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101 views

Trivialising cover for étale morphisms

Let $f:Y \to X$ be a finite étale morphism of smooth and proper schemes over a field $k$ (not necessarily separable closed). Is there a geometrically connected étale cover $\{U_i\}$ of $X$ which ...
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If $X\times C$ is isomorphic to $Y\times C$, does it follow that $X$ is isomorphic to $Y$

Let $k$ be a field. Let $X$, $Y$ and $C$ be varieties over $k$ such that $X\times_k C $ is isomorphic to $Y\times_k C$. Assume that $C$ is a curve. (The varieties $X$, $Y$ and $C$ are assumed to be ...
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Curve - non singular curve and its genus

Help me please with this problem:$ X \subset \mathbb{P}^{2}$ defined as $x^{3}y+y^{3}z+z^{3}x=0$ 1.Prove X - non singular curve and find its genus. 2.Prove X - maximal curve over $F_{8}$ field, and ...
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On local rings of a normalization

Let $X$ an irreducible singular curve with a singular point $x$. Consider $A$, the normalization of $\mathcal{O}_{X,x}$, and $x_1,\dots,x_r$ the points over $x$ in the normalization of $X$. Why $A\...
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find valuations

consider $f(x)=x^4+5x^3+1$. Let $\alpha$ be a root of this polynomial and consider $K=\mathbb{Q}(\alpha)$. Which are the absolute values on $K$ extending the usual absolute values on $\mathbb{Q}$? Are ...
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457 views

What is the motivation behind the Hilbert Symbol?

The Hilbert Symbol it superficially similar to the Legendre Symbol: it measures whether or not solutions to some polynomial exist. In the case of the Legendre Symbol it was clear for me that it is a ...
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Bad reduction at prime numbers for the elliptic curve $y^2+y=x^3-x^2+2x-2$

Consider the elliptic curve $$E:y^2+y=x^3-x^2+2x-2.$$ My goal is to compute the conductor of the elliptic curve, the example is from https://planetmath.org/conductorofanellipticcurve. My problem isn'...
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How can function fields have different degrees over the projective line

I'm confused. Let $X$ be a curve over a field $k$. Let $K= K(X)$ be its function field. Then, $K(X)$ is a field. Each non-constant morphism $f:X\to \mathbf{P}^1_k$ gives a field extension $K(\...
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147 views

For curves, is being defined over a number field invariant under birational equivalence

Suppose that a (connected) Riemann surface $X$ is birational to a Riemann surface $Y$ which can be defined (algebraically) over the field of algebraic numbers. Does this imply that $X$ itself can be ...
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226 views

Does there exist a finite morphism of algebraic curves such that…

Let $K\subset L$ be a finite field extension. Let $X$ and $Y$ be (smooth projective geometrically connected) curves over $L$. Let $f:X\to Y$ be a finite morphism of curves over $L$. Assume that $Y$...
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Infinite family of genus one non-elliptic curves over the rationals

How easy is it to write down genus one curves over $\mathbf Q$ without a rational point? Can we write down an infinite family?
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594 views

the elliptic curves with j-invariant zero

Let $B\in K^\ast$, where $K$ is a number field. Let $y^2=X^3+B$ be the Weierstrass equation for an elliptic curve $E_B$ over $K$. Note that the $j$-invariant of $E$ is zero. When is $E_B$ ...
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When is this quotient by an action on the product of a variety with itself non-singular

Let $X$ be a smooth projective geometrically connected variety over a field $k$. Let the cyclic group $G=\{e,a\}$ with two elements act on $X \times X$ via $a\cdot (x_1,x_2) = (x_2,x_1)$. When is ...
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900 views

Is there a fundamental domain for $\Gamma(2)$ contained in the following strip

Let $\Gamma(2)$ be the subgroup of $\mathrm{SL}_2(\mathbf{Z})$ satisfying the usual congruence conditions. It acts on the complex upper half-plane. Does it have a fundamental domain contained in the ...
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853 views

Reduction map from the generic to the special fibre

I have a few basic questions about Liu's [1, Section 10.1.3] description of the reduction map from the closed points of the generic fibre of a proper scheme over a complete DVR to its special fibre. ...
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Construction of arbitrary regular polygons with ruler and compass

A lot of arithmetic geometry has to do and took off with constructions or proofs of non-constructibility of specific figures only with ruler and compass. See e.g. the non-constructibility of the ...
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Analogies between Hodge conjecture and Tate conjecture

I hear sometimes that there is many analogies between the Hodge conjecture and the Tate conjecture. If we take a look at the statements of this two conjectures, we have the followings : The Tate ...
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1answer
129 views

Showing that a Severi-Brauer Variety with a point is trivial

Let $X/k$ be a variety over a field such that $X_{\overline k} \cong \mathbb P^n$ over $\overline k$ for some $n$. Suppose moreover that $X$ has a rational $k$-point $P$. Then, I know that $X \cong \...
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Reduction map of elliptic curves

$\newcommand{\b}[1]{\overline{#1}}$ $\newcommand{\m}[1]{\,(\text{mod }#1)}$ $\def\red{\tilde{E}_{\mathfrak{p}}}$ $\def\p{\mathfrak{p}}$ $\def\at{_{\mathfrak{p}}}$ $\def\comp{K_{\mathfrak{p}}}$ I'm ...
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237 views

Torsors under elliptic curves splitting over the same fields

I have a question somewhat related to my last question. Suppose $C$ and $C'$ are two genus $1$ curves (smooth, projective, geom conn.) over a perfect field $k$ with no $k$-rational points and that $C$ ...
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774 views

genus of normalization of stable curve

Let $X$ be a stable curve of genus $g$ over the field $k$, i.e., a $k$-rational point of the Deligne-Mumford stack $M_g$. What is the genus of the normalization of $X$? Does it depend on the number ...
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125 views

Twists of rational points

Let $X$ be a "nice" algebraic curve over some field $K$, say characteristic zero. The twists of $X$, i.e., the curves $Y$ over $K$ such that $X_{\overline K} \cong Y_{\overline K}$, are in bijection ...
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345 views

“Reduction modulo $p$” of a scheme

Suppose $\chi$ is a scheme of finite type over $\mathbb{Z}$. I sometimes see the notion of "reduction modulo $p$ of $\chi$" (which I will denote by $\chi_p$). What is meant here ? Is it just $\chi \...
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80 views

Diophantine equation related to coset representatives of $\Gamma_0(N)\backslash SL_2(\mathbb Z)$

I am trying to verify the coset representatives of $\Gamma_0(N)\backslash SL_2(\mathbb Z)$ used in the proof of Proposition 1.43 in Introduction to the Arithmetic Theory of Automorphic Functions by ...
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89 views

Varieties with the same universal cover

Let $X$ and $Y$ be smooth projective varieties over $\mathbb C$ which have the isomorphic universal covers (in the analytic category). It's simple to see that $\dim X = \dim Y$. Is the Kodaira ...
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Irreducible Plane Curves Over the Algebraic $p$-adic Numbers [closed]

Let $A=\overline{\mathbb{Q}}\cap \mathbb{Q}_p$ be the relative algebraic closure of $\mathbb{Q}$ in the field of $p$-adics. Is there an example of an irreducible plane curve $C$ given by $f(x,y)=0$ ...
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156 views

Divisor on an arithmetic surface and “base change”

Fix a number field $K$ with ring of integers $O_K$; moreover $\sigma:K\to \mathbb C$ is an embedding of fields. Let $X\to\operatorname{Spec} O_K$ be an arithmetic surface ($X$ is regular and ...
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112 views

Elliptic curves over $\mathbb C$ have same endomorphism ring but not isomorphic

I am finding elliptic curves over $\mathbb C$ have same endomorphism ring but not isomorphic. Elliptic curves over $\mathbb C$ can be identified with $\mathbb C/\land$ for some lattice $\land$. And ...
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105 views

Lifting points of étale group scheme.

Consider the following setting: let $(R,\mathfrak{m})$ be a Noetherian complete discrete valuation ring (with maximal ideal $\mathfrak{m}$) and let $K$ be its field of fractions. Now take $L$ be the ...
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Size of automorphism group of element of $Y_0(5)$

The modular curve $Y_0(5)$ parametrizes elliptic curves $E$ with an isogeny of degree five. So an element of $Y_0(5)$ can be interpreted as $E \xrightarrow{\phi} E'$. Suppose we are working over a ...
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74 views

Does this diagram of Chern classes and push forwards commute

Let $p:Y\to X$ be a birational proper surjective morphism of regular surfaces, and let $D$ be a divisor on $Y$ such that $p(D)$ is a point. Then $p_\ast D =0$ by definition. Is there an easy way to ...
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413 views

Formula for index of $\Gamma_1(n)$ in SL$_2(\mathbf Z)$

Is there a precise formula for the index of the congruence group $\Gamma_1(n)$ in SL$_2(\mathbf Z)$? I couldn't find it in Diamond and Shurman, and neither could I find an explicit formula with a ...
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134 views

Does de Franchis' theorem hold over any base field

Let $k$ be a field and let $X$ be a hyperbolic curve over $k$. Then, there are only finitely many hyperbolic curves $Y$ over $k$ dominated by $X$. I know this statement holds over $k=\mathbf{C}$. In ...
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281 views

units in discrete valuation rings

Let $B/A$ be an extension of discrete valuation rings such that the purely inseparable extension of residue fields $l/k$ has a primitive element, i.e., $l=k(y)$ for some element $y$ in $l$. I want to ...
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52 views

Quotients and exact sequences of algebraic group schemes

Now I study group schemes to understand fundamental properties about semi abelian varieties and generalized jacobian varieties. Let $G$ be an algebraic group scheme over a field $k$ ($=$ $k$ group ...
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1answer
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Finding the integral closure of $k[x]$ in $k(x)(\sqrt{f})$, where $f(x)=x^6+tx^5+t^2x^3+t\in k[x]$.

This is an exercise (1.15) of Dino Lorenzini' s An Invitation to Arithmetic Geometry. Let $F$ be a field of characteristic $2$. Let $k:=F(t)$, the field of rational functions in the variable $t$. Let ...
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Questions about the Neron-Ogg-Shafarevich criterion

One version of the Neron-Ogg-Shafarevich criterion for abelian varieties says that for a local field $K$ with valuation ring $R$ and perfect residue field $k$ and an abelian variety over $A$, $A$ has ...
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456 views

Multiplication by $m$ isogenies of elliptic curves in characteristic $p$

I've been attempting to prove some comments I've read on MO by myself for my undergrad thesis regarding étale morphisms of elliptic curves. My definition of an étale morphism is taken from Milne's ...
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Why is $\mathcal{O}_F \otimes \mathbb{Z}_p$ a Dedekind ring?

In the paper Compactifications de l'espace de modules de Hilbert-Blumenthal (Compositio. '78), M. Rapoport relates some assertions from a letter of Deligne to Serre, dated 1972. At the very beginning, ...
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195 views

Punctured Elliptic Curve

I've come across the word "punctured elliptic curve" here and there, but none of the basic texts on the topic (Husemoller, Silverman) define or mention it. What point is removed from the curve (the ...
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Can someone suggest a path to study Mordell-Weil theorem for someone studying on their own?

So, I want to read the proof of Mordell-Weil theorem and so, I picked up the book 'Arithmetic of Elliptic Curves' by J. Silverman and J.S. Milne's Elliptic Curves book. But after going through both ...
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Formal completion of modular curves

Let $N\geq 4$ be an integer coprime with $p$, where $p$ is a fixed prime number. Then we know that there exists a scheme $Y_N$ over $\text{Spec}(\mathbb{Z}_p)$ whose $\text{Spec}(R)$-points are ...
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Pushforward of sheaf of relative differentials in family of elliptic curves [duplicate]

Update: never mind, This question has been asked before here Let $f:E \to S$ be an elliptic curve (precise definition is given below from Hida’s book Geometric Modular Forms and Elliptic Curves). Is $...
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Qing Liu exercise 4.1.3: Non-affine Dedekind scheme

Exercise 4.1.3 of Qing Liu's Algebraic Geometry and Arithmetic Curves asks of us to prove, given a normal Noetherian local scheme $X$ of dimension $2$ with closed point $s$, that $X \backslash \{ s \}$...
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171 views

4-Torsion on an elliptic curve embedded in $\mathbb{P}^{3}$ as the intersection of two quadrics

I'm having trouble with exercise 3.10 from Silverman's Arithmetic of Elliptic Curves. I include part (a) for context and part (d) because that's the part I'm stuck on. All fields in this section are ...
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126 views

Néron-Severi groups

Let $X$ be a smooth projective variety over the complex numbers. Let $$c_1 : \text{Pic}(X)\to H^2(X,\mathbf{Z}(1))$$ Its kernel is the subgroup of homologically trivial divisor classes on $X$. How ...