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4
votes
1answer
86 views

A Guide to the Proof of Fermat's Last Theorem from the Modularity Theorem

A few years ago, a friend of mine told me that he had taken an advanced undergraduate number theory class (so something that assumed only a knowledge of algebra and mathematical maturity) which ended ...
2
votes
0answers
24 views

Arakelov Theory and Arakelov curves

There exists a definition of Arakelov Curve in Arakelov theory? My question is because Neukirch (Algebraic Number Theory, Chapter III) defined Arakelov divisors in the set $X=Spec(\mathcal ...
2
votes
0answers
28 views

Why do we assume the ring to be torsion free when dealing with formal logarithms in the context of formal group laws?

Let $F$ be a formal group over a ring $R$. Why do we require that $R$ has no additive torsion before we discuss formal logarithms?
0
votes
1answer
16 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
66 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 ...
2
votes
1answer
51 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 ...
5
votes
0answers
70 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
32 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 ...
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} ...
3
votes
0answers
57 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 ...
1
vote
1answer
48 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
0answers
89 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
19 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
21 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
29 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
0answers
80 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
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
2answers
165 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?
1
vote
0answers
50 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 ...
2
votes
1answer
83 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 ...
1
vote
1answer
64 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 ...
0
votes
1answer
57 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
162 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
89 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 ...
2
votes
0answers
138 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
88 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, ...
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 ...
16
votes
0answers
179 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 ...
1
vote
1answer
109 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 ...
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. ...
4
votes
1answer
36 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. ...
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?
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 ...
7
votes
1answer
78 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
383 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 ? ...
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 ...
2
votes
1answer
72 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 ...
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
40 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
3answers
125 views

Examples of smooth curves of genus $0$ and degree $d>2$. [on hold]

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.
1
vote
1answer
126 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 ...
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 ...
3
votes
2answers
76 views

Prove that $\frac{a^3}{x} + \frac{b^3}{y} + \frac{c^3}{z} \ge \frac{(a+b+c)^3}{3(x+y+z)}$ a,b,c,x,y,z are positive real numbers.

I stumbled upon it on some olympiad papers. Tried to AM>GM but didn't get any idea to move forward.
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0answers
43 views

Are there any linkage or transformation between some period transcendental numbers and algebraic irrational numbers?

Are there any linkage or transformation by combination of integral and algebraic function like in the definition of period number between some period transcendental numbers and algebraic irrational ...
7
votes
1answer
60 views

Torsion of finitely generated $\mathbb Z_ \ell$-module finite?

Let $M$ be a finitely generated $\mathbb Z_ \ell$-module, where $\ell$ is a prime number and $\mathbb Z_\ell$ is the ring of $\ell$-adic integers. Let $T$ be its torsion submodule. Is $T$ finite? In ...
1
vote
1answer
20 views

Find the first $3$ terms of the two possible geometric progressions.

The fourth term of a G.P is $3$ and the sixth term is $147$. Find the first $3$ terms of the two possible geometric progressions. Can you help me find $a$ and $r$? It is too complicated. I took two ...
0
votes
1answer
55 views

Find the term of the G.P.

Find the term of the G.P. $1, 1.2, 1.44, 1.728,... $ which is just greater than $100$. Please somebody explain me the question? All i know is $a=1$ and $r=1.2$ Help :( should it be $T_{n} > ...
1
vote
1answer
65 views

Chow motives of quadratic fields

Let us write $CM_k$ for the category of effective Chow motives up to rational equivalence over $k$. Let $k = \mathbb{Q}$. We consider for different primes $p,q$ the Varieties $X = ...