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1answer
12 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 ...
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1answer
42 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
53 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 = ...
3
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2answers
288 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 ...
3
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1answer
40 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 ...
0
votes
1answer
30 views

Tate module of linear algebraic group

Let $G$ be a (smooth, connected, geometrically integral) commutative linear algebraic group over $\mathbf F_q$. Just as for abelian varieties, we can define the $\ell$-adic Tate module $$ T_\ell G ...
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1answer
55 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 ...
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0answers
12 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 ...
8
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1answer
215 views

Where does this elliptic curve come from?

In Zeta functions of an infinite family of K3 surfaces, Scott Alhgren, Ken Ono and David Penniston compute the zeta functions (given a good reduction restriction mentioned below) of the K3 surfaces ...
8
votes
1answer
127 views

Geometric intuition behind the Hasse principle

Let $f(X,Y) \in \mathbb{Q}[X,Y]$ be a quadratic polynomial. The Hasse-Minkowski theorem says that $f(X,Y) = 0$ has a solution $(x,y) \in \mathbb{Q}^2$ iff it has a solution in $\mathbb{R}^2$ and ...
2
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1answer
69 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 ...
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2answers
188 views

Families of curves over number fields

Is there known a number field $K$ and a curve $F(x, y) \in K(t)[x, y]$ such that $F(x, y)$ does not have points over the field of rational functions $K(t)$ but for all but finitely many positive ...
4
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1answer
96 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
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0answers
33 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
47 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 ...
4
votes
2answers
79 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
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0answers
26 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 ...
0
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0answers
27 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
30 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
95 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
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1answer
68 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
44 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 ...
3
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2answers
61 views

Infinite family of genus one non-elliptic curves over the rationals

How easy is it to write down genus one curves over $\mathbf Q$ without a rational point? Can we write down an infinite family?
3
votes
1answer
36 views

Size of automorphism group of element of $Y_0(5)$

The modular curve $Y_0(5)$ parametrizes elliptic curves $E$ with an isogeny of degree five. So an element of $Y_0(5)$ can be interpreted as $E \xrightarrow{\phi} E'$. Suppose we are working over a ...
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 ...
3
votes
0answers
681 views

Does Wiles's proof of Fermat's last theorem essentially use axiom of choice? [closed]

Does Wiles's proof of Fermat's last theorem essentially use axiom of choice? In other words, if it dose use AC, can we get rid of its use from his proof? EDIT Since some people seem to misunderstand ...
27
votes
1answer
485 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.
3
votes
1answer
66 views

Etale cohomology and algebraic closure

$\DeclareMathOperator{\h}{H}$Apologies in advance if this is overly stupid. Let $k$ be a field and $X$ a variety over $k$. Let $n$ be an integer which is invertible in $k$. One often looks at the ...
3
votes
2answers
53 views

Non-iterative solution for $(a + nb)\mod c < d$

With the given parameters $a$, $b$, $c$, and $d$ I'm looking for a solution of the formula $(a + nb)\mod c < d$. The smallest positive $n$ is the value I want to determine. I can easily solve ...
3
votes
1answer
112 views

Arithmetic and Geometric Mean Inequality

Use the AM - GM inequality (no other method is acceptable), to prove that for all positive integers $n$: $$\left(1 +\dfrac{1}{n}\right)^n \leq \left(1 + \dfrac{1}{n+1}\right)^{n+1}$$ I see that it ...
21
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1answer
317 views

Intuition for Class Numbers

So I've been thinking about the analytic class number formula lately, and class numbers in general and I'm trying to develop a good intuition for them. My basic question, which may be too ...
9
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2answers
231 views

Some basic examples of étale fundamental groups

I'm trying to get a better understanding of étale fundamental groups, and I think that the overall idea -- the big picture -- is beginning to become clear, but my computational ability seems to be ...
1
vote
1answer
105 views

State of the art in arithmetic moduli of elliptic curves?

In trying to get into the topic of moduli spaces of elliptic curves, the following question arises: What is the state of the art in the topic right now? Deligne and Rapoport describes how the ...
23
votes
4answers
645 views

Why is it “easier” to work with function fields than with algebraic number fields?

I just bought a copy of Jürgen Neukirch's book Algebraic Number Theory. While browsing through it I found a section titled § 14. Function Fields in chapter I. In it the author describes ...
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1answer
146 views

Dimension of family of hyperelliptic curves

Suppose we have an elliptic curve E with a point $P$ of order $5$ over a field of characteristic $0$. Denote $E'$ the curve $E/\langle P\rangle$. Now let $x$ (resp. $x'$) be a function on $E$ (resp. ...
10
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1answer
132 views

Which number fields allow higher genus curves with everywhere good reduction

The field of rational numbers is not such a number field. That is, there does not exist a smooth projective morphism $X\to\text{Spec } \mathbf{Z}$ such that the generic fibre is a curve of genus ...
0
votes
1answer
77 views

Project point from larger box to smaller box

I have two boxes as follows: http://i.imgur.com/D0c5Cr7.png I'm trying to take any point from the larger box and find its corresponding location in the smaller box. Basically, the larger box is a ...
3
votes
1answer
76 views

Krull's intersection theorem in the q-expansion principle

I'm currently reading the proof of the q-expansion principle in Katz'73 paper "p-adic properties of modular schemes and modular forms" . The principle itself is a Corollary (1.6.2) of Theorem 1.6.1, ...
7
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1answer
133 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 ...
8
votes
1answer
103 views

Existence of smooth elliptic curves with complex multiplication

this is my first question ever on a platform like this so please forgive me any kind of unintended misbehaving. In Kudla, Rapoport and Yang "On the derivative of an Eisenstein series of weight one" ...
9
votes
1answer
174 views

Primes of ramification index 1 with inseparable residue field extension

I've been reading through Neukirch's Algebraic Number Theory, and I'm a little puzzled about a possibility with ramification of primes. As usual, let $\mathcal{O}_K$ be a Dedekind domain with field ...
5
votes
1answer
68 views

Problems on showing that a reduction map is defined, and that a certain scheme is finite.

I am currently on the last chapters in Liu's book and I am trying to solve the following problem, which is the first step in showing that a certian reduction map is well-defined: Let $X \rightarrow T$ ...
2
votes
1answer
436 views

Who are considered to be masters of arithmetic geometry?

After reading this question I was wondering who are considered to be masters of arithmetic geometry and where can I find the papers which initiated the field arithmetic geometry.
1
vote
1answer
75 views

On an openess property

Let $S=Spec(A)$ where $A$ is a noetherian integral domain. Let $f:X\rightarrow S$ be a flat, proper morphism of schemes. Let $U\subset X$ be an open and $V=f(U)$ (in particular $V$ is open by ...
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5answers
311 views

Derived category and so on

I am looking for an introductive reference to the theory of derived categories. Especially I need to start from the very beginning and I need to know how to use this in examples which comes from ...
9
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1answer
270 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 ...
2
votes
1answer
43 views

Extend maps between etale groups

Let $V$ be a discrete valuation ring, $S=\operatorname{Spec}(V)$ and $\eta$ (resp. $s$) be the generic (resp. closed) point of $S$. Let $G$ and $H$ be flat group schemes over $S$ and assume I know ...
4
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0answers
83 views

Computing a contraction of an exceptional divisor.

For a few days, I have been working on the following problem, from Qing Liu's book: Let $\mathcal{O_K}$ be a discrete valuation ring with uniformizing parameter t and residue characteristic $\neq ...
3
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0answers
54 views

étale cohomology of valuation rings

Let $S$ be the spectrum of a discrete valuation ring (we can assume complete or henselian if necessary). Is it true that the étale cohomology group $H_{et}^2(S,\mathbb{Z})$ is zero?If not in general ...
15
votes
1answer
841 views

Learning path to the proof of the Weil Conjectures and étale topology

Assume you are an algebraic geometry advanced student who has mastered Hartshorne's book supplemented on the arithmetic side by the introduction of Lorenzini - "An Invitation to Arithmetic Geometry" ...