The tag has no wiki summary.

learn more… | top users | synonyms

0
votes
1answer
15 views

connection between absolute irreducibility and smooth+geometrically connected

Let $C_1$ and $C_2$ be two smooth, projective, and geometrically connected curves over a field $K$ of characteristic $0$. Assume there are two non-constant branch mapping of $K$-curves, $\phi_1\colon ...
1
vote
0answers
56 views

Fano surfaces all of whose rational points lie on some geometric line

Are there any ? Namely let $X$ be a smooth del Pezzo surface defined over $\mathbb{Q}$ that has rational points and such that the degree of the del Pezzo is small, say $d=3$ or $4$. Is it possible ...
1
vote
1answer
39 views

Extension of a complete discrete valuation ring

My question came when I was reading the famous Tate's paper on $p$-divisible groups. At the beginning of chapter $(2.4)$ he cites this fact as obvious. If you take a complete discrete valuation ring ...
2
votes
1answer
49 views

Lifting points of étale group scheme.

Consider the following setting: let $(R,\mathfrak{m})$ be a Noetherian complete discrete valuation ring (with maximal ideal $\mathfrak{m}$) and let $K$ be its field of fractions. Now take $L$ be the ...
8
votes
2answers
180 views

Completion along zero section of an elliptic curve.

I am trying to understand the intuition that I should have about the formal group of an elliptic curve. Say that I have an elliptic curve $E\to \text{Spec} R$ for some ring $R$, with section $0\colon ...
0
votes
1answer
47 views

field of rational functions of a curve

Let $C$ be the algebraic curve defined by the modular polynomial $\phi_N$ of order $N>1$ over the rational numbers, i.e. \begin{equation}C:=\text{specm}(\mathbb{Q}[X,Y]/\phi_N(X,Y)). \end{equation} ...
5
votes
0answers
63 views

Some questions about reduction of elliptic curves

Let $E \rightarrow S$ be an elliptic curve (i.e, a smooth proper curve of genus 1). If $S = \text{Spec (K)}$ where $K$ is a local field, the usual way of doing a reduction at a prime $\mathfrak{p} = ...
0
votes
1answer
29 views

Incommensurable units as ratios

I am having a bit of trouble understanding the concept of an incommensurable unit. From what I have gathered so far, it is simply a magnitude that cannot be expressed as the ratio of two natural ...
3
votes
0answers
56 views

degrees of L-functions and dimensions of Shimura Varieties

I try to grab a (very) little understanding of what a Shimura variety is, and although I still don't understand the formal definition of that notion, a few vague ideas have come to my mind. Hence ...
2
votes
0answers
85 views

Understanding the Falting's Theorem

I'm an undergraduate student of mathematics, but soon I'll graduate, and as a personal project I want to understand Falting's Theorem, specifically I want to understand Falting's proof; but yet I have ...
0
votes
0answers
17 views

Cartier dual of an exact sequence

Suppose we have an exact sequence of affine finite flat commutative group schemes over an arbitrary ring $R$: \begin{equation} 0\rightarrow H\xrightarrow{i} G\xrightarrow{j} K\rightarrow 0 ...
1
vote
0answers
20 views

Rationality of intersection of algebraic groups

Suppose that $G$ (defined over $\mathbb{Q}$) and $H$ (defined over $\mathbb{R}$) are two algebraic subgroups of a larger algebraic group defined over $\mathbb{Q}$. Assume that $G(\mathbb{R})$ and ...
1
vote
0answers
28 views

Explicit value of $\overline{\mathcal{O}}_{\mathbb{P}^1(\mathbb{Z})}(1).\overline{\mathcal{O}}_{\mathbb{P}^1(\mathbb{Z})}(1) $

Let us consider the Fubini-Study metric on the part at infinity of the line bundle $\mathcal{O}_{\mathbb{P}^1(\mathbb{Z})}(1)$ to obtain the Hermitian line bundle ...
2
votes
1answer
21 views

Invertible sheaves on arithmetic surfaces and gcd.

this is exercise 9.1.12 b) in Qing Liu's book "Algebraic geometry and Arithmetic curves". Let $\pi:X \rightarrow S$ be an arithmetic surface with smooth and geometrically connected generic fiber ...
2
votes
0answers
78 views

Witt vector question

I've started reading various papers and notes on Schemes over the Witt Vectors. In example 8.8 of these: https://www.uni-due.de/~mat903/books/esvibuch.pdf W2 has addition defined as $k \oplus k\cdot ...
0
votes
1answer
70 views

Points on an elliptic curve over $\mathbb F_p$

Let $E$ be an elliptic curve over $\mathbb F_p$ (the finite field of $p$ elements) defined by $y^2=x(x-n)(x-m)$ where $p\nmid nm(n-m)$. Let $N$ be the number of $\mathbb F_p$-valued points of $E$. ...
2
votes
2answers
157 views

Does ABC implies Fermat's last theorem?

I read from the newspaper that Mochizuki's proof of the ABC conjecture implies the Fermat's last theorem. Is it true? I think it implies the proof only for large enough exponents?
18
votes
1answer
1k 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 ...
0
votes
1answer
41 views

why complitcated construction of PD-defferential operator in Berthelot and Ogus's book

In the book " Notes on Crystalline Cohomology" by P. Berthelot and A. Ogus, they introduced the cencept of PD-defferential operators in a complicate way, i.e. using dividied power hull. However if I ...
1
vote
0answers
49 views

Rational points and resolution of singularities

Suppose $X$ is an algebraic variety over a field $F$ of characteristic 0. By resolution of singularities, there is a nonsingular variety $Y$ over $F$ with a proper birational morphism $Y \rightarrow ...
1
vote
1answer
61 views

Is the module of invariant differential forms of a Neron model of an abelian variety a projective module?

Let $A$ be an abelian variety of dimension $d$ over a number field $K$. Let $\mathcal{A}$ be its Neron model over the ring of integers $O_K$. Let $\Omega_{\mathcal{A}/O_K}$ and $\Omega_{A/K}$ be the ...
2
votes
1answer
75 views

kernel of the exponential map is isomorphic to the singular homology group

Let $G$ be an algebraic torus or an abelian variety over the complex numbers. Then $G(\mathbb{C})$ is a complex Lie group. Is it true that we have the following exact sequence ? $ 0 \to ...
0
votes
1answer
53 views

Are there Neron models for algebraic groups of multiplicative type?

Let $K$ be a number field with Galois group $G$ and $N$ be a finitely generated abelian group which is also a discrete $G$-module. Let $D(N)$ be the algebraic group defined as ...
1
vote
2answers
130 views

Arithmetic mean <= Quadratic mean, proof?

I tried to solve this for hours but no success. Prove, that the arithmetic mean is <= quadratic mean. I am in front of this form: $$ \left(\frac{a_1 + ... + a_n} { n}\right)^2 <= \frac{a_1^2 + ...
4
votes
0answers
87 views

Contraction of curves on a surface and stalks

Let $$f:X \rightarrow S$$ be a fibered surface over a Dedekind scheme of dimension $1.$ Let $$s_1, \ldots, s_n$$ be closed points of S and $\{E_{ij}\}$ irreducible vertical divisors of $X$ with ...
1
vote
0answers
178 views

Normalization of a node

How do you describe the normalization of $\mathrm{Spec}(k[[x,y]]/(xy))$? In particular, call this $N$. Why can $N$ be written as the intersection of 2 local rings and the same holds for the maximal ...
2
votes
0answers
133 views

What would be the most rigorous book to stydy algebraic geometry and arithmetic curves on my own?

I would like to study algebraic geometry and arithmetic curves on my own but are there suggestions where to start? Namely, I like very rigorous way to do mathematics and I was suggested Liu's book ...
1
vote
0answers
81 views

Component Group Neron Model Elliptic Curve Cyclic

I'm studying the chapter on Neron Models in Silverman's book "Advanced Topics in the Arithmetic of Elliptic Curves" at the moment, and I do not quite understand why in the split multiplicative case, ...
15
votes
0answers
173 views

Why are period integrals naïve periods?

Apologies for the long question. I recall the definition of a (naïve) period according to Kontsevitch and Zagier [KS]: A (naïve) period is a complex number whose real and imaginary parts are ...
2
votes
0answers
90 views

Deep questions in number theory not accessible by combinatorial results

Number theory and arithmetic geometry were invented to solve many questions about properties of numbers. What are the some of the foundational results or estimates that are accessible to powerful ...
1
vote
1answer
103 views

Galois theory on curves

Context: Let $\mathbb{F}$ be the algebraic closure of $\mathbb{F}_q$ for $q$ prime. We know that $\mathbb{F}(t)$ for $t$ transcendental is the function field of the projective line ...
4
votes
1answer
34 views

coefficients of the zeta function of curve over a finite field $\mathbb{F}_q$

Let $C$ be a non-singular curve over $\mathbb{F}_q$. Denote by $d$ the degree map from the group of divisors to $\mathbb{Z}$ and denote by $P$ the set of prime divisors w.r.t. to the function field. ...
2
votes
0answers
28 views

another representation of the zeta function of a curve over a finite field

Let $C$ be a non-singular curve over $\mathbb{F}_q$. Denote by $d$ the degree map from the group of divisors to $\mathbb{Z}$ and denote by $P$ the set of prime divisors w.r.t. to the function field. ...
5
votes
1answer
91 views

Differential of a morphism of abelian varieties

I am reading the lecture notes of J.S. Milne on Abelian varieties and I got stuck at some point. Let $\alpha,\beta\colon X\rightarrow Y$ be homomorphisms of abelian varieties $X$ and $Y$. Then for ...
0
votes
3answers
21 views

Trouble with finding geometric progression pattern

I have this system: $$ b_2-b_1 = 18 $$ $$ b_4-b_3 = 162 $$ I have to find $b_1$ (the first element) and $q$ (common ratio). Any ideas how to solve it?
2
votes
1answer
71 views

Punctured Elliptic Curve

I've come across the word "punctured elliptic curve" here and there, but none of the basic texts on the topic (Husemoller, Silverman) define or mention it. What point is removed from the curve (the ...
7
votes
2answers
295 views

Why should automorphism groups of compact hyperbolic curves be finite

Let $X$ be a compact connected Riemann surface of genus at least two, or let $X$ be a smooth projective connected curve over an algebraically closed field of characateristic zero. Then Hurwitz proved ...
7
votes
1answer
77 views

Isogenies of abelian varieties

Let $\pi:X\to Y$ be a finite morphism of smooth projective curves over an algebraically closed field (of characteristic zero if necessary) which are both of genus $>1$. We have two "natural" maps ...
1
vote
2answers
320 views

Geometric progression of 1 and 1/3tan^2θ

The first two terms of a geometric progression are where 0<θ<π/2 (i) Find the set of values of θ for which the progression is convergent. [2] What does convergent mean and how to solve this ? ...
7
votes
1answer
149 views

Do K3 surfaces with an Enriques involution have a polarization of bounded degree

Does there exists a real number $C$ with the following property. For any Enriques surface $E$ over a number field $K$ with K3 cover $X\to E$, there exists an ample divisor $H$ on $X$ such that $H^2 ...
4
votes
1answer
2k views

What is the status of the purported proof of the ABC conjecture?

Back in August 2012, Japanese mathematician Shinichi Mochizuki announced a proof of the abc conjecture using Inter-universal Teichmüller Theory. What has been the status of his proof? Has there been ...
4
votes
0answers
52 views

When does a variety have a point over a finite field for sufficiently large primes p?

Let $X$ be an algebraic variety over the rational numbers. Suppose that $X$ has positive dimension. I would like to say that $X(\mathbb{F}_p)$ is non-empty for sufficiently large primes $p$. One idea ...
3
votes
1answer
183 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 ...
31
votes
1answer
877 views

Has SGA 4$\frac 1 2$ been typeset in TeX?

The title says it all. I've CW'd the question since I'm answering it - this seemed like the best way to get the news out.
0
votes
0answers
31 views

An isogeny from a split algebraic torus

Suppose that there is an isogeny (in the category of commutative algebraic groups) from a split algebraic torus to a semi-abelian variety. Does it follows that this semi-abelian variety is also an ...
3
votes
0answers
39 views

trivialising cover for etale morphisms

Let $f:Y \to X$ be a finite etale morphism of smooth and proper schemes over a field $k$ (not necessarily sep closed). Is there a geometrically connected etale cover $\{U_i\}$ of $X$ which ...
1
vote
1answer
114 views

etale neighborhoods

I've read that quasi-compact etale morphisms of schemes over a not necessarily algebraiclly closed field $F$ (I'm happy to take $F$ a field of char $0$) are the algebraic analogs of local ...
1
vote
3answers
112 views

Examples of smooth curves of genus $0$ and degree $d>2$.

Can we provide a source of explicit examples ? The degree assumption $d>2$ means that I would like to see examples which are not conics.
2
votes
0answers
51 views

What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers and the integer

As we know,$\sqrt{2} = [1;2,2,2,2,…]$; while $\sqrt{14}= [3;1,2,1,6,1,2,1,6…]$,let $L$ the length of period of simple continued fraction expansion of quadratic algebraic numbers be the number of ...
1
vote
0answers
40 views

$p$-divisible group of tori

I am looking for a reference of the following question which should be well known. Let $k$ be any field and $T$ an algebraic torus over $k$ which is not necessarily split. Let $T(l)$ be the ...