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 $$ ...
Jeremy Rouse's user avatar
32 votes
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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 ...
Joe Silverman's user avatar
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 ...
Adam P. Goucher's user avatar
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 (...
Nicolas Mascot's user avatar
23 votes
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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 ...
Filippo Alberto Edoardo's user avatar
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
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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 ...
David Loeffler's user avatar
22 votes
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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 ...
GH from MO's user avatar
  • 98.2k
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 ...
Joe Silverman's user avatar
22 votes
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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 ...
David Loeffler's user avatar
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 ...
Myshkin's user avatar
  • 17.4k
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 ...
Joe Silverman's user avatar
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 ...
François Brunault's user avatar
18 votes
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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 ...
Myshkin's user avatar
  • 17.4k
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 ...
Abdelmalek Abdesselam's user avatar
18 votes
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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 ...
Franz Lemmermeyer's user avatar
18 votes
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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 ...
Joe Silverman's user avatar
18 votes
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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 ...
Joe Silverman's user avatar
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 ...
Jeremy Rouse's user avatar
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 ...
Will Sawin's user avatar
  • 135k
17 votes
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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. ...
Chris Wuthrich's user avatar
17 votes
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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 ...
Zev Klagsbrun's user avatar
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 ...
Pete L. Clark's user avatar
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)^...
Myshkin's user avatar
  • 17.4k
16 votes
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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 ...
Myshkin's user avatar
  • 17.4k
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 ...
Pete L. Clark's user avatar
16 votes
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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 ...
Joël's user avatar
  • 25.7k
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/...
ThiKu's user avatar
  • 10.3k
16 votes
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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 ...
Joe Silverman's user avatar
16 votes
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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 ...
Joe Silverman's user avatar

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