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2
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0answers
61 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
40 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
0answers
15 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 ...
1
vote
2answers
93 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
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0answers
41 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
55 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
41 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
47 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
52 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
76 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
99 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
54 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
70 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 ...
15
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0answers
141 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
1answer
83 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
24 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
29 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
83 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 ...
6
votes
1answer
64 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
62 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
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0answers
51 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
64 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
23 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
37 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
91 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.
1
vote
1answer
86 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
50 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
38 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
75 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
38 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
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1answer
59 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
15 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
48 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
62 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 = ...
1
vote
0answers
22 views

Jacobi Form and its Fourier expansion

Let k,m be non negative integers. A Jacobi form of weight k and index m is a holomorphic function f on $\mathbb{H} x \mathbb{C}$ (where $\mathbb{H}$ denotes the upper half plane) satisfying the ...
7
votes
1answer
158 views

What is the intuition behind the Fontaine-Mazur Conjecture?

The Fontaine-Mazur conjecture (over $\textbf{Q}$ for simplicity) says that a (continuous irreducible) Galois representation $$ \rho: \text{Gal}(\overline{\textbf{Q}}/\textbf{Q}) \to ...
3
votes
1answer
105 views

Question about Qing Liu's Algebraic Geometry book

I was just wondering what the real prerequisites are for reading Qing Liu's 'Algebraic Geometry and Arithmetic Curves', and if it is a good first book on the subject. In his preface he states that the ...
3
votes
1answer
56 views

Reduction map on torsion of elliptic curves

Let $E$ be an elliptic curve over $\mathbb{Q}$ with good reduction at a prime $p$. It is well-known that the map $$E[N]\to E_p[N]$$ is injective when $p\nmid N$. It is even a bijection since both ...
4
votes
1answer
129 views

Group law for an elliptic curve using schemes

I was trying to understand better the definition of the group law for an elliptic curve given in Katz and Mazur's book ...
0
votes
0answers
38 views

Galois representation associated to a number field

I think I'm missing something completely trivial. I want to know how to compute the Galois representation associated to an extension of $p$-adic fields. Let $p$ and $q$ be odd prime numbers. Fix ...
2
votes
1answer
56 views

hasse principle for a number field

I am a little bit of a learner in number theory/arithmetic geometry, so feel free to redirect this question to math.stackexchange if necessary. The question is the following: I know the Hasse ...
5
votes
2answers
88 views

How does $\text{Gal}(L/K)$ act on the automorphism group of an elliptic curve?

Let $L/K$ be a finite Galois extension of number fields; I'm interested mainly in the case $K = \Bbb{Q}$ and $L= \Bbb{Q}(\sqrt{d})$. Let $X$ be an elliptic curve over $K$ and $\text{Aut}(X_L)$ the ...
2
votes
0answers
32 views

The characteristic polynomial of Frobenius of an RM curve

Let $C$ be a genus two curve over $\mathbb{Q}$. We can reduce $C$ modulo a prime $p$ to obtain a curve $\bar{C}$ over $\mathbb{F}_p$. By counting points of $\bar{C}$ over $\mathbb{F}_p$ and ...
4
votes
1answer
1k 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 ...
0
votes
1answer
38 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 ...
2
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0answers
37 views

Etale cohomology and restricted direct product

[migrated to mathoverflow: http://mathoverflow.net/questions/161734/etale-cohomology-and-restricted-direct-product] $\newcommand{\h}{\operatorname{H}}$ Let $k$ be a global field, $A$ an abelian ...
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0answers
139 views

A question in Chapter III.4 of Dino Lorenzini's “An Invitation to Arithmetic Geometry”

Question 1 I am studying in the book "An Invitation to Arithmetic Geometry" by Prof. Dino Lorenzini. In Chapter III Section 4, we consider the following condition: Let $A$ be a Dedekind domain ...
2
votes
1answer
103 views

Hilbert modular forms and Hecke operators over Q

Let F be a totally real field. We know that we can define a Hecke operator $T_\mathfrak{m}$ on the space of Hilbert modular forms over $F$, say with some level structure, for any ideal $\mathfrak{m}$ ...
2
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0answers
53 views

Elliptic curve which attains potential good reduction over an Artin-Schreier extension.

I am looking for an elliptic curve $E$ over the field $\overline{\mathbb{F}_{p}}((t))$, which attains good reduction over an Artin-Schreier extension of $\overline{\mathbb{F}_{p}}((t))$, i.e.: an ...