Tagged Questions

6
votes
1answer
46 views

Existence of a variety with prescribed properties

In these notes that give a proof of the Weil conjectures for curves, the author writes on page 17 that given a smooth projective curve $X$ over a finite field $k = \mathbb{F}_q$ for a fixed prime $q$, ...
1
vote
0answers
22 views

deg of composition on supersingular curve

Let we have supersingular curve $E(\bar{\mathbb{F}_q})$. Let we have algebraic function $f \in \bar{\mathbb{F}_q}(E)$ with div($f) = \sum_{i=0}^{i=n}n_iP_i$. Then div$(f) \circ [q] = ...
2
votes
0answers
31 views

number of solutions eqauation on supersingular elliptic curve

To Frobenius endomorphism on supersingular elliptic curve I want to prove that equation $\pi_q(X) = A$ has 1 solution for any point $A \in E(\bar{\mathbb{F}_q}))$ where $E$ is supersingular. Is it ...
1
vote
0answers
34 views

Frobenius endomorphism on supersingular elliptic curve

Let we have supersingular curve $E(\bar{\mathbb{F}_q})$. Is it true that for every point $P$ $q$-Frobenius endomorphism $\pi_q$ can be write as $A + [q]B$ where $P = A + B$? It is true if ...
5
votes
1answer
59 views

Function field question from Silverman's AEC

Just before Proposition 1.7 on page 5 of AEC (2nd ed), Silverman defines $M_P$ as an ideal in the affine coordinate ring. Then he states Proposition 1.7 (the intrinsic characterization of ...
4
votes
1answer
125 views

Fermat's Last Theorem in multiple variables

I was wondering if there was anything we could say about when, given $m$, $\exists n (\forall x_1,\dots,x_m \in \mathbb{N} ( x_1^n + x_2^n + \dots + x_{m-1}^n \neq x_m^n))$ Fermat's Last Theorem ...
5
votes
1answer
50 views

Why do varieties with torsion canonical sheaf have finite etale covers with trivial canonical sheaf

Let $B$ be a variety with torsion canonical sheaf, i.e., $\omega^{\otimes n}_B \cong \mathcal O_B$ for some $n>0$. Then, why does there exist a finite (etale?) morphism $X\to B$ such that $K_X$ is ...
3
votes
1answer
91 views

sum of three cubes and parametric solutions

The first paragraph in the following link asserts that the equation $x^3+y^3+z^3=2$ has finite many parametric solutions over $\mathbb{Q}$. In other words, there are finite many polynomial triples ...
4
votes
0answers
75 views

When does a modular form satisfy a differential equation with rational coefficients?

Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ for $\Gamma$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least ...
5
votes
2answers
192 views

A cohomological statement equivalent to the Riemann Hypothesis

Is there a possibility for looking for a theory of cohomology and an equivalent cohomological statement for Riemann hypothesis over $\mathbb{Z}$?
8
votes
1answer
280 views

Undergraduate roadmap for Langlands program and its geometric counterpart

What are the topics which an undergraduate with knowledge of algebra, galois theory and analysis learn to understand Langlands program and its goemetric counterpart? I would also like to know what are ...
3
votes
1answer
79 views

Does the Euler characteristic increase if I add an effective divisor

Let $D$ be an effective divisor on a smooth projective connected complex algebraic variety $X$. Suppose that $D\leq E$. Is it true that $$\chi(X,\mathcal{O}_X(D)) \leq \chi(X,\mathcal{O}_X(E))?$$
3
votes
0answers
31 views

What are the easiest surfaces of general type

The "easiest type" of curves of general type are those of genus two. In this case $\chi(X,\mathcal O_X) = -1$ and $\deg K = 2$, where $K$ is the canonical sheaf. I'm a bit lost when it comes to ...
5
votes
1answer
69 views

Ramification indices and residue degrees of a finite Galois extension

Let $A$ be a dvr with fraction field $K$ of characteristic zero. Let $L/K$ be a finite Galois extension and let $B$ be the integral closure of $A$ in $L$. For a prime $b$ of $B$, let $e_b$ be its ...
5
votes
1answer
78 views

Definition of tamely ramified

I think I can show that the following definitions of "tamely ramified" coincide. I thought it would be good to be sure. Sorry for the easy questions. Let $O_K$ be a dvr with maximal ideal $\mathfrak ...
3
votes
1answer
45 views

Showing that the map on $\mbox{Div}^0(E)$ induced by an isogeny takes principal divisors to principal divisors.

I'm doing a course on elliptic curves. An isogeny $\phi:E_1 \rightarrow E_2$ induces a map $$\begin{array}{llll}\phi_*: & \mbox{Div}^0(E_1) & \rightarrow & \mbox{Div}^0(E_2) \\ \\ & ...
2
votes
1answer
71 views

Proving the condition for two elliptic curves given in Weierstrass form to be isomorphic

I'm taking a course on elliptic curves and trying to understand the proof of Proposition 3.2. Let $E$, $E'$ be elliptic curves over $K$ in Weierstrass form: ...
4
votes
0answers
111 views

Defining the Riemann-Roch space of a divisor

I'm doing a course on elliptic curves. It starts with a bit of a crash course in algebraic geometry, giving statements alone. We were given the following definition The Riemann-Roch space of $D$ ...
6
votes
1answer
134 views

Doesn't Abhyankar's lemma contradict faithful flat descent (no, but I'm confused)

Let $O_K$ be a dvr with fraction field $K$. Let $L/K$ be a tamely ramified finite Galois extension. Then, Abhyankar's Lemma implies that there exists a finite Galois extension $K^\prime/K$ such that ...
2
votes
0answers
95 views

Diophantine equations/Diophantine Geometry

I am very knew to this site and I am eagerly waiting for solutions of: (1) Let $x$ be an algebraic number with degree $n > 1$. Then there exists only finitely many rational numbers $p/q$ (in ...
4
votes
1answer
52 views

What is Weil paring computing really?

I have trouble in understanding Weil paring on $N$-torsion points on an elliptic curve. Please see Wikipedia for the definition of Weil paring. I would like to know what Weil paring is computing ...
3
votes
1answer
104 views

In one version of the Modularity Theorem, what does “arise from modular forms” mean?

One version of Modularity Theorem says that The elliptic curves with rational $j$-values arise from modular forms. Where $$j(\tau)=1728\frac{g_2^3(\tau)}{\Delta(\tau)}$$ I know every ...
1
vote
1answer
32 views

Question about a property of elliptic function

Let $E=\mathbb{C}/\Lambda$ be a complex elliptic curve where $\Lambda=\omega_1\mathbb{Z}\oplus\omega_2\mathbb{Z}$ Let $f$ be a nonconstant elliptic function with respect to $\Lambda$. Let ...
2
votes
2answers
94 views

What does degree of an isogeny mean?

The book I'm reading doesn't provide the definition of degree of an isogeny and I failed to google it. Can anyone tell me?
5
votes
1answer
55 views

A question about complex tori.

Let $\Lambda$,$\Lambda'$ be two lattice in $\mathbb{C}$ and $m\neq 0\in\mathbb{C}$ satisfying $$ m\Lambda\subset\Lambda' $$ The, the book I'm reading says that by the theory of finite Abelian groups ...
1
vote
1answer
50 views

Why the kernel of isogeny is finite?

It is said that the kernel of a isogeny is finite because it is discrete and complex tori are compact. I have some questions about this. 1. Following is my reason for the kernel is discrete. ...
4
votes
2answers
137 views

Does this equation have integer solutions

Let $g\geq 2$ be an integer. (It will be the genus of some curve.) Are there positive integers $d$ and $e$ such that the equality $$ (e-2)(e-1) = 2d(g-1)+2$$ holds?
2
votes
1answer
108 views

Nef divisors on the compactified modular curve level $N$

Consider the compactified modular curve with full level structure $X=\overline{\Gamma(N)\setminus \mathcal{H}}$. We know the Hodge bundle (the extension of the hodge bundle to the compactification) ...
6
votes
1answer
119 views

Are there infinitely many pairs of rational numbers such that…

Are there infinitely many pairs of rational numbers $(a,b)$ such that $a^3+1$ is not a square in $\mathbf{Q}$, $b^3+2$ is not a square in $\mathbf{Q}$ and $b^3+2 = x^2(a^3+1)$ for some $x$ in ...
1
vote
0answers
17 views

group law on weil-chatelet group

Is there a reference for the gemetric definition of the group law on the Weil-Châtelet group of an Abelian variety more recent than the original Weil's paper ("On algebraic groups and homogeneous ...
2
votes
0answers
51 views

On CM Jacobians

I am looking for an example of a curve whose Jacobian is an abelian variety with complex multiplication by a non-abelian number field. Does anybody know such an example? Does it exist? Thanks
3
votes
0answers
117 views

Extensions of Mixed Hodge Structures

The analogy on the front page of this paper by Bloch and Kriz seems like it's going to be lovely, but I don't get it, because I don't know how to view a torsor for $\mathbb{Q}(1)$ as an extension of ...
2
votes
1answer
80 views

Primes of good reduction for varieties

Suppose I have a (smooth, projective) variety $X/\mathbb{Q}$. Is the notion of a primes of bad/good reduction well-defined from this data? Motivation and attempt at an answer: The question should be ...
4
votes
2answers
149 views

For what algebraic curves do rational points form a group?

For what real algebraic curves do rational points form a group ? How does this relate to Jacobian Varieties ?
12
votes
2answers
203 views

Existence of divisors of degree one on a curve over a finite field

Let $C$ be a smooth, geometrically irreducible projective curve defined over a finite field $\mathbb{F}_q$. Given a (scheme-theoretic) point $x \in C$, define the degree of $x$ to be the degree of ...
3
votes
1answer
119 views

Roots of rational equation with multiple variables?

Let's say we have a rational polynomial in $k$ variables. We are only interested in rational solutions. If $k = 1$, name the variables ${x}$, if $k = 2$, name them ${x,y}$. For $k = 1$, it can be ...
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 ...
6
votes
1answer
94 views

Relation between zeta value and genus of modular curve

This question is sort of vague, so I don't mind a vague answer. We have the special value formula $\zeta(-1)=-B_2/2 = -1/12$, where $\zeta$ is the Riemann zeta function. Also, the "genus" of the ...
12
votes
2answers
329 views

Motivation for stable curves

I was looking at Deligne-Mumford's paper on the irreducibility of the space of curves of a given genus, and it seems that they generalize the notion of a smooth curve to a "stable curve." I'm a little ...
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
136 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
228 views

modular forms and line bundle

Let $\Gamma \leq SL_2(\mathbb{Z})$ be a congruence subgroup, $X$ the corresponding compact modular curve. I often see the statement (for example, in many posts here on SE) that modular forms (of ...
9
votes
1answer
311 views

Stacks in arithmetic geometry [closed]

Stacks, of varying kinds, appear in algebraic geometry whenever we have moduli problems, most famously the stacks of (marked) curves. But these seem to be to be very geometric in motivation, so I was ...
5
votes
1answer
152 views

Extending Galois automorphism to group automorphism

Let $F \subset K$ be a field extension of degree $n$, then $F^n \cong K$ as $F$-vectorspaces. Now $K^\times$ acts on $F^n$, by multiplication on $K$, and so $K^\times$ embeds into $GL_n(F)$, and every ...
2
votes
1answer
124 views

Strong approximation in function fields

How does the strong approximation theorem for global function fields looks like? For the number field $\mathbb{Q}$ it can be expressed as the surjection $$ \mathbb{Q}^\times \times \mathbb{R}^\times ...
13
votes
1answer
942 views

Geometric Explanation of Tamagawa Numbers

Sometimes in order to understand a concept thoroughly we need to have a algebraic view ( in terms of equations ) and corresponding geometric view. My interest always lies with understanding the ...
0
votes
1answer
87 views

Number of points on a curve in a finite field

From Ireland and Rosen Number theory book(ch11.#11) Consider the curve $y^2=x^{3}-Dx$ over $\mathbb{F}_{p}$, where $D \not= 0$. Call this curve $C_{1}$. It can be shown that the substitution ...
8
votes
0answers
97 views

Homogeneous forms of degree $n$ in $n$ indeterminates over $\mathbb{Z}$: which ones come from the norm of a number field?

Is there a characterization of the homogeneous forms of degree $n$ in $n$ indeterminates over $\mathbb{Z}$ which occur as the norm of some algebraic number ring with a suitable $\mathbb{Z}$-basis? ...
21
votes
1answer
561 views

Deligne, elliptic curves and modular forms

I'm trying to understand an argument of Deligne (in Courbes elliptiques: Formulaire d'après J. Tate), but I'm not familiar enough with algebraic geometry, so I'm getting quite confused. So even in my ...
4
votes
1answer
269 views

Special privilege enjoyed by Elliptic Curves with Complex Multiplication

I think after reading the title one may understand the intention of me, this question is concerned about the Elliptic curves having a Complex Multiplication. I have been reading many theorems, ( ...

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