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14
votes
3answers
935 views

Important papers in arithmetic geometry and number theory

Having been inspired by this question I was wondering, what are some important papers in arithmetic geometry and number theory that should be read by students interested in these fields? There is a ...
1
vote
1answer
68 views

automorphisms of varieties with respect to a cover

Let $X$ and $Y$ be (smooth projective connected) varieties over $\mathbf{C}$. Let $\pi:X\to Y$ be a finite surjective flat morphism. Does this induce (by base change) a map $\mathrm{Aut}(Y) \to ...
1
vote
0answers
78 views

invert Grothendieck spectral sequence

I have 4 topoi $A,B,C,D$ (these are associated to abelian sheaves on some sites) and functors (which corresponds to some push-forward of sheaves) $F: A\rightarrow B$ $G: B \rightarrow C $ $H: A ...
5
votes
1answer
237 views

A question about modular curves and base change

Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Suppose that the curve $X\times_{K,\sigma} \mathbf{C}$ is a modular curve for some $\sigma:K\to \mathbf{C}$. Can ...
7
votes
0answers
235 views

Do Neron models of hyperbolic curves exist

Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Assume that $g\geq 2$. Does there exist a Neron model $\mathcal X$ for $X$ over $O_K$? By a Neron model, I mean ...
2
votes
1answer
61 views

If the reduction is smooth and projective, can I conclude the same about the scheme

Let $X$ be a $R$-scheme, where $R$ is a dvr. Suppose that the reduction of $X$ (over the closed point of $\mathrm{Spec} \ R$) is smooth and projective. Does this imply that $X$ is smooth and ...
5
votes
1answer
133 views

Very special rational points on curves over number fields

For some reason, I'm convinced the answer to the following question should be (obviously) negative, but I can't come up with a good reason. Does there exist a number field $K$, a smooth projective ...
7
votes
1answer
216 views

Does every curve over a number field have infinitely many rational functions of fixed degree

Let $X$ be a curve over a number field $K$ of genus $g\geq 2$. Does there exist an integer $d$ such that $X$ has infinitely many rational functions (i.e., finite morphisms $f:X\to \mathbf{P}^1_K$) of ...
8
votes
1answer
3k views

Translation for EGA/SGA

People often recommend Grothendieck's EGA (Elements de Geometrie Algebrique) and SGA (seminaire de geometrie algebrique) as a good reference for learning arithmetic geometry. However, as the title ...
4
votes
1answer
90 views

The universal cover of the multiplicative group over the field of algebraic numbers

Let $X=\mathbf{A}^1_{\overline{\mathbf{Q}}}-\{0\} = \mathbf{G}_{m,\overline{\mathbf{Q}}}$ be the multiplicative over the field of algebraic numbers. Each finite etale cover $Y\to X$ (with $Y$ ...
1
vote
1answer
67 views

Defining invariants of varieties over fields

Let $f$ be a real-valued function on the set of curves over $\overline{\mathbf{Q}}$. Assume that isomorphism curves give rise to the same value for $f$. Let $K$ be a number field and let $X$ be a ...
8
votes
1answer
135 views

Flatness of residual representations associated to modular forms

Let $f\in S_k(\Gamma_1(N),\chi)$ be a Hecke eigenform of weight $k\geq 2$, $p$ an odd prime not dividing $N$, and $K_f$ the number field generated by the Hecke eigenvalues of $f$. Fixing a prime ...
12
votes
2answers
1k views

Why is Hodge more difficult than Tate?

There are strong connections between the Hodge and the Tate conjectures, mainly at the level of similarities and analogies. To quote from an answer of Matthew Emerton on MathOverflow: "[...] we ...
3
votes
1answer
83 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 ...
3
votes
1answer
389 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 ...
1
vote
1answer
140 views

Can a non-proper variety contain a proper curve

Let $f:X\to S$ be a finite type, separated but non-proper morphism of schemes. Can there be a projective curve $g:C\to S$ and a closed immersion $C\to X$ over $S$? Just to be clear: A projective ...
7
votes
1answer
200 views

Is the Fermat scheme $x^p+y^p=z^p$ always normal

Let $K$ be a number field with ring of integers $O_K$. Is the closed subscheme $X$ of $\mathbf{P}^2_{O_K}$ given by the homogeneous equation $x^p+y^p=z^p$ normal? I know that this is true if ...
2
votes
0answers
57 views

What are the branch points of $X(n)\to X(1)$

Let $\Gamma \subset \mathrm{SL}_2(\mathbf{Z})$ be a finite index subgroup. Let $X_\Gamma \to X(1)$ be the corresponding morphism of compact connected Riemann surfaces (obtained by adding the cusps). ...
3
votes
0answers
64 views

What applications does the theory of fibered surfaces have

Let $C$ be a smooth projective connected curve over $\mathbf{C}$. Let $X$ be a curve over the function field of $C$. Arakelov and Parshin proved the Mordell conjecture by considering a model for $X$ ...
2
votes
0answers
80 views

Is the degree of a Galois morphism bounded by $84(g-1)$

Let $X\to Y$ be a finite morphism which is Galois in the category of smooth projective connected curves over $\mathbf{C}$. Assume $g=g(X) \geq 2$. Is the degree of $X\to Y$ bounded by $84(g-1)$? I ...
0
votes
1answer
154 views

Does the absolute Galois group act on the moduli space of curves

Do elements of the absolute Galois group $G=\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ of $\mathbf{Q}$ induce automorphisms of $\mathbf{C}$ if we extend the morphisms trivially to the rest of ...
3
votes
1answer
108 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 ...
3
votes
2answers
181 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 ...
3
votes
1answer
137 views

Are these two notions of Galois morphism the same

Let $f:X\to Y$ be a finite morphism of integral schemes. Let $G$ be the automorphism group of $X$ over $Y$. Are the following two conditions equivalent? The function field extension $K(Y)\subset ...
4
votes
0answers
96 views

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 ...
11
votes
3answers
481 views

Is a cover Galois if and only if it is geometrically Galois

Let $K$ be a number field and let $\pi:X\to \mathbf{P}^1_K$ be a finite morphism, where $X$ is a smooth projective geometrically connected curve. Is $\pi$ a Galois cover if and only if the base ...
3
votes
0answers
82 views

Does composing the Frobenius with an automorphism give another Frobenius

Let $X_0$ be a variety over $\mathbf{F}_q$. Consider the Frobenius $F_0:X_0\to X_0$. Let $X= X_0\times \bar{\mathbf{F}_q}$ and let $F:X\to X$ be $F_0 \times \textrm{id}$. Let $f:X\to X$ be an ...
3
votes
0answers
119 views

Why is the trace map on an abelian variety continuous

Let $X$ be a (EDIT) variety with a group structure. For $a\in A$, let $t_a$ be the translation on $X$: $t_a(x) = a+x$. Why is the function $f:X\to \mathbf{C}$ given by $$f(a) = \sum_{i} (-1)^i ...
2
votes
1answer
151 views

how does one intersect the diagonal with a graph on the surface $X\times X$

I want to do a concrete example of an intersection product for myself. Consider the endomorphism $f:\mathbf{P}^1_k\to \mathbf{P}^1_k$ given by $(x:y)\to (y:x)$. It has precisely two fixed points: ...
5
votes
1answer
253 views

The normalization of the peculiar curve $x^p + y^p - (x+y)^p = 0$

Fix a prime number $p$. Consider the affine curve $C$ in $\mathbf{A}^2$ over a number field $K$ given by the equation $x^p+y^p - (x+y)^p =0$. Its Jacobi matrix is $(px^{p-1} -p(x+y)^{p-1} \ py^{p-1} ...
3
votes
0answers
180 views

Minimal resolution of singularities of Fermat curve

Fix a prime number $p$. Let $F$ be the geometrically connected Fermat curve of exponent $p$ over a number field $K$, i.e., $F$ is given by the equation $x^p+y^p = z^p$. Consider the same equation ...
3
votes
1answer
185 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 ...
6
votes
1answer
173 views

Computing the trace of the following automorphism of the elliptic curve $y^2 = x^3+x$

Consider the elliptic curve $E$ defined by $y^2z= x^3 +xz^2$ over an algebraic closure $\overline{\mathbf{Q}}$ of $\mathbf{Q}$. Consider the endomorphism $f:E\to E$ given by $(x:y:z)\mapsto ...
12
votes
1answer
370 views

Working with Morphisms in Local Coordinates

In light of the holiday, I would like to air a grievance. I have no good way to recoordinatize a morphism of varieties as I move between coordinate neighborhoods. Let me explain what I mean with ...
13
votes
1answer
382 views

2-Torsion Group Scheme

Consider the elliptic curve $zy^2 + z^2y = x^3.$ I would like to explictly compute the 2-torsion group scheme, $E[2],$ over $\mathbf{Spec}(\mathbb{Z}_2),$ but I'm having a tough time writing down the ...
1
vote
2answers
132 views

can singular points become nonsingular after a base change

Let $X$ be a normal surface over a field $k$. Assume that $X$ is singular. Does there exist a field extension $L/k$ (finite or infinite) such that $X_L$ is nonsingular? The answer is no in general. ...
10
votes
2answers
578 views

In what senses are archimedean places infinite?

According to Bjorn Poonen's notes here (§2.6), we should add the archimedean places of a number field $K$ to $\operatorname{Spec} \mathscr{O}_K$ in order to get a good analogy with smooth projective ...
3
votes
1answer
220 views

Characterization of Horizontal Irreducible Divisors

I´m asking for a proof of a fact used by Arakelov in his paper: Intersection Theory of Divisors on an Arithmetic Surface (page 1169 row 16). He gives no references or explanations for this fact. The ...
10
votes
1answer
135 views

Which number fields allow higher genus curves with everywhere good reduction

The field of rational numbers is not such a number field. That is, there does not exist a smooth projective morphism $X\to\text{Spec } \mathbf{Z}$ such that the generic fibre is a curve of genus ...
4
votes
1answer
213 views

discriminant of an étale cover of an elliptic curve

Let $\pi:X\to E$ be a finite étale morphism, where $E$ is an elliptic curve over a number field $K$. Assume $X$ to be connected, and to be of genus 1. Edit: Assume $X$ and $E$ have semi-stable ...
7
votes
1answer
182 views

Twists of curves over number fields

Let $X$ be a curve over $\overline{\mathbf{Q}}$. I can prove that $X$ can be defined over some number field. (Take two equations defining $X$ in $\mathbf{P}^3$ and consider the number field ...
2
votes
1answer
82 views

local field extension

let $k$ a perfect field and consider the following field $F= k((\omega))$, complete w.r.t a valuation for which $\omega$ is a uniformizer. Consider the field extension $E=F[T]/(T^2-\omega^3)$. Is it ...
4
votes
2answers
212 views

isomorphisms of algebraic closures

let $K$ be an algebraically closed field. Consider the algebraic closure $\overline{K(X)}$ of $K(X)$, with $X$ trascendent over $K$. Are there cases in which $\overline{K(X)}\cong K$? where $\cong$ is ...
23
votes
4answers
799 views

Why is it “easier” to work with function fields than with algebraic number fields?

I just bought a copy of Jürgen Neukirch's book Algebraic Number Theory. While browsing through it I found a section titled § 14. Function Fields in chapter I. In it the author describes ...
7
votes
3answers
824 views

What is the operation $\boxtimes$?

Reading papers about $p$-adic analysis and Galois representations, I have found objects like this $D \boxtimes \mathbb{Q}_p$. So my question is what is $\boxtimes$ and how do we read it ?
4
votes
1answer
185 views

$\pi^{tame}(\mathbb{A}^1_k)$ is trivial

Fixed an algebraically closed field of characteristic $p>0$, it is well known the result of the title: $\pi^{tame}(\mathbb{A}^1_k)\simeq 1$. Where the tame fundamental group, in this situation, ...
0
votes
1answer
78 views

theory for contraction maps on curves

do u know any reference for studying the maps $\bar{M}_{i,n}\rightarrow \bar{M}_{j,n-2}$ (moduli space of stable curves with marked points)where the map is given by identifying two marked points? ...
4
votes
5answers
358 views

Derived category and so on

I am looking for an introductive reference to the theory of derived categories. Especially I need to start from the very beginning and I need to know how to use this in examples which comes from ...
6
votes
3answers
352 views

Analogies between Prime Ideals and Knots

While reading this question posted at this link: Subjects studied in number theory i interestingly landed up on this Wikipedia page, and was quite amazed to see the variety of branches opening up. ...
36
votes
1answer
783 views

Geometric intuition behind The Mordell Conjecture

The Mordell Conjecture/Faltings Theorem says roughly that if $K$ is an algebraic number field and $X$ is an algebraic curve defined over $K$ of genus $g >1$ then the set of $K$-rational points ...