40
votes
Accepted
When $\frac{a^2}{b+c}+\frac{b^2}{a+c}+\frac{c^2}{a+b}$ is integer and $a,b,c$ are coprime natural numbers, is there a solution except (183,77,13)?
Yes, there is another solution. The next one I found is a bit big, namely
$$ a = 15349474555424019, b = 35633837601183731, c = 105699057106239769. $$
This solution also satisfies the property that
$$ ...
32
votes
Accepted
Cryptography and elliptic curves
Not directly, as far as I know, since explicitly computing large multiples of points in $E(\mathbb Q)$ is infeasible. However, people have considered lifting points from $E(\mathbb F_p)$ to $E(\mathbb ...
30
votes
Is there any theory why (for Bitcoin) the discrete logarithm problem is so hard to solve?
Yes. I'll talk about why elliptic curve discrete log is harder than ordinary discrete log.
Suppose we have $g, h$ and want to find $n$ such that $g^n = h$.
The usual methods for solving the discrete ...
24
votes
Why is this "the first elliptic curve in nature"?
I actually only wrote the part that says that this curve is a model for $X_1(11)$, not the first part, which I think was written by John Cremona.
It is standard to order elliptic curves by conductor (...
23
votes
Accepted
Proof of Prop 1.1 in Wiles' "Modular elliptic curves and Fermat's last theorem"
Well, although there is a typo (Wiles forgot to close his parenthesis, and wrote $H^2(G,\mu_{p^r}\to H^2(G,\mu_{p^s})$ in his proof), his claim is correct. Let, as ibid. $F$ be the finite extension of ...
22
votes
congruent number problem
The fatal error in the reasoning in the paper "L-functions and rational points on a CM elliptic curve via the classical number theory" at the top of the list of papers at
http://kazuma-morita.jimdo....
22
votes
Accepted
Modular forms from counting points on algebraic varieties over a finite field
The correct setting for this construction turns out to be projective varieties, so let me suppose we have a smooth variety $X$ inside $\mathbf{P}^N$, for some $N \ge 1$, defined by the vanishing of ...
22
votes
Accepted
Conjecture: The number of points modulo $p$ of certain elliptic curve is $p$ or $p+2$ for $p$ of form $p=27a^2+27a+7$
The conjecture is true. To prove it, we can restrict to $k=1$ as explained in the comments. Let $\chi$ denote a cubic Dirichlet character modulo $p$; this exists as $p\equiv 1\pmod{3}$, and it is ...
22
votes
Is there any theory why (for Bitcoin) the discrete logarithm problem is so hard to solve?
As many people have said, the reason that people believe that factoring and discrete log in $\mathbb F_p^*$ are hard, and that discrete log in $E(\mathbb F_p)$ is even harder, is because a lot of ...
22
votes
Accepted
Quaternionic and octonionic analogues of the Basel problem
This isn't really a full answer, but it's too long for a comment, and perhaps it's informative all the same.
Your sum $S_k[\mathcal{O}]$ can be written as the value at $s = k$ of the sum
$$\sum_{0 \ne ...
21
votes
Recent progress toward Birch and Swinnerton-Dyer conjecture
No, the conjecture is still wide open for rank $r\geq 2$.
The closest thing to progress is the work of Bhargava and Shankar that quantifies the rank $0$ case and shows that BSD holds for a positive ...
19
votes
Possible groups of K-rational points for elliptic curves over arbitrary fields
The structure of $E(K)$ for $K$ a complete local field, say a finite extension of $\mathbb Q_p$ or $\mathbb C_p$, is quite standard. Let $E_0(K)$ denote the set of points with good reduction. Then ...
19
votes
Accepted
Is there a substitution that relates every Fermat curve to an elliptic curve?
The following reference seems to answer the question of which Fermat curves admit a non-constant map to an elliptic curve:
Neal Koblitz, David Rohrlich, Simple factors in the Jacobian of a Fermat ...
18
votes
Accepted
Is the Tate-Shafarevich group of a rational elliptic curve finite?
MO is not the place to discuss the validity of preprints, but I think it is safe to say that the finitiness of the Tate-Shafarevich group for elliptic curves over $\mathbb{Q}$ is considered an open ...
18
votes
Recent progress toward Birch and Swinnerton-Dyer conjecture
Benedict Gross recently gave a series of lectures here at the University of Virginia on things related to the Birch and Swinnerton-Dyer Conjecture. One of the recent notable developments he mentioned ...
18
votes
Accepted
When did people start thinking of elliptic curves as groups?
The first mathematician who talked about groups of points on elliptic curves (in the sense of Galois, i.e., in the modern sense of the word group) was
Juel [Ueber die Parameterbestimmung von Punkten ...
18
votes
Accepted
Why do Bhargava-Skinner-Zhang consider the ordering by height?
Roughly speaking, the height of an arithmetic object (number, variety, ...) is a natural measure of its complexity, say in the sense of "how many bits does it take to describe the object." (This is ...
18
votes
Accepted
Can someone suggest a path to study Mordell-Weil theorem for someone studying on their own?
Milne's and Knapp's books are excellent. As for my AEC, it covers lots of stuff that's not needed if you just want to get to Mordell-Weil. So for AEC, here's a path to MW:
If you know some algebraic ...
17
votes
Accepted
Are there integer solutions to $3y^2 = 4x^3-1$ other than $(1,1)$ and $(1,-1)$?
The projective form of your curve is $3y^{2} z = 4x^{3} - z^{3}$. This has three obvious points: $(1 : 1 : 1)$, $(1 : -1 : 1)$, and $(0 : 1 : 0)$.
Your curve is isomorphic over $\mathbb{Q}$ to the ...
17
votes
Accepted
The valuation of j-functions vs number of isomorphisms for an elliptic curve
Yes, let's use the fact that the moduli stack of elliptic curves is etale-locally a scheme. We could also use the formal deformation space of the elliptic curve mod $\pi$ (of course we may assume $E_1 ...
17
votes
Accepted
Integer points of one Mordell equation
This curve has rank 0 over $\mathbb{Q}$. The 2-descent fails to determine this, because the $2$-torsion subgroup of the Tate-Shafarevich group is non-trivial. Instead, one can compute the $L$-value. ...
17
votes
Accepted
Hard: One more generator needed for a Z/6 elliptic curve
A set of points that generate $E(\mathbb{Q})$ modulo torsion is given by
...
16
votes
CM $j$-invariants in $p$-adic fields
Good question. I am leaving some first thoughts now; I hope to have a chance to think more about it later.
Because CM elliptic curves have potentially good reduction, the $j$-invariant is an ...
16
votes
Birationally transforming a quartic elliptic curve
The method explained in Husemöller's book on elliptic curves is as follows:
Take a general quartic $v^2=f_4(u)=a_ou^4+a_1u^3+a_2u^2+a_3u+a_4$, and let
$$u=\frac{ax+b}{cx+d}\qquad v=\frac{ad-bc}{(cx+d)^...
16
votes
Accepted
Possible groups of K-rational points for elliptic curves over arbitrary fields
By Mordell-Weil, for any number field $K$ we have
$$C(K)=\mathbb{Z}^r \times E(K)_{\mathrm{tors}}$$
As you mention, Mazur showed all the possible options for $E(\mathbb{Q})_{\mathrm{tors}}$ in his ...
16
votes
Possible groups of K-rational points for elliptic curves over arbitrary fields
The answer to one possible interpretation of the title question -- vary over all elliptic curves over all fields and ask which groups arise -- is given in this paper.
With regard to the structure of ...
16
votes
Accepted
Tate-Shafarevich group over number fields
No.
It is always difficult to "prove" that something is "not known", but this may do: I claim it is not even known when $K=\mathbb Q$, $A$ is an elliptic curve $E$. In fact in this case, the result ...
16
votes
When did people start thinking of elliptic curves as groups?
H. POINCARÉ
Sur les propriétés arithmétiques des courbes algébriques
Journal de mathématiques pures et appliquées 5e série, tome 7 (1901), p. 161-234.
http://sites.mathdoc.fr/JMPA/PDF/...
16
votes
Accepted
Bounded Torsion, without Mazur’s Theorem
By all means hold out hope, but I don't think that the ideas in Dem'janenko's papers are going to work. I spent a lot of time in grad school looking at them.
If I remember correctly, Dem'janenko ...
16
votes
Accepted
Counterpart of cyclotomic polynomials for elliptic divisibility sequences
The counterpart of the cyclotomic polynomials are elliptic division polynomials, which can be defined recursively by a non-linear recursion (usually presented as a pair of recursions, one for odd ...
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