# Tagged Questions

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### Number of Orbits of symmetric group acting on $(\mathbb{Z}/n)^{l}$

I have encountered a problem that I suspect has been thoroughly studied but I have not been able to find references. Can anyone point me to a published reference dealing with this or a closely related ...
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### 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 ...
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### 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 ...
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### Question regarding function field

I have learned in my algebraic curves class that the function field is the field of rational functions on a curve $C$ (or some variety). I was at a number theory talk, where the person counted the ...
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### Number of points on an elliptic curve over $\mathbb{F}_{q}$.

I have the following elliptic curve: $$E: \quad Y^{2} = X^{3} + 1 ~ \text{over} ~ \mathbb{F}_{q}, ~ \text{where} ~ q \equiv 1 ~ (\text{mod} ~ 3).$$ I want to know the number of points on this curve. ...
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### Lattices and Elliptic curves and number fields

Let $K$ be a number field with ring of integers $O_K$. If $K$ is totally real, then $O_K$ is a lattice in $\mathbb R$. If $K$ is imaginary quadratic, then $O_K$ is a lattice in $\mathbb C$. If $K$ ...
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### Some questions related to Iwasawa invariants of elliptic curves

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at an odd prime $p$. Let $\mathbb{Z}_{p}$ denote the ring of $p$-adic integers, and $\mathbb{Q}^{cyc}$ be the ...
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### Intuition behind using projective geometry for defining the addition on an elliptic curve

We already had the chord-and-tangent construction that can be used to define a way of "adding" points on an elliptic curve. Also this addition satisfies all the group laws. Still why one needs to ...
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### 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 ...
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### So can anybody indicate whether it is worthwhile trying to understand what Mochizuki did?

So I am looking at some math stuff and I start looking at the abc-conjecture. Naturally I run into the name Mochizuki and so start trying to see what he did. Well, he is starting look like another ...
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### Proving the Uniformization Theorem for Elliptic Curves (An Exercise from Silverman's Advanced Topics on Elliptic Curves )

In Silverman's Advanced Topics in the Arithmetic of Elliptic Curves there are two demonstrations of the Uniformization Theorem for the Elliptic Curves (It says that, given an Elliptic Curve $E$, ...
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### Is there a general way to have a polynomial in two variables over C (a plane curve) be irreducible?

Is there a general way to have a plane curve be irreducible? If the curve $C \in \mathbb{C} [x,y]$, would it be sufficient for it to factor into linear terms? What about if I have an equation of the ...
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### 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 $\mathbb{C}$ ...
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### Confusion with computing kernel of an isogeny between two elliptic curves

Consider the two elliptic curves $$E_3: y^2+y=x^3+x^2+x \enspace [Cremona:19A3]$$ and $$E_1: y^2+y=x^3+x^2−9x−15 \enspace [Cremona:19A1]$$ Let $\varphi$ be the $3$-isogeny from $E_3$ to $E_1$. I want ...
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### Computing the kernel of an isogeny between two elliptic curves

Consider the two rational elliptic curves - $E_{1}: y^{2}+y=x^{3}+x^{2}-131x-650$ $[\text{Cremona}:35a2]$ $E_{2}: y^{2}+y=x^{3}+x^{2}-x$ $[\text{Cremona}:35a3]$ We know that ...
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### Isogeny of an elliptic curve

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p$ be a prime. Then what does it mean by "$E$ has a $\mathbb{Q}$-isogeny of degree $p$"?
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### 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 ...
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### Kernel of a map from degree $r$ forms

Let $k$ be a finite field of order $q$. Let $P_r$ be the all the forms (in two variables) of degree $r$ with coefficients in $k$. And let $A_{D,N}$ be all the forms of degree $N$, which vanish on an ...
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### Question about a remark in Serre's Local Fields

I am reading Serre's Local Fields. In Section V.4, Serre considers a finite totally ramified extension of local fields $L/K$ with the residue field $\bar{L}=\bar{K}$ a perfect field. For $\bar{K}'$ a ...
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### Twist of elliptic curve

It is continuation of this question: explict form of the equation of elliptic curve Let $p$ is prime and $p = 3 ($mod $4)$. $q = p^n$. It is easy to see that $E: y^2 = x^3 + x$ has $1 \pm 2q + q^2$ ...
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### 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" ...
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### explict form of the equation of elliptic curve

Let $E(\mathbb{F}_{q^2})$ is elliptic curve with #$E(\mathbb{F}_{q^2}) =q^2 + q + 1$. Can we write equation of this curve in the explicit form?
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### 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 ...
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### Relative density of images of diophantine polynomials

My current research project involves sampling a sequence along non-constant polynomials with non-negative integer coefficients defined over the natural numbers, and I'd like to be able to say when two ...
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### Analogy between Picard group and Ideal class group

Can you give a reference where the conformity between Picard group and Ideal class group is explained? What is analogy of Picard group of elliptic curve over finite field?
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### When a value of a polynomial over $\mathbb Z$ is a perfect square

For which values of $x\in\mathbb{Z}$ the polynomial $16x^3-24x+9$ is a perfect square? I don't know if this question has a solution, but Wolfram Alpha says that the answer is $x=0$ (click), even if ...
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### 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 ...
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### What is Spec of the Adeles?

Let $K$ be a global field and $A_K$ the ring of adeles. What are the prime ideals of $A_K$? I have been told that a full proof of this is quite subtle, but have been unable to find a reference ...
In these notes that give a proof of the Weil conjectures for curves, the author writes on page 17 that given a smooth projective curve $X$ over a finite field $k = \mathbb{F}_q$ for a fixed prime $q$, ...