All Questions
1,978 questions
20
votes
0
answers
408
views
Ado's theorem and the reduction to positive characteristic
The synopsis: proofs of Ado theorem in positive characteristic are simple, and in characteristic $0$ are difficult. Can one infer the characteristic $0$ case from the positive characteristic case?
The ...
- 2,934
2
votes
1
answer
256
views
Twist of the Tate Curve
Suppose we have an elliptic curve $E$ over $K$, an $l$-adic field. Say that $|j(E)|>1$ where $|.|$ is the $l$-adic valuation. By the theory of the Tate curve $E$ is isomorphic over $L$ to a Tate ...
- 637
1
vote
0
answers
66
views
The modular forms of cubic twist of elliptic curves [duplicate]
I want to ask the same question with Does the modular form associated to cubic twist of a elliptic curve $E$ corresponds to some twist of $f_E$?
Let $E$ be an elliptic curve defined over $\Bbb Q$ and $...
- 390
10
votes
1
answer
614
views
With 6 inverted, is the ring of Weierstrass curves a quotient of the Lazard ring by a regular sequence?
Let $L$ be the Lazard ring, i.e., the underlying ring of the universal one-dimensional formal group law. Let $M$ be the ring $\mathbb{Z}[c_4, c_6, 1/6]$ of Weierstrass curves over $\mathbb{Z}[1/6]$. ...
- 1,335
2
votes
1
answer
170
views
Automorphism of positive characteristic field
Suppose $K$ is a field with $\text{char}(K) \geq 0$. Let $L$ be a cyclic extension of $K$ with degree $2n$. We consider the generator $\sigma$ of the Galois group $\text{Gal}(L/K)$.
I am interested in ...
- 923
1
vote
0
answers
88
views
Bad primes of twists of modular curves $X_E^{-1}(p)$
I am interested in a good reference for reading about primes of bad reduction for the modular curve $X_E^{-}(N)$, which is a twist of $X(N)$ parametrizing all elliptic curves $E'$ whose $N$-torsion ...
- 637
19
votes
1
answer
711
views
Discrepancy in Magma's calculation and Sage's of elliptic curve?
$\DeclareMathOperator\Sha{Sha}$I calculated the Tate–Shafarevich group $\Sha(E/K)[2]$ of the elliptic curve $E:y^2=x^3+17x$ over $K=\Bbb{Q}(\sqrt{-37})$.
I calculated that by hand and I reached the ...
- 1,541
1
vote
1
answer
238
views
When $E_D:y^2=x^3+17D^2x$ has even rank?
Let $E:y^2=x^3+17x$ be an elliptic curve.
In this MO page(Infinitely many elliptic curve with twist rank more than $1$ in specific case), Nulhomologous's and other's comment reads from parity ...
- 1,541
3
votes
0
answers
125
views
How many elliptic curves over a finite field have a square discriminant?
$\newcommand{\char}{\operatorname{char}}$Given a finite field $F_q$ with $q\equiv 1 \bmod 3$ and $\char(F_q)>3$, I need to figure out how many isomorphism classes of elliptic curves $E/F_q$ have a ...
- 31
5
votes
1
answer
210
views
Persistent homology of $\mathbb{F}_p$-points of elliptic curves
I'm currently teaching a short summer course on cryptography to high school students. Today, I taught them about elliptic curves. After spending some time playing around with their graphs over $\...
- 573
1
vote
0
answers
162
views
Exercise in Cassels's book
I am trying to prove the following theorem:
Theorem. Let $d=q_1q_2$ where $q_1> 0$, $q_2>0$ are rational primes, with $q_1\equiv 2\mod 9$ and $q_2 \equiv 5 \mod 9$. Then the only rational point ...
- 11
3
votes
2
answers
412
views
Infinitely many elliptic curve with twist rank more than $1$ in specific case
Let $E/\Bbb{Q}$ be an elliptic curve. Let $D$ be a square free negative integer.
It is conjectured that 50％ of twist of elliptic curve $E_D$ has rank $0$ and $50％$ has rank $1$.
But is some particular ...
- 1,541
22
votes
1
answer
770
views
Writing $3p$ when $p \equiv 1 \pmod{3}$ as a sum of two rational cubes. Is this result new? And what about its converse?
I recently discovered a proof of the following. Let $p$ be a prime that's $1 \bmod {3}$. Suppose that $p$ is not represented by the principal quadratic form $(1,9,81)$ of discriminant $-243$ (The ...
- 5,422
6
votes
1
answer
442
views
Ker of corestriction of Galois cohomology
Let $G$ be a Galois group and $H$ be its normal subgroup. Let $M$ be a $G$-module.
Consider the restriction map $res: H^1(G,M) \to H^1(H,M)$. Its kernel is given by $H^1(G/H,M^H)$.
On the other hand, ...
- 1,541
1
vote
0
answers
320
views
Tate-Shafarevich group of Elkies curve
The Elkies curve
$$
E:y^2+xy+y=x^3−x^2−20067762415575526585033208209338542750930230312178956502x+34481611795030556467032985690390720374855944359319180361266008296291939448732243429
$$
conductor of ...
- 245
0
votes
0
answers
162
views
Norm map of Tate-Shafarevich group $\mathrm{Sha}(E/K)\to \mathrm{Sha}(E/\Bbb{Q})$
Let $K$ be a quadratic number field. Let $\sigma$ be a generator of Galois group of $K/\Bbb{Q}$. Let $E$ be an elliptic curve defined over $\mathbb{Q}$.
Let $\mathrm{Sha}(E/K)$ denote the Tate-...
- 1,541
1
vote
0
answers
91
views
Is it possible to lift a pair of points on an elliptic $\mathbb{F}_{\!q}$-curve to a pair of short points on an elliptic $\mathbb{F}_{\!q}(t)$-curve?
Let $E$ be an (ordinary) elliptic curve over a finite field $\mathbb{F}_{\!q}$ of a (quite large) characteristic. For simplicity, suppose that $E(\mathbb{F}_{\!q})$ is a prime group. In addition, let $...
- 1,833
4
votes
1
answer
292
views
Discrepancy in the calculation of $2$-Selmer group by Magma and LMFDB
The result of LMFDB claims (https://www.lmfdb.org/EllipticCurve/Q/1640/c/1 )
that (2-part of) Tate-Shafarevich group $\mathrm{Sha}(E/\Bbb{Q})$ of elliptic curve $y^2=x^3-8747x-314874$ has order $16$. ...
- 1,541
5
votes
1
answer
249
views
Integral points near elliptic curves
This question is an extension of my earlier question here, answered by Noam Elkies.
Let $A,B \in \mathbb{Z}$. Consider the inequality
$$\displaystyle |y^2 - x^3 - Ax - B| = O(|x|^{1/2 + \theta}).$$
...
- 27k
1
vote
0
answers
129
views
How Galois group acts on Tate-Shafarevich group?
Let $L/K$ be a quadratic number field extension. Let $\operatorname{Sha}(E/L)$ be Tate-Shafarevich group of elliptic curve $E/L$.
How $\sigma \in \operatorname{Gal}(L/K)$ acts on $\operatorname{Sha}(E/...
- 1,541
5
votes
1
answer
750
views
Upper bound for Hall's conjecture on separation of squares and cubes
Hall's (weak) conjecture is the statement that for all $\varepsilon > 0$ there exists a positive number $c(\varepsilon) > 0$ such that for all $x,y \in \mathbb{Z}$ with $y^2 \ne x^3$, that
$$\...
- 27k
1
vote
0
answers
187
views
Supersingular points on the modular curve $X_0(p)$ over $\mathbb{F}_p$ and their Frobenius action
I am trying to understand certain points on the modular curve $X_0(p)$ over $\mathbb{F}_p$ (where $p$ is a rational prime) via their moduli description.
A point $P \in X_0(p)$ can be viewed as $P:=(E, ...
- 637
3
votes
0
answers
177
views
Does every (ordinary) elliptic curve over a quite large prime field $\mathbb{F}_{p}$ have a lift to the function field with huge Mordell-Weil rank?
Ulmer (and then other authors) showed existence of elliptic curves $E$ over the function field $\mathbb{F}_{p}(t)$ (where $p$ is a prime) with huge Mordell-Weil ranks (i.e., depending on $p$). The ...
- 1,833
1
vote
0
answers
150
views
Relative compactification without resolutions of singularities
Let $Y$ be a smooth proper variety over a field $k$, let $X$ be a smooth variety over $k$, $U\hookrightarrow X$ the complement of a strict normal crossing divisor and $\phi\colon U\to Y$ a map. By ...
- 309
5
votes
0
answers
168
views
Generalization of Deuring's theorem
Deuring has the following important theorem on elliptic curves over a finite field: Let $p$ be a prime at least $5$, and $N=p+1-a$ an integer with $|a|\leq 2\sqrt{p}$, then the number of elliptic ...
- 213
3
votes
0
answers
112
views
What are the possibilities of the general fibres in an Iitaka fibration?
This question is motivated by complex algebraic geometry.
If $X$ is a complex algebraic variety with Kodaira dimension in $[1,\dim X-1]$, then the Iitaka fibration (the rational map induced by the ...
- 9,501
2
votes
0
answers
153
views
Order $4$ element of Tate-Shafarevich group
Let $E/\Bbb{Q}$ be an elliptic curve defined over $\Bbb{Q}$. Tate-Shafarevich group $\mathit{Sha}(E/\Bbb{Q})$ is defined as follows.
$$\mathit{Sha}(E/\Bbb{Q})\stackrel{\text{def}}{=} \operatorname{Ker}...
- 1,541
0
votes
0
answers
319
views
Percent of rational coordinates that is a multiple of another point on the elliptic curve
Consider elliptic curves $E:= y^2=x^3+Ax+B $ (A, B are integers) which have points $P, Q$ with rational coordinates and satisfy $P=[n]Q, n>1$. Now consider the below problem:
Input: Rational ...
3
votes
1
answer
272
views
Computing explicit isogenies between elliptic curves over different kinds of fields
I have some questions about isogenies of elliptic curves in two settings:
1. Elliptic curves defined over the rationals.
1.1. Given two elliptic curves $E/\mathbb{Q}$ and $E'/\mathbb{Q}$ we can decide ...
- 637
1
vote
0
answers
144
views
Characterization of Selmer group in terms of two descent
This question is about p 337 of Silverman's book ''The arithmetic of elliptic curves'', p 337, http://www.pdmi.ras.ru/~lowdimma/BSD/Silverman-Arithmetic_of_EC.pdf.
Let $E:y^2=x^3+ax^2+bx$ and $E':Y^2=...
- 1,541
6
votes
1
answer
305
views
Definition of modular curve associated to $\Gamma(N)$
For a positive integer $N$, we define $$\Gamma(N)=\big \{ \begin{bmatrix} a & b \newline c & d\end{bmatrix}\in \operatorname{SL}_2(\mathbb{Z}): \begin{bmatrix} a & b \newline c & d\end{...
- 831
2
votes
0
answers
142
views
$K(S,2)=\{b \in K^{\times}/{K^{\times}}^2\mid v(b)≡0 \bmod2, \forall v \notin S\}$ and Selmer group
This question is essentially related to the theory of elliptic curves (Selmer group), but this question itself is just a field theoretic one.
To calculate the Selmer group of given elliptic curve, we ...
- 1,541
0
votes
1
answer
107
views
Is there a method to check if two sections of an elliptic surface are dependent over the endomorphism ring or not?
Let $E\!: y^2 = x^3 + a(t)x + b(t)$ be an elliptic curve over the function field $\mathbb{F}_{q}(t)$ over a finite field $\mathbb{F}_{q}$ of characteristic $5$ or greater. For simplicity, let $E$ be ...
- 1,833
1
vote
0
answers
156
views
About the exact sequence $0\to $$Ш(E/\Bbb{Q})[\phi]\to Ш(E/\Bbb{Q})[2]\to Ш(E'/\Bbb{Q})[\hat{\phi}]$ in Silverman's book of elliptic curves
This question is about Silverman's book,$7$-th line from the bottom of $350$ page of 'The arithmetic of elliptic curves' (http://www.pdmi.ras.ru/~lowdimma/BSD/Silverman-Arithmetic_of_EC.pdf) .
Let $E$ ...
- 1,541
2
votes
1
answer
411
views
Galois cohomology of Tate modules
Let $E,E'$ be a pair of elliptic curves defined over $\mathbb{Z}$. Let $T_p[E], T_p[E']$ be their associated ($p$-adic) Tate modules. These are Galois representations for the absolute Galois group of $...
- 2,907
4
votes
0
answers
103
views
Boundary divisor of an equivariant compactification of a non-trivial twist of $\mathbb{G}_a^n$
Let $K$ be a non-perfect field of positive characteristic. Then there exist non-trivial forms of $\mathbb{G}_a^n$ which split over finite purely inseparable extensions of $K$, i.e., algebraic groups $...
3
votes
0
answers
94
views
Dimension of a kernel of a cocycle map
Inspired by a previous question (Dimension of a kernel of a linear map) and some of the answers I was given I thought wheter I can generalize the question to the following:
Compute the kernel (or at ...
- 911
3
votes
0
answers
176
views
Extending a theorem of Washington
In Class numbers of the simplest cubic fields, Larry Washington states the following theorem (I have added some hypotheses to make the statement more self-contained), which is Theorem 2 in said paper:
...
- 27k
0
votes
0
answers
124
views
Rank growth of elliptic curve $E:y^2=x^3-17$ in quadratic number field
Let $E:y^2=x^3-17$ be an elliptic curve.
It is known that rank$(E/\Bbb{Q})=0$.
(For example, prop $6.5$, $362$p in Silverman's book 'The arithmetic of elliptic curves')
Over $K=\Bbb{Q}(i)$, what is ...
- 1,541
0
votes
0
answers
76
views
Rank of infinite family of elliptic curves over the rationals without assuming finiteness of Sha
Are there any known infinite families of elliptic curves over the rationals, that are proved to have rank $\geq 2$, without assuming finiteness of their Tate-Shafarevich group?
- 71
3
votes
0
answers
183
views
discriminant of the division field of an elliptic curve
Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and $\ell$ be a prime number. Let $\bar{\rho}$ be the representation of $Gal(\bar{\mathbb{Q}}/\mathbb{Q})$ on $E[\ell]$. Let $L:=\mathbb{Q}(E[\...
- 685
3
votes
0
answers
122
views
Torsion of Fermat hypersurfaces
An interesting invariant of a rationally chain-connected variety $X/k$ is the exponent of the group,
$$ \ker{(\mathrm{CH}_0(X_K) \xrightarrow{\deg} \mathbb{Z})} $$
where $K = k(X)$ is the function ...
- 3,730
1
vote
1
answer
484
views
Why an elliptic curve can be defined as an abelian variety of dimension 1?
Now we define an elliptic curve as "a smooth projective curve of genus one with a specified base point". (A little question by the way: Is the requirement "with a specified base point&...
- 65
8
votes
1
answer
579
views
Possible extensions of a conjecture (perhaps now a theorem) proposed by Sylvester
Let $\omega$ be a primitive cube root of unity, $\mathcal O$ be $\Bbb Z[\omega]$, and $K$ be $\Bbb Q[\omega]$.
I am asking about the $K$-rank of the elliptic curve $x^3 + y^3 = M$ for certain cube-...
- 5,422
1
vote
1
answer
246
views
Cremona transformation of (real) elliptic curves
I am reading a preprint and I do not quite follow a seemingly known fact:
Let $E$ be an elliptic curve on the real projective plane with two real components. Let $O\subset E$ the oval component (...
- 2,275
5
votes
2
answers
659
views
Why all supersingular elliptic curves over $\bar{\mathbb{F}_p}$ are isogenous?
Lemma 3.2.1 in Baker, González-Jiménez, González, Poonen, "Finiteness theorems for modular curves of genus at least 2", Amer. J. Math. 127 (2005), 1325–1387.
enter image description here
I ...
- 65
1
vote
1
answer
193
views
Are morphism from $\mathbb{P}^{1}$ to itself often liftable to a morphism from a curve to an elliptic curve with bounded degree?
(all morphism here means birational)
(the ground field is "small", but I don't think it should matter)
Here is the picture. I have a morphism $f:\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}$. I ...
9
votes
0
answers
180
views
When is the solution to a linear system of ODEs an algebraic variety?
Question: Are the following observations well known, and in what general context?
Let $A$ be a diagonalizable $n\times n$ matrix over $\mathbb{C}$ and consider the following system of differential ...
- 3,761
3
votes
0
answers
87
views
Obstruction for points to be contained in smooth hypersurfaces in tterms of inseparability degree of residue field
Let $k$ be an imperfect field of char $p>0$ and
$x \in \mathbb{P}^n_k$ be closed point of projective space.
In this discussion Qing Liu wrote that
Over an imperfect field, a reduced point can not ...
- 619
4
votes
0
answers
105
views
Matrix description for automorphisms of genus $2$ curve split into two copies of an elliptic curve
Consider a genus $2$ curve $C$ and its Jacobian $J$ (for simplicity, over the field $\overline{\mathbb{Q}}$ or $\mathbb{C}$). Assume that $J$ is $(2,2)$-isogenous to the direct square $E^2$ of an ...
- 1,833