1
vote
0answers
37 views

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 ...
2
votes
1answer
52 views

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 $\mathbb{C}$ ...
4
votes
1answer
35 views

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 ...
4
votes
1answer
79 views

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 ...
3
votes
1answer
51 views

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 $p$"?
6
votes
2answers
69 views

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 ...
5
votes
0answers
92 views

Number of points on an affine hypersurface

I am curious to know the following and I would appreciate any help! Let $k$ be a finite field of $q$ elements. Let $X \subseteq \mathbb{A}^{n+1}_k $ be the affine hypersurface defined by $f = ...
1
vote
1answer
32 views

How to prove $-P = (x, -a_1x - a_3 - y)$ for an Elliptic Curve of the General Weirstrass equation for P not the identity?

Let $P = (x,y) \ne \{\infty\}$. Then $-P$ is the other finite point of intersection of the curve and the vertical line through $P$. General Weirstrass equation: E: $a_1y^2+a_3xy+a_5y = ...
1
vote
0answers
28 views

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 ...
6
votes
1answer
59 views

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 ...
8
votes
0answers
76 views

What is the connection between Weil's character bound and Riemann Hypothesis over finite fields

Weil's character bound states that: Let $\mathbb{F}_{q}$ be a finite field of size $q$. Let $\chi$ be a multiplicative character of order $m$. Let $f(x)$ be a polynomial of degree $d$ such that $f(x) ...
1
vote
1answer
70 views

A special cubic curve

How can I transfer following cubic curve to a Weierstrass normal form? $$2x^2y+4xy^2+2y^3-2axy-ay^2+a=0,$$ where $a$ is a fixed rational number.
10
votes
0answers
168 views

Important topics for arithmetic geometry (esp Arakelov geometry)?

Seeing other past successful 'roadmap' questions, I hope this question is acceptable and not too vague. I know I'd like to eventually study arithmetic algebraic geometry - but I also know that it's a ...
1
vote
1answer
52 views

Scheme over a DVR

Let $\mathcal{O}$ be a discrete valuation ring with finite residue field $k$ of characteristic $p$. Let $S=\mathrm{Spec}(R)$ be a noetherian scheme over $\mathrm{Spec}(\mathcal{O})$. Are there ...
3
votes
1answer
86 views

The group of $\mathbb{K}$--rational points for isomorphic elliptic curves

The Springer text by Tom Apostol on Dirichlet series and modular forms, which I have, defines modular functions and modular forms on page 34 and on page 114 respectively, not to mention the Springer ...
0
votes
1answer
89 views

Points on elliptic curves

I am learning elliptic curves theorem and I have read in more papers that for two distinct points $P$ and $Q$ there is always point $R$ such that $P+Q+R = \infty$. I know that this point should be ...
9
votes
1answer
182 views

Tate's Thesis: in what sense is Tate's Theorem 4.2.1 the Riemann-Roch theorem for curves?

I am reading Tate's Thesis. Tate derives a theorem which he calls "the number-theoretic analogue of the Riemann-Roch theorem" from an abstract Poisson summation formula. I am accustomed to thinking of ...
6
votes
2answers
91 views

finite generation of unit group of global function field

Let $X/\mathbb{F}_q$ be a smooth projective geometrically connected curve. Why do we have $\Gamma(X - \{p_1,\ldots,p_n\},\mathcal{O}_X)^\times$ is finitely generated of rank $n$? (analogue of the ...
18
votes
1answer
232 views

Why is there “no analogue of $2i\pi$ in $\mathbf C_p$”?

In his paper Fonctions L p-adiques, Pierre Colmez says: Tate a montré qu'il n'existait pas dans $\mathbf C_p$ d'analogue $p$-adique de $2i \pi$ et donc par conséquent que les périodes $p$-adiques ...
9
votes
1answer
636 views

Explicit Derivation of Weierstrass Normal Form for Cubic Curve

In page 22-23 of Rational Points on Elliptic Curves by Silverman and Tate, authors explain why is it possible to put every cubic curve into Weierstrass Normal Form. Here are relevant pages: (My ...
3
votes
0answers
161 views

Rational Points on Elliptic Curves

I have this homework problem: Can there be an elliptic curve, view as a projective curve, with no rational points with at least one 0 as a coordinate?
3
votes
1answer
68 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, ...
1
vote
0answers
46 views

Algebraic curve over rings $\mathbb{Z}/n\mathbb{Z}$

Can you give a reference about algebraic curve over rings $\mathbb{Z}/n\mathbb{Z}$? I'm very interested in analogy Hasse-Weil theorem, R-R theorem... Thank you.
0
votes
0answers
99 views

number points and twist of elliptic curve

Let elliptic curve $y^2 = x^3 + 1$ has ($q^2 + 2q + 1$) $\mathbb{F}_{q^2}$-rational points. Let $\zeta$ is generator of $\mathbb{F}_{q^2}^*/\mathbb{F}_{q^2}^{*6}$. Curve $E'$ with equation $y^2 = x^3 ...
1
vote
0answers
102 views

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 $1 \pm 2q + q^2$ ...
8
votes
1answer
94 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" ...
6
votes
1answer
130 views

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?
4
votes
0answers
89 views

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 ...
4
votes
0answers
91 views

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 ...
0
votes
0answers
123 views

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?
6
votes
2answers
87 views

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 ...
3
votes
0answers
53 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 ...
12
votes
0answers
155 views

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 ...
6
votes
1answer
52 views

Existence of a variety with prescribed properties

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 $q$, ...
1
vote
0answers
31 views

deg of composition on supersingular curve

Let we have supersingular curve $E(\bar{\mathbb{F}_q})$. Let we have algebraic function $f \in \bar{\mathbb{F}_q}(E)$ with div($f) = \sum_{i=0}^{i=n}n_iP_i$. Then div$(f) \circ [q] = ...
2
votes
0answers
33 views

number of solutions eqauation on supersingular elliptic curve

To Frobenius endomorphism on supersingular elliptic curve I want to prove that equation $\pi_q(X) = A$ has 1 solution for any point $A \in E(\bar{\mathbb{F}_q}))$ where $E$ is supersingular. Is it ...
1
vote
0answers
50 views

Frobenius endomorphism on supersingular elliptic curve

Let we have supersingular curve $E(\bar{\mathbb{F}_q})$. Is it true that for every point $P$ $q$-Frobenius endomorphism $\pi_q$ can be write as $A + [q]B$ where $P = A + B$? It is true if ...
6
votes
1answer
69 views

Function field question from Silverman's AEC

Just before Proposition 1.7 on page 5 of AEC (2nd ed), Silverman defines $M_P$ as an ideal in the affine coordinate ring. Then he states Proposition 1.7 (the intrinsic characterization of ...
4
votes
1answer
159 views

Fermat's Last Theorem in multiple variables

I was wondering if there was anything we could say about when, given $m$, $\exists n (\forall x_1,\dots,x_m \in \mathbb{N} ( x_1^n + x_2^n + \dots + x_{m-1}^n \neq x_m^n))$ Fermat's Last Theorem ...
7
votes
1answer
116 views

Why do varieties with torsion canonical sheaf have finite etale covers with trivial canonical sheaf

Let $B$ be a variety with torsion canonical sheaf, i.e., $\omega^{\otimes n}_B \cong \mathcal O_B$ for some $n>0$. Then, why does there exist a finite (etale?) morphism $X\to B$ such that $K_X$ is ...
3
votes
1answer
138 views

sum of three cubes and parametric solutions

The first paragraph in the following link asserts that the equation $x^3+y^3+z^3=2$ has finite many parametric solutions over $\mathbb{Q}$. In other words, there are finite many polynomial triples ...
5
votes
0answers
142 views

When does a modular form satisfy a differential equation with rational coefficients?

Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ for $\Gamma$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least ...
7
votes
3answers
427 views

A cohomological statement equivalent to the Riemann Hypothesis

Is there a possibility for looking for a theory of cohomology and an equivalent cohomological statement for Riemann hypothesis over $\mathbb{Z}$?
11
votes
1answer
839 views

Undergraduate roadmap for Langlands program and its geometric counterpart

What are the topics which an undergraduate with knowledge of algebra, galois theory and analysis learn to understand Langlands program and its goemetric counterpart? I would also like to know what are ...
3
votes
1answer
97 views

Does the Euler characteristic increase if I add an effective divisor

Let $D$ be an effective divisor on a smooth projective connected complex algebraic variety $X$. Suppose that $D\leq E$. Is it true that $$\chi(X,\mathcal{O}_X(D)) \leq \chi(X,\mathcal{O}_X(E))?$$
3
votes
0answers
41 views

What are the easiest surfaces of general type

The "easiest type" of curves of general type are those of genus two. In this case $\chi(X,\mathcal O_X) = -1$ and $\deg K = 2$, where $K$ is the canonical sheaf. I'm a bit lost when it comes to ...
6
votes
1answer
107 views

Ramification indices and residue degrees of a finite Galois extension

Let $A$ be a dvr with fraction field $K$ of characteristic zero. Let $L/K$ be a finite Galois extension and let $B$ be the integral closure of $A$ in $L$. For a prime $b$ of $B$, let $e_b$ be its ...
5
votes
1answer
273 views

Definition of tamely ramified

I think I can show that the following definitions of "tamely ramified" coincide. I thought it would be good to be sure. Sorry for the easy questions. Let $O_K$ be a dvr with maximal ideal $\mathfrak ...
3
votes
1answer
50 views

Showing that the map on $\mbox{Div}^0(E)$ induced by an isogeny takes principal divisors to principal divisors.

I'm doing a course on elliptic curves. An isogeny $\phi:E_1 \rightarrow E_2$ induces a map $$\begin{array}{llll}\phi_*: & \mbox{Div}^0(E_1) & \rightarrow & \mbox{Div}^0(E_2) \\ \\ & ...
2
votes
1answer
116 views

Proving the condition for two elliptic curves given in Weierstrass form to be isomorphic

I'm taking a course on elliptic curves and trying to understand the proof of Proposition 3.2. Let $E$, $E'$ be elliptic curves over $K$ in Weierstrass form: ...