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19
votes
0answers
330 views

Tate conjecture for Fermat varieties

I've been looking at Tate's Algebraic Cycles and Poles of Zeta Functions (hard to find online... Google books outline here) and have a question about his work on (conjecturing!) the Tate conjecture ...
2
votes
1answer
118 views

Can we descend field extensions of prime degree of number fields to number fields of the same degree

Let $K$ be a number field and let $p$ be a prime number. Let $L$ be a degree $p$ field extension of $K$. Does there exist a degree $p$ field extension $M$ of $\mathbf{Q}$ such that ...
4
votes
3answers
172 views

How do I write down a curve with exactly one rational point

Let $g\geq 1$. I would like to write down (for all $g$) a smooth projective geometrically connected curve $X$ over $\mathbf{Q}$ of genus $g$ with precisely one rational point. Is this possible? For ...
4
votes
2answers
191 views

Genus of curves embedded into some projective space

The Plucker formula allows one to calculate the genus of a curve embedded into $\mathbf{P}^2$. Does there exist a "higher-dimensional" analogue of the Plucker formula? More precisely, let ...
2
votes
2answers
198 views

Rational points on singular curves and their normalization

Let $X$ be a curve over a field $k$. Assume that $X$ is geometrically connected, geometrically reduced and stable. Let $Y\to X$ be the normalization. Is $Y(k) = X(k)$?
3
votes
2answers
174 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 ...
0
votes
2answers
146 views

Why are these curves not defined over a smaller field

Let $K$ be a number field and let $\pi$ be an element in $K$. Assume that $\pi$ is not contained in a subfield of $K$. Consider the curve $y^2 = x^{2g+1}+\pi$. This defines (after homogenization and ...
3
votes
1answer
78 views

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 ...
0
votes
1answer
142 views

Smallest genus example of a non planar curve

A curve is a smooth projective connected curve over an algebraically closed field. Every curve of genus 2 is planar. Also, every curve of genus 3 is planar. But what about curves of genus 4? What ...
5
votes
1answer
214 views

trivial Picard group

let $S=\operatorname{Spec}(A)$ be an affine scheme. For which ring $A$, not field is it known that $H^1(S,\mathcal{O}_S^{*})$ is trivial? If $X\to S$ is a finite map and $H^1(S,\mathcal{O}_S^{*})$ is ...
13
votes
1answer
872 views

What is an intuitive meaning of genus?

I read from the Finnish version of the book "Fermat's last theorem, Unlocking the Secret of an Ancient Mathematical Problem", written by Amir D. Aczel, that genus describes how many handles there are ...
14
votes
3answers
726 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
60 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
68 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
207 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
212 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
57 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 ...
4
votes
1answer
130 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 ...
6
votes
1answer
189 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 ...
7
votes
1answer
2k 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
78 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
56 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
113 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 ...
11
votes
2answers
837 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 ...
2
votes
1answer
75 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 ...
1
vote
1answer
239 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
112 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 ...
6
votes
1answer
183 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
52 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
58 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
73 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
136 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
98 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
160 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
115 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
94 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
439 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 ...
0
votes
0answers
131 views

What is the Hurwitz number of an elliptic curve

One can associate a Hurwitz number to any rational function $f:X\to \mathbf{P}^1$ on a compact connected Riemann surface $X$ which ramifies over precisely FOUR points. Suppose that $X$ is an elliptic ...
0
votes
0answers
73 views

What is the moduli space of smooth AFFINE curves of given genus

It is interesting to study the moduli space $M_g$ of smooth projective curves of genus $g$. Why not smooth affine curves of genus $g$?
3
votes
0answers
72 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 ...
2
votes
0answers
108 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
131 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
228 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
165 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 ...
1
vote
0answers
149 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
145 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
350 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
329 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
126 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. ...
9
votes
2answers
362 views

In what senses are archimedean places infinite?

According to Bjorn Poonen's notes here (ยง2.6), 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 ...