1
vote
0answers
44 views

Trivial Rost-Motive of a quadric

Let $q$ be the anisotropic,quadratic form of rank two corresponding to $\alpha = d(q) \in H^1(k,\mu_2)$. In his lecture notes "Topics in quadratic Forms" Vishik writes: For $n=1$ we get the ...
1
vote
0answers
23 views

Invariant of ramification under Mobius transformations

Is there some way of quantifying ramification of a cover up to Mobius transformations? i.e. If $K$ is an algebraically closed field of characteristic $p>0$, we can consider the Artin-Schreier ...
0
votes
1answer
53 views

Counting solutions mod p of a polynomial equation

Hello: Does somebody know if the following is true?: Let $f\in \mathbb{Z}[X]$ be a monic irreducible polynomial of degree $n$. Then there exists a positive integer $N$ and ...
5
votes
0answers
75 views

27 lines on a smooth cubic surface

It is known that every smooth cubic surface with coefficients in $\mathbb{Q}$ has $27$ lines defined over a number field extension of $\mathbb{Q}$ of degree at most $51840$ as the group ...
1
vote
1answer
54 views

Geometric meaning of being integrally closed in some overring

The geometric counterpart of integrally closed rings (in their fraction fields) are normal varieties, as described in this MathOverflow post. Is their a similar notion in algebraic geometry for being ...
2
votes
1answer
57 views

Existence of finite morphism to projective line

As we all know, Belyi's theorem says: A complex curve $X$ is defined over a number field, if and only if there exists a finite morphism $t:X\to \mathbb{P}^1_\mathbb{C}$ of varieties over $\mathbb{C}$ ...
6
votes
2answers
80 views

References for elliptic curves over schemes

As in the title, I want some references about theories for elliptic curves over rings(not fields) or over schemes. I heard that behaviours(?) of such elliptic curves are not as simple as elliptic ...
2
votes
1answer
108 views

Is there a “geometric” interpretation of inert primes?

I am currently learning about étale morphisms and how they behave a lot like covering maps. Now if I'm in the example of the ring of integers of a number field, it is possible to think of it as a ...
3
votes
1answer
59 views

Example 4.3.19 in Liu: unramification with schemes and numbers

In exemple 4.3.19 of Liu's book one hase $L/K$ an extension of number fields with integer rings $\mathcal{O}_L$ and $\mathcal{O}_K$, $\mathfrak{q}\subseteq\mathcal{O}_L$ a prime ideal and ...
4
votes
0answers
133 views

Difficulty of exercises in several books

Since I am teaching myself, I have no idea of the difficulties of exercises in many books. I want to know whether I am not understanding the content of these books if I cannot easily solve the ...
5
votes
1answer
122 views

How should I think about Ihara's Lemma?

I am trying to learn about Ihara's lemma because I see it mentioned in many papers in arithmetic geometry. To be more precise, I would like to know: What is the significance of this result? Why is ...
0
votes
1answer
95 views

Points on elliptic curves

I am learning elliptic curves theorem and I have read in more papers that for two distinct points $P$ and $Q$ there is always point $R$ such that $P+Q+R = \infty$. I know that this point should be ...
4
votes
1answer
146 views

Frey Curve as a Solution to FLT

I have read in many places that the Frey curve (if it existed) $y^2=x(x-A)(x+B)$ (or equivalently, $y^2=x(x-A)(x-C)$ corresponds to the solutions of $a^n+b^n=c^n$, where $A=a^n/c^n$ and $B=b^n/c^n$. ...
12
votes
2answers
169 views

Are there applications of noncommutative geometry to number theory?

The marriage of algebraic geometry and number theory was celebrated in the twentieth century by the school of Grothendieck. As a consequence, number theory has been completely transformed. On the ...
10
votes
1answer
229 views

Tate's Thesis: in what sense is Tate's Theorem 4.2.1 the Riemann-Roch theorem for curves?

I am reading Tate's Thesis. Tate derives a theorem which he calls "the number-theoretic analogue of the Riemann-Roch theorem" from an abstract Poisson summation formula. I am accustomed to thinking of ...
2
votes
1answer
67 views

How deep is the connection between $\mathcal{O}$ for a structure sheaf and $\mathcal{O}$ for a ring of algebraic integers?

So clearly there's some connection because in both contexts you have a ring and its function field involved; in one setting that's $\mathcal{O}_X(U)$ and $\mathcal{O}_{X, \eta}$ and in the other it's ...
4
votes
1answer
190 views

Rational Points on Elliptic Curves

I have this homework problem: Can there be an elliptic curve, view as a projective curve, with no rational points with at least one 0 as a coordinate?
5
votes
1answer
52 views

How to relate the valuation of x/y (For a minimal Weierstrass equation)

I'm reading an article about elliptic curves, but since I'm not very experienced on this subject, I ended up getting stuck. The problem starts as: "Let $K/\mathbb{Q}$ be a number field and $E/K$ an ...
4
votes
0answers
104 views

Modular Functions with Rational Fourier Expansions

I have been reading the paper of Cox, McKay and Stevenhagen "Principal Moduli and Class Fields", http://arxiv.org/pdf/math/0311202v1.pdf, and I have a question regarding the nature of the function ...
2
votes
1answer
62 views

Automorphism action on Artin map

I would like to refer you to Serge Lang's book on Elliptic Functions $\S10.4$ (page 140). I'm trying to understand a part in theorem 10 where he proves that $$\mu(\mathfrak{p})'=\mu(\mathfrak{p}')$$ ...
1
vote
0answers
49 views

What “projective space” is this?

Let $(a,b,c)\in \mathbb{R}^3$. Then given, $x, y, z$, let $\{(ap^x, bp^y, cp^z): p \in \mathbb{R}\}$ be an equivalence class. Has this space of points been studied in relation to solutions of ...
9
votes
1answer
272 views

what is a “dévissage” argument?

In Serre's "Local Fields" (Chapter 2, section 2, proposition 3) he proves something about field extensions which he breaks into parts: first he dealt with the separable case, and then with the ...
5
votes
1answer
129 views

Reduction modulo $p$ in number fields

For every prime number $p$, there exist a map $$f:\mathbb{P}^n(\mathbb{Q})\to\mathbb{P}^n(\mathbb{F}_p)$$ defined by: for $P\in \mathbb{P}^n(\mathbb{Q})$, we can find a unique tuple ...
12
votes
1answer
191 views

Introduction to the trace formula for people outside number theory

I am looking for references on the trace formula, by which I mean the Selberg trace formula and its successor the Arthur-Selberg trace formula. I am aware that there are "standard references" on the ...
4
votes
1answer
160 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 ...
6
votes
2answers
139 views

Usage of algebraic geometry in understanding the total Galois group of the rational

A professor of mine remarked in class that: "The tools via which we understand the total Galois group of the rationals comes from algebraic geometry". Could anyone shed some light on this remark, or ...
3
votes
0answers
46 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 ...
6
votes
1answer
115 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 ...
6
votes
1answer
406 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 ...
7
votes
1answer
218 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 ...
1
vote
0answers
37 views

Is $M_g$ a subvariety of $M_{h}$ for some $h>g$

Let $g\geq 24$. Then $M_g$ is of general type. Does there exist $h>g$ such that $M_g$ is a subvariety of $M_h$? That is, does there exist an immersion $M_g \to M_g$? If the answer is not known, ...
2
votes
0answers
31 views

Are there generalizations of Prym varieties to higher dimensions

Prym varieties are abelian varieties that are associated to a double cover of algebraic curves. Can we also associate an abelian variety to a double cover of algebraic surfaces in a reasonable way? ...
3
votes
2answers
390 views

Book recommendations for commutative algebra and algebraic number theory

Are there any books which teach commutative algebra and algebraic number theory at the same time. Many commutative algebra books contain few chapters on algebraic number theory at end. But I don't ...
4
votes
2answers
175 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?
7
votes
1answer
133 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
24 views

Can one determine in finite time whether a point is $S$-integral

Let $x$ be a $\mathbf{Q}$-rational point of $\mathbf{P}^1-\{0,1,\infty\}$. Let $S$ be a finite set of primes. How do I check in finite time whether $x$ is $S$-integral or not? I know how to do this ...
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 ...
2
votes
2answers
193 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 $Y(k) = X(k)$?
22
votes
1answer
513 views

Connectedness of the spectrum of a tensor product.

Let $A$, $B$ be finite free $\mathbb{Z}$-algebras such that $\operatorname{Spec}(A)$ and $\operatorname{Spec}(B)$ are both connected. Is $\operatorname{Spec}(A\otimes_{\mathbb{Z}} B)$ connected?
4
votes
1answer
99 views

DVR-valued points of schemes

Let $X$ be a scheme of finite type over a discrete valuation ring $R$ with fraction field $K$, such that the generic fibre $X_K$ is smooth over $K$. Let $Y$ be a closed subscheme of $X$ which contains ...
5
votes
1answer
194 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 ...
0
votes
1answer
97 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 ...
13
votes
0answers
182 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? ...
0
votes
0answers
308 views

What is a global section, and why do we use cohomology?

I have a few questions about the use of cohomology. Firstly,we use cohomology to measure the obstruction of a section from a global section, so what can we do about a global section? I got very ...
2
votes
1answer
176 views

Conjugacy classes in GL(n)

Given an element $\gamma$ in $GL(n,F)$, where $F$ is either a global field or a non archimedean local field. Assume $\gamma$ is elliptic, i.e. its characteristic polynomial irreducible. Let $Z(F)$ be ...
7
votes
1answer
346 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, ( ...
3
votes
1answer
123 views

Computing the free-part

I wanted to ask about some existing algorithms for computing points over elliptic curves. Background : We know that the famous theorem of Mordell and Weil says that " Group of rational points on an ...
8
votes
1answer
366 views

How does topology enter Number theory and how we can grasp its essence?

In infinite Galois theory,main theorem failed and we get a "Krull topology" to mend the main theorem, we even generalized that to make definition for profinite group. In local class field theory, the ...
6
votes
2answers
1k views

Local-Global Principle and the Cassels statement.

In a recent article I have read, i.e. " Lecture notes on elliptic curves ", Prof.Cassels remarks in page-110 that There is not merely a local-global principle for curves of genus-$0$, but ...
1
vote
1answer
223 views

Constructing a homomorphism from class group to Sha.

In response to my previous question I got a wonderful answer from Prof.Emerton explaining about the similarities between $–®$ and class group. In order to add something the comments I got from Mr. B R ...