6
votes
1answer
53 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 ...
2
votes
0answers
38 views

Brauer groups of curves and base change

Let $X/k$ be a smooth, projective curve over $k$ and let $L/k$ be a finite extension of fields, where $k$ is a finite extension of $\mathbb{Q}_p$, $p \not=2$. Suppose $k(X)$ contains no elements ...
0
votes
1answer
20 views

maximal subtorus of a connected commutative algebraic linear group [closed]

I'm wondering the following: is the maximal subtorus of a connected commutative algebraic linear group over $k$ a) normal and closed b) defined over $k$ (for $k$ a field of characteristic zero, ...
3
votes
1answer
44 views

Some questions on the formation of the BSD conjecture

I'm quite curious how Birch and Swinnerton-Dyer formed their famous conjecture in the beginning of 1960s. I read some paper of Birch and Swinnerton-Dyer, as well as some paper of Tate and several ...
2
votes
1answer
54 views

When is “being a linear algebraic $k$-group” preserved?

Let $G$ be a linear algebraic group over a field $k$, with Char$(k)=0$. What "group-theoretical operations" preserve the property of "being a $k$-linear algebraic group"? For example When ...
4
votes
0answers
47 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 ...
0
votes
0answers
20 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 ...
1
vote
0answers
35 views

Counting the Number of Points in an Algebraic Variety

How can we count the number of points in $$S = \{(x,y) \in \mathbb{Z_m}^2: x^2+ky^2 = c\}$$ where $k,c$ are some positive integers?
1
vote
0answers
37 views

a question on torsors

Let $\phi : Y \to X$ be an $X$-torsor under $G_X = G \times_k X$, where $Y,X$ are $k$-schemes and $G$ is an algebraic $k$-group. Let $U \subseteq X$ be an open subset, and let $W = \phi^{-1}(U)$. I'm ...
0
votes
0answers
42 views

Integral point on certain cubic surfaces and rational parametric solutions

The motivation of the following question comes from the Problem D4 in the book "unsolved problems in number theory" by Richard Guy. No integral points on the surface $S:x^3+y^3+z^3=3$ is known other ...
3
votes
1answer
106 views

Shtukas?$\mbox{}$

Does there exist an exposition of the significance of shtukas for someone who is mathematically literate but is largelly ignorant of Drinfeld modules? This arises in the work of Peter Scholze among ...
4
votes
2answers
85 views

what is the most easy to read Algebraic Geometry book? [duplicate]

All: what is the most easy to read (most accessible) Algebraic Geometry book ? (If possible, I am looking for an introduction book, maybe for undergraduate, and maybe similar to A Friendly ...
2
votes
1answer
65 views

How to determine if an equation is a rational variety?

Is there an easy way to determine if a given equation is a rational variety? For example, is $x^3+y^3+z^3=3$ a rational variety? I have maxima installed on my computer, if that helps.
1
vote
0answers
48 views

Could any one explain the difference between the theorems?

In the paper http://annals.math.princeton.edu/2007/165-2/p04 Theorem 2. Let $b \ge 2$ be an integer. The b-ary expansion of any irrational algebraic number cannot be generated by a finite automaton. ...
2
votes
0answers
57 views

Are there any fundamental new discoveries in Algebraic Number Theory than in ‘traditional’ number theory ? [closed]

I am new to Algebraic Number Theory. I wonder if there is any fundamental new discoveries in Algebraic Number Theory than in ‘traditional’ number theory ? I want to know, beside ‘generalizing’ or ...
4
votes
1answer
47 views

If $G/H$ and $G$ are connected linear algebraic groups must $H$ also be connected?

Let $k$ be a perfect field (e.g of characteristic zero) and let $G$ and $H$ be linear algebraic groups over $k$, with $H$ a normal subgroup of $G$. If both $G$ and $G/H$ are connected, must $H$ ...
0
votes
0answers
30 views

how to lift geometrically integralness using etale(+something else) morphisms

Take $X$ and $Y$ to be $k$-varieties, where $k$ is a field of characteristic 0. Assume also that $X$ is geometrically integral, and let $f: Y \to X$ be an etale a morphism of $k$-varieties. Question: ...
1
vote
1answer
62 views

Algebraic groups of multiplicative type in char 0

For a number field $k$ (so of char 0), are algebraic $k$-groups of multiplicative type always linear?
0
votes
0answers
37 views

What is $H_1(A_\mathbb{C}^{top},\mathbb{Q})$

Let $A$ be an abelian variety defined over a number field. I have seen in a few papers the singular homology $H_1(A_\mathbb{C}^{top},\mathbb{Q})$ being used. I read up on the singular homology but it ...
6
votes
1answer
54 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 ...
2
votes
1answer
42 views

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 ...
0
votes
1answer
62 views

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 ...
5
votes
0answers
85 views

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 ...
0
votes
1answer
39 views

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 ...
1
vote
1answer
84 views

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. ...
1
vote
1answer
39 views

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 $K$ ...
6
votes
0answers
115 views

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 ...
1
vote
0answers
75 views

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 ...
4
votes
2answers
84 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 ...
9
votes
2answers
360 views

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 ...
5
votes
1answer
81 views

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 $E$, ...
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
61 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
53 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
103 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
78 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
102 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
98 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
38 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
63 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 ...
14
votes
0answers
150 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
71 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.
1
vote
1answer
68 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
96 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
96 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 ...
11
votes
1answer
260 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
105 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
272 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 ...
10
votes
1answer
943 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 ...