The arithmetic-geometry tag has no wiki summary.
16
votes
0answers
166 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
105 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 ...
3
votes
3answers
152 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
114 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
122 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 $Y(k) = X(k)$?
2
votes
2answers
87 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
125 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
69 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
93 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 ...
4
votes
1answer
130 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 ...
9
votes
1answer
423 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
501 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
42 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 ...
0
votes
0answers
59 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
158 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
160 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
47 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 ...
3
votes
1answer
114 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 ...
4
votes
1answer
135 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 ...
6
votes
1answer
339 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 ...
3
votes
1answer
64 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 $Y$ ...
0
votes
1answer
47 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
81 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 ...
10
votes
2answers
494 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
59 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
117 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
74 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
157 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
45 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
43 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
63 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
99 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 ...
2
votes
1answer
71 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
117 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
77 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 $K(Y)\subset ...
4
votes
0answers
69 views
Curve - non singular curve and its genus
Help me please with this problem:$ 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
342 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
96 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
56 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
68 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
69 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
88 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: ...
4
votes
1answer
167 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} ...
2
votes
0answers
130 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
116 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 ...
4
votes
1answer
112 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
321 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 ...
12
votes
1answer
245 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
117 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. ...
6
votes
2answers
235 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 ...
