**3**

votes

**1**answer

154 views

### Identifying the canonical principal polarization of a Jacobian

Let $X$ be a curve over an algebraically closed field $k$ (even over $k = \mathbb{C}$ if you want), let $J = Pic^0_{X/k}$ be its Jacobian, let $P \in X(k)$ be a point, and let $i \colon X \...

**5**

votes

**3**answers

449 views

### Extending rational Diophantine triples to sextuples

(This is a follow-up to a previous post.) A rational Diophantine $m$-tuple is a set of rationals {$a_1,a_2,\dots a_m$} such that (with $i\neq j$), all $a_i a_j+1$ is a square. Problem: Find a class of ...

**4**

votes

**1**answer

148 views

### Lifting of Frobenius on torsors over abelian varieties

This is related to my previous question Assume that $A$ is an abelian variety over a field $k$ of characteristic $p$, $\mathcal{L}$ is a line bundle on $A$. Assume that $A$ is ordinary and $\mathcal{L}...

**3**

votes

**1**answer

178 views

### Lifting of Frobenius on semi-abelian varieties

Let $A$ be a semi-abelian variety over a field $k$($char\, k=p$). Namely, there is an exact sequence of group schemes $$0\to T\to A\to B\to 0$$ where $T$ is a torus, $B$ an abelian variety. Assume ...

**8**

votes

**0**answers

147 views

### Hilbert scheme of projectively normal elliptic curves

Consider the Hilbert scheme of degree $n$, genus $1$ curves in $\mathbb P^{n-1}$. It contains the locus of smooth curves embedded by the complete linear system of a degree $n$ divisor. Let $X_n$ be ...

**6**

votes

**0**answers

128 views

### Is there a prime degree endomorphism on supersingular elliptic curves?

Let $E$ be a supersingular elliptic curve which is defined over field $\mathbb{F}_{p^2}$ and $l$ be a prime such that $gcd(l,p)=1$.
Is there an endomorphism $\phi\in End(E)$ such that $deg(\phi)=l$?
...

**12**

votes

**1**answer

341 views

### Weak Mordell-Weil over number fields

I have a question regarding the Mordell Weil theorem a number field $K$.
I read the proof of the Mordell Weil theorem in "rational points on elliptic curves" by Tate and Silverman.
They presented a ...

**1**

vote

**0**answers

66 views

### Complex Structure Moduli of Elliptic Fibrations

Given an elliptically fibered Calabi-Yau threefold in Weierstrass form I want to compute the number of complex structure moduli of the fibration.
I know how it is done for the generic Weierstrass ...

**0**

votes

**1**answer

141 views

### Find special elliptic curves from j-invariant

Let $E$ be the elliptic curve defined over $GF(p)$ and $j$ be j-invariant of $E$, where $p$ is a big prime number. Also suppose $l$ be small prime number (for example $l<5000$) and $\#E$ denote ...

**5**

votes

**0**answers

223 views

### An example of a coarse moduli space of elliptic curves

Let $\mathcal{S}$ be the moduli stack (over $\text{Spec }\mathbb{Z}$) of elliptic curves endowed with a trivialisation of its Hodge bundle.
Classically, we know that $\mathcal{S}\otimes \mathbb{Z}[1/...

**0**

votes

**0**answers

68 views

### Algebraic operations with memory hardness properties

In cryptography, there are password hash functions like scrypt and argon2 for which the fastest known algorithms employ large ...

**6**

votes

**3**answers

257 views

### Uniform bounds on the number of integer points on a family of elliptic curves

Let $P(x,y)$ be a binary cubic polynomial with integer coefficient. Let $n$ be an integer. Suppose the (complex) curve $P(x,y)=n$ is nonsingular, so is an elliptic curve. Is there any bound on the ...

**1**

vote

**0**answers

358 views

### The Sato-Tate conjecture (Frobenius eigenvalues)

Another question crossed my mind.
In the statement of the Sato-Tate conjecture, one usually assumes that
the elliptic curve has no CM. But, I read the Morita's paper for the BSD in the CM
case and ...

**2**

votes

**2**answers

515 views

### Curves of higher genus

I saw the question:
Abelian varieties with CM
and though I know that there are rare CM elliptic curves, I wonder
what kind of curves with higher genus have the CM Jacobians?

**1**

vote

**1**answer

209 views

### Finding cyclic subgroups of points on elliptic curves for isogeny based cryptography

Isogeny based cryptography is one of the newest post-quantum cryptography. Hardness of this system is based on finding isogeny between two elliptic curves. Also this is a theorem:
Elliptic curves ...

**0**

votes

**0**answers

43 views

### lower bound for solve ECDLP

Let $P$ and $Q$ are two points of NIST elliptic curve $E$ (defined over $F_{2^m}$ with prime $m$) and $k$ is a private key such that $k.P=Q$. Suppose $\ell$ be the number of bits in $k$, and let $k_i$ ...

**3**

votes

**1**answer

749 views

### Abelian varieties with CM

In this site, I looked at a paper of Kazuma Morita claiming the BSD conjecture for the CM case
posted on his homepage (he made a mistake three years ago for full BSD).
But, I am interested in this ...

**3**

votes

**0**answers

284 views

### Lifting a real quadratic twist of an Elliptic Curve to the modular curve

Let $E$ be an elliptic curve of conductor $N\cdot p^2$ over $\mathbb{Q}$, defined by the equation
$$y^2=x^3+p^2b\cdot x + p^3\cdot c$$
and parametrized by a map
$$X_{0}(N\cdot {p}^{2})\rightarrow E$$
...

**17**

votes

**1**answer

496 views

### Why do the $2$-Selmer ranks of $y^2 = x^3 + p^3 $ and $y^2 = x^3 - p^3 $ agree?

I was playing around with sage, when I found that the ranks (over $\mathbf{Q}$) of the elliptic curves $y^2=x^3+p^3$ and $y^2=x^3-p^3 $ almost always agree, for $p$ prime. The first few exceptions ...

**31**

votes

**2**answers

1k views

### Estimating the size of solutions of a diophantine equation

A. Is there natural numbers $a,b,c$ such that $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b}$ is equal to an odd natural number ?
(I do not know any such numbers).
B. Suppose that $\frac{a}{b+c} + \...

**5**

votes

**1**answer

185 views

### Is there $t\in\operatorname{Gal}(\overline{K}/K)$ s.t. $\operatorname{rank}_{\mathbf{Z}_p}((t-1)E_{p^\infty}(\overline{K}))=1$?

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, let $$K:=\varinjlim_{k\in\mathbb{Q}[\mu_{p^\infty}]} \mathbb{Q}\left[\mu_{p^\infty},k^{1/p^\infty}\right]$$
and $G:=\operatorname{Gal}(\overline{...

**5**

votes

**1**answer

287 views

### Division by $n$ in elliptic curves

Let $E/\mathbb F_{p^m}$ be an arbitrary elliptic curve over the Galois field $\mathbb F_{p^m}$, and let $$[n]^{-1}(P)\cap E(\mathbb F_{p^m})=\{Q\in E(\mathbb F_{p^m})\mid nQ=P\}.$$ Also let $N=\#E(\...

**5**

votes

**1**answer

548 views

### Elliptic curves and prime numbers

Let $p_n$ be the $n^{th}$ prime number. Suppose $E(F_{p_n})$ denotes an elliptic curve over the Galois field $GF(p_n)$ which is defined by $y^2=x^3+ax+b$. Is the below claim true?
For each integer ...

**17**

votes

**2**answers

917 views

### elliptic curves and group cohomology

Recently, I've been trying to understand Jacob Lurie's 2-equivariant elliptic cohomology a bit better than I had in the past.
From what I can tell, the fragment of the story that only deals with ...

**8**

votes

**2**answers

409 views

### What do we know about the structure of $J_{0}(N)$ over $\mathbb{Q}[{\mu}_{{p}^{\infty}},{{k}}^{\frac{{1}}{{p}^{n}}}])$?

What is known about the structure of $J_{0}(N)$ over $\mathbb{Q}[\mu_{p^{\infty}}]$?
More generally, what do we know about $J_{0}(N)$ over
$\mathbb{Q}[\mu_{p^{\infty}},k^{1/p^{n}}]$, where $k\in\...

**19**

votes

**0**answers

214 views

### “High-concept” explanation for proof of a theorem of Ochanine?

See Akhil Mathew's notes on Ochanine's theorem for elliptic genera here and here.
Let $\phi: \Omega_{SO} \to \Lambda$ be a genus. We might ask when $\phi$ satisfies the following multiplicative ...

**10**

votes

**1**answer

432 views

### Galois representations for the curve $y^{2} = x^{3} - x$

Let $E / \mathbb{Q}$ be the elliptic curve given by $y^{2} = x^{3} - x$. I would like to know explicitly what the field of all $2$-power torsion looks like, as well as the image in $\mathrm{GL}(T_{2}(...

**6**

votes

**1**answer

247 views

### Why are some solutions of these diophantine equations off the usual patterns?

This is inspired by a recent question about complete multipartite integral graphs. I am wondering if more can be said about tripartite integral graphs with block sizes $a<b<c$. It is easy to see ...

**7**

votes

**1**answer

325 views

### What are some open problems regarding elliptic curves in finite fields?

I accept that my question seems so vague and broad, and I already looked into some similar questions in MO. But I would like to learn specifically about some open problems and conjectures regarding ...

**4**

votes

**0**answers

123 views

### Ranks of elliptic curves over Q(t)

I have an elliptic curve $E/\mathbb{Q}(t)$, and I want to compute its rank. Does knowing the rank over $\mathbb{F}_p(t)$ for some prime of good reduction give a bound on the rank over $\mathbb{Q}(t)$?...

**5**

votes

**1**answer

375 views

### The Weil numbers and modulus of an elliptic curve

I have an ignorant question about elliptic curves which I'll be slightly imprecise about. If I have an elliptic curve $X$ defined over $\mathbb Z$, I can base change to $\mathbb C$, and then $X(\...

**0**

votes

**1**answer

165 views

### Rational maps between elliptic curves [closed]

I am studying Silverman's "The Arithmetic of Elliptic Curves" and I got the following question:
In the first chapters he defines rational between projective varieties (see the first definition in I.3)...

**6**

votes

**1**answer

404 views

### realizing uniform boundedness of Galois representations associated to elliptic curves

This is less of a direct question and more of an argument that I've been worried about for a while and want to check (apologies for the length and if my writing is unclear).
Suppose I have an ...

**24**

votes

**4**answers

856 views

### Deep/precise relationship between two approaches to FLT for polynomials, $n = 3$

David Speyer commented the following here.
I saw Brian Conrad give an excellent one hour talk to undergraduates where he proved that there do not exist nonconstant, relatively prime, polynomials $...

**7**

votes

**1**answer

498 views

### Analytic continuation for $L$-functions of elliptic curves

Let $E$ be an elliptic curve over a number field.
When $E$ has no CM and is a $\mathbb Q$-curve (i.e. it is $\overline{\mathbb Q}$-isogenous to all of its conjugates), it is nowadays known that $E$ ...

**9**

votes

**2**answers

590 views

### BSD and congruent numbers

Let $n$ be a positive integer, and let $E_n$ denote the elliptic curve $y^2=x^3-n^2x$. By work of Tunnell, it's known that if $E_n$ satisfies the BSD conjecture, then there is an algorithm to decide ...

**-1**

votes

**1**answer

89 views

### Is there an algorithm to find a linear dependence between points on elliptic curves?

Let $E$ be an elliptic curve over a finite field $\mathbb{F}_p$ of characteristic $p$. Let $P,Q\in E(\mathbb{F}_q)$, such that $Q=mP+n\tau(P)$, where $\tau$ is the p-th power of frobenious map and $m$ ...

**3**

votes

**2**answers

286 views

### Congruent numbers and elliptic curves

Who first explicitly stated the link between $N$ being a congruent number and the existence of rational points of infinite order on $y^2=x(x^2-N^2)$?

**1**

vote

**0**answers

135 views

### Elliptic curves with potential good reduction over a prescribed extension

Notation: Let $K/\mathbb{Q}$ be a quadratic number field; let $p\geq 3$ be a rational prime and let $\mathfrak{p}$ denote a prime lying above $p$; let $K_{\mathfrak{p}}$ denote the completion of $K$ ...

**4**

votes

**1**answer

214 views

### Checking whether two rational points of infinite order are generating the torsion free part of an elliptic curve

Let an elliptic curve be given.
As the title says I want to know if we can show that two independent points $P$ and $Q$ are generators of the torsion free part of $E$.
For instance let $E:y^2=x^3-...

**1**

vote

**1**answer

157 views

### Is an Isomorphism from an Abelian variety to a Shimura variety always defined over a solvable extension?

Are there counterexamples to the following:
Given two varieties $A$, $\tilde{A}$, both defined over $\mathbb{Q}$, one of which, say $A$, is a Shimura variety. Then, every isomorphism defined over $\...

**8**

votes

**1**answer

244 views

### Intuitive reasons for the existence of modular parametrizations

Whenever I encounter anything about modular parametrizations, I have a feeling it is something very unnatural: you have some kind of moduli space and all of a sudden it parametrizes an object ...

**13**

votes

**1**answer

670 views

### Reference to a Don Zagier Result and the Congruent Number Problem

I was looking for a reference/explanation as to how Don Zagier managed to find the side lengths of a rational right triangle with area 157. There have been many literature references to the fact that ...

**7**

votes

**2**answers

460 views

### Galois cohomologies of an elliptic curve

I asked this question at math stackexchange but did not get any answer and I was suggested to post the question here.
I am studying basic theory of elliptic curves. I encountered Galois cohomology. ...

**12**

votes

**1**answer

807 views

### What's wrong with my understanding of the scheme $\text{Isom}(E_\lambda, E_{\lambda'})$?

Let $\mathcal{M}_{1,1}$ be the moduli stack of elliptic curves (over the complex numbers). Define
$$\begin{eqnarray*} X &:=& \Bbb{A}^1_{\lambda} - \{0,1\}\\
X' &=& \Bbb{A}^1_{\lambda'} ...

**6**

votes

**1**answer

600 views

### Why are integer points on elliptic curves interesting and useful?

I read some papers which dealed with integer points on elliptic curves. One of these papers are
http://projecteuclid.org/euclid.rmjm/1214947612.
My question is: Why are integer points on elliptic ...

**1**

vote

**2**answers

346 views

### For what integer $n$ are there infinitely many $-a+nb+c = -d+ne+f$ where $a^6+b^6+c^6 = d^6+e^6+f^6$?

(Much revised for clarity.) I was considering the system of equations,
$$-a+nb+c = -d+ne+f\tag1$$
$$a+b+c = d+e+f\tag2$$
$$a^2+b^2+c^2 = d^2+e^2+f^2\tag3$$
$$a^6+b^6+c^6 = d^6+e^6+f^6\tag4$$
...

**4**

votes

**0**answers

96 views

### Minimal discriminant of an elliptic curve in terms of its Galois representation

From the Galois representation of an elliptic curve $E$ we can read the conductor of $E$, and further some information about the minimal discriminant. So is there any more information about the ...

**4**

votes

**2**answers

263 views

### Explicit $2$-descent on elliptic curves

Let $k$ be a field of characteristic $0$ and let
$$E: y^2 = f(x)$$
be an elliptic curve over $k$, with $\mathrm{deg}(f) = 3$. Kummer theory yields a map
$$\varphi:\mathrm{H}^1(k, E[2]) \to \mathrm{H}^...

**10**

votes

**1**answer

311 views

### Integral points on elliptic curves of the form $y^2=x^3+px$

As the title says. Can we determine all the integral points on elliptic curves of the form
$$y^2=x^3+px$$
for a prime $p$? If yes, can someone explain me how? A good reference would also be ...