All Questions
221 questions
68
votes
0
answers
4k
views
Constructing non-torsion rational points (over $\mathbb{Q}$) on elliptic curves of rank $>1$
Consider an elliptic curve $E$ defined over $\mathbb Q$. Assume that the rank of $E(\mathbb Q)$ is $\geq2$. (Assume the Birch-Swinnerton-Dyer conjecture if needed, so that analytic rank $=$ algebraic ...
- 20.2k
63
votes
3
answers
8k
views
Is there a "Basic Number Theory" for elliptic curves?
Tate's thesis showed how to profitably analyze $\zeta$ functions of number fields in terms of adelic points on the multiplicative group. In particular, combining Fourier analysis and topology, Tate ...
- 156k
37
votes
3
answers
5k
views
Is there a nice proof of the fact that there are (p-1)/24 supersingular elliptic curves in characteristic p?
If $k$ is a characteristic $p$ field containing a subfield with $p^2$ elements (e.g., an algebraic closure of $\mathbb{F}_p$), then the number of isomorphism classes of supersingular elliptic curves ...
- 45.7k
34
votes
7
answers
3k
views
What heuristic evidence is there concerning the unboundedness or boundedness of Mordell-Weil ranks of elliptic curves over $\Bbb Q$?
Some experts have a hunch that for any nonnegative integer $r$ there are infinitely many elliptic curves over $\Bbb Q$ with Mordell-Weil rank at least $r$.
The best empirical evidence for this hunch ...
- 3,268
33
votes
4
answers
8k
views
Etale cohomology and l-adic Tate modules
$\newcommand{\bb}{\mathbb}\DeclareMathOperator{\gal}{Gal}$
Before stating my question I should remark that I know almost nothing about etale cohomology - all that I know, I've gleaned from hearing off ...
- 7,072
29
votes
1
answer
3k
views
Is the Modularity Theorem (currently) effective?
The Modularity Theorem says every elliptic curve over $\mathbb{Q}$ can be gotten from the classic modular curve $X_0(N)$ by a rational map. Here $N$ is the conductor, easily calculable from a ...
- 11.2k
28
votes
3
answers
2k
views
Intuitive pictures in characteristic p
This is a tough one, but does anyone know of any images that recall characteristic p geometry (over algebraically closed fields) in some sense? It is not enough if it is some picture that can be also ...
- 2,215
27
votes
2
answers
3k
views
Reference for de Rham cohomology in positive characteristic
It is known in characteristic $0$ that (algebraic) de Rham cohomology is a Weil cohomology theory. However, in characteristic $p > 0$ it isn't, if only because it has mod $p$ coefficients, whereas ...
- 28.8k
27
votes
1
answer
3k
views
Fundamental group of the moduli stack of elliptic curves
I've heard that the étale fundamental group of the moduli stack of elliptic curves (over $\mathbb{Z}$) is trivial. Is there an easy proof of that? (Note that there are plenty of étale covers once one ...
- 25.6k
25
votes
5
answers
9k
views
Algorithms for finding rational points on an elliptic curve?
I am looking for algorithms on how to find rational points on an elliptic curve $$y^2 = x^3 + a x + b$$ where $a$ and $b$ are integers. Any sort of ideas on how to proceed are welcome. For example, ...
- 48.8k
25
votes
3
answers
5k
views
Conceptual understanding of the Gross-Zagier theorem.
The Gross-Zagier paper "Heegner points and derivatives of $L$-series", is really computational and hard to plow through. It seems it is futile to read it as such and one must look for a more ...
- 1,417
25
votes
0
answers
1k
views
Status of the Euler characteristic in characteristic p
In the introduction to the Asterisque 82-83 volume on `Caractérisque d'Euler-Poincaré, Verdier writes:
Enfin signalons que la situation en caractéristique positive est loin
d'être aussi ...
- 8,723
25
votes
3
answers
3k
views
Background for the Elkies-Klagsbrun curve of rank 29
Elkies and Klagsbrun have recently announced an elliptic curve over Q of rank (at least) 29, improving on the previous record from 2006!
https://web.math.pmf.unizg.hr/~duje/tors/z1.html
It has trivial ...
- 1,139
24
votes
3
answers
3k
views
Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in\mathbb Z$ be unbounded as $n$ varies?
Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in\mathbb Z$ be unbounded as $n$ varies?
$x,y$ are integral points on an Elliptic Curve and are easy to find using enumeration of divisors of $n$ (...
- 454
22
votes
1
answer
2k
views
Can one prove complex multiplication without assuming CFT?
The Kronecker-Weber Theorem, stating that any abelian extension of $\mathbb Q$ is contained in a cyclotomic extension, is a fairly easy consequence of Artin reciprocity in class field theory (one just ...
- 15.4k
22
votes
1
answer
2k
views
Which elliptic curves over totally real fields are modular these days?
As the title says. In particular, every elliptic curve over $\mathbb{Q}$ is modular; but what is the current state of the art for general totally real number fields? I assume the answer is ...
- 13.1k
21
votes
5
answers
5k
views
Mirror symmetry mod p?! ... Physics mod p?!
In his answer to this question, Scott Carnahan mentions "mirror symmetry mod p". What is that?
(Some kind of) Gromov-Witten invariants can be defined for varieties over fields other than $\mathbb{C}$...
- 21k
20
votes
5
answers
4k
views
Equivalent statements of the Riemann hypothesis in the Weil conjectures
In the cohomological incarnation, the Riemann hypothesis part of the Weil conjectures for a smooth proper scheme of finite type over a finite field with $q$ elements says that: the eigenvalues of ...
- 671
19
votes
1
answer
5k
views
How do you describe vector bundles on elliptic curves?
Throughout "curve" means smooth projective curve over an algebraically closed field.
Motivation and Background
I read somewhere that Atiyah has classified vector bundles on elliptic curves. My ...
- 3,968
19
votes
1
answer
1k
views
Are Q-curves now known to be modular?
I really should know the answer to this, but I don't, so I'll ask here.
A Q-curve is an elliptic curve E over Q-bar which is isogenous to all its Galois conjugates. A Q-curve is modular if it's ...
- 41.4k
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
19
votes
3
answers
3k
views
Can you show rank E(Q) = 1 exactly for infinitely many elliptic curves E over Q without using BSD?
Let $K$ be a number field and let $\mathcal O_K$ be the ring of integers. Following this paper of Cornelissen, Pheidas, and Zahidi, a key ingredient needed to show that Hilbert's tenth problem has a ...
- 3,268
19
votes
4
answers
5k
views
Birationally transforming a quartic elliptic curve
Consider the elliptic curve
$$y^2=ax^4+cx^2+dx+f$$
where I assume complex coefficients for the purposes of this question.
I am aware that there are algorithmic methods for birationally transforming ...
18
votes
1
answer
1k
views
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)?
Given $a,b,c\in \Bbb{N}$ such that $\{a,b,c\}$ are coprime natural numbers and $a,b,c>1$. When
$$\frac{a^2}{b+c}＋\frac{b^2}{c+a}＋\frac{c^2}{a+b}\in\mathbb Z\,?$$
I know the solution $\{183,77,13\}$...
- 183
18
votes
1
answer
1k
views
Largest rank assumed by infinitely many elliptic curves
One of the most interesting questions in Mathematics concerns the Mordell-Weil rank of the group of rational points on elliptic curves $E/\mathbb{Q}$, namely whether this quantity is bounded as one ...
- 27k
18
votes
3
answers
3k
views
Lifting varieties to characteristic zero.
If you want to compute crystalline cohomology of a smooth proper variety $X$ over a perfect field $k$ of characteristic $p$, the first thing you might want to try is to lift $X$ to the Witt ring $W_k$ ...
- 4,015
18
votes
1
answer
5k
views
About isogeny theorem for elliptic curves
$K$ a number field, $G_K$ its Galois group, $E_1, E_2$ two elliptic curves defined over $K$. The isogeny theorem says that if for some prime number $\ell$, The Tate modules (tensored with $\mathbb{Q}$)...
- 1,503
16
votes
2
answers
1k
views
Congruence for the number of points in the elliptic curve $y^2 = x^3+b \pmod{p}$
Let $E$ be the elliptic curve $y^2=x^3+1$ and $p \equiv 1 \pmod{3}$ a prime. Computing the number of points mod $p$ of $E$ using the naive method gives:
$$ \#E(\mathbb F_p) = 1+ \sum_{x=0}^{p-1} \left(...
- 854
16
votes
4
answers
3k
views
Torsion subgroups in families of twists of elliptic curves
Here is the short version:
Fix an elliptic curve $E/\mathbb{Q}$. How does the torsion structure $E_d(\mathbb{Q})_{tors}$ vary, as $E_d$ runs through the quadratic twists of $E$?
Here is the longer ...
- 831
16
votes
4
answers
1k
views
Geometric meaning of fiber of modular parameterization over a point of an elliptic curve?
Given an elliptic curve $E/\mathbb{Q}$ of conductor $N$, parameterization $\psi : X_0(N) \rightarrow E$, and a point $P \in E$, take the fiber $\psi^{-1}(P)$. Its points, being on $X_0(N)$, correspond ...
- 4,593
16
votes
2
answers
3k
views
The parity conjecture
The parity conjecture for elliptic curves predicts that the rank of an elliptic curve
defined over the rationals has the same parity as the p-Selmer rank for a prime number p. Could anyone familiar ...
- 32.6k
16
votes
1
answer
1k
views
Examples of elliptic curves over $\mathbb{Q}$
I need examples of two non-isogenous elliptic curves $E_{1}, E_{2}$ over $\mathbb{Q}$ having the following 2 properties -
1) $E_{1}, E_{2}$ have no rational torsion points.
2) $E_1[9] \cong E_2[9]$ ...
- 1,209
16
votes
2
answers
4k
views
Elliptic Curves over Rings?
So an elliptic curve $E$ over a field $K$ is a smooth projective nonsingular curve of genus $1$ together with a point $O \in E$.
I was reading Silverman's "Arithmetic of Elliptic Curves" and it ...
- 1,458
15
votes
2
answers
845
views
Elements of arbitrary large order in the first Galois cohomology of an elliptic curve
Let $E$ be an elliptic curve over $k=\mathbb{Q}$. Consider $H^1(k,E)$.
In this answer Daniel Loughran writes: "I'm pretty sure that this cohomology group has elements of arbitrarily large order". I ...
- 14.2k
15
votes
0
answers
591
views
For how many primes does an elliptic curve over a totally imaginary field have supersingular reduction?
An elliptic curve over a finite field, $k$, of characteristic p is called supersingular if it has no $p$-torsion over $k^{\mathrm{alg}}$, or equivalently, if $\mathrm{End}(E)$ is an order in a ...
- 605
15
votes
1
answer
858
views
components of E[p], E universal in char p.
I have just realised that a group scheme I've known and loved for years is probably a bit wackier than I'd realised.
In this question, in Charles Rezk's answer, I erroneously claim that his ...
- 41.4k
15
votes
3
answers
3k
views
Rational Points on $y^2=x^3-86069^5$
The analytic rank of the Mordell elliptic curve $y^2=x^3-86069^5$ indicates that it has rank 2. However, deriving a set of generators, and hence the regulator, is proving to be a little bit of an ...
- 791
15
votes
1
answer
2k
views
Fermat's Bachet-Mordell Equation
Fermat once claimed that the only integral solutions to $y^2 = x^3 - 2$ are $(3, \pm 5)$.
Fermat knew Bachet's duplication formulas (more precisely, Bachet had a formula for computing what we call $-...
- 32.6k
15
votes
2
answers
814
views
Can the failure of the multiplicativity of Euler factors at bad primes be corrected?
Warning: This one of those does-anyone-know-how-to-fix-this-vague-problem questions, and not an actual mathematics question at all.
If $X$ is a scheme of finite type over a finite field, then the ...
- 9,418
15
votes
2
answers
1k
views
Structure of $E(Q_p)$ for elliptic curves with anomalous reduction modulo $p$
For simplicity, take $p\ge7$ a prime and $E/\mathbb{Q}$ an elliptic curve with good anomalous reduction at $p$, i.e., $|E(\mathbb{F}_p)|=p$. There is a standard exact sequence for the group of points ...
- 47.4k
15
votes
1
answer
1k
views
Number of curves over a finite field
Let $K$ be a finite field. Is there a formula for the number of isomorphism classes of genus $g$ smooth curves over $K$?
In other words does there exists a formula for the number of rational points ...
- 8,998
14
votes
3
answers
4k
views
Recent progress toward Birch and Swinnerton-Dyer conjecture
Has there been any progress toward the Birch and Swinnerton-Dyer conjecture after
The current status of the Birch & Swinnerton-Dyer Conjecture
- 141
14
votes
3
answers
2k
views
A question on K_1 of an elliptic curve
Consider an elliptic curve $E/ \mathbb{Q}$, with a regular model $\mathcal{E} / \mathbb{Z}$. We have (Beilinson regulator) maps
$$ K_1(\mathcal{E})^{(2)} \to K_1(E)^{(2)} \to H_D^3(E_{/ \mathbb{R}} , \...
- 5,637
14
votes
1
answer
1k
views
Do varieties with ample canonical bundle have finite automorphism group in small characteristic?
Suppose $X$ is a smooth projective variety over a field $k$, with ample canonical bundle. If $\operatorname{char}(k)=0$ or $\operatorname{char}(k)>\dim(X)$ and $X$ lifts to $W_2(k)$ (thanks ...
- 23k
13
votes
3
answers
709
views
For an elliptic curve $E/\mathbb{Q}$ can the cohomology group $H^1(\text{Gal}(\mathbb{Q}(E[p])/\mathbb{Q}), E[p])$ be nontrivial?
Suppose that $E$ an elliptic curve defined over $\mathbb{Q}$ and $p$ an odd prime. Let $G=\text{Gal}(\mathbb{Q}(E[p])/\mathbb{Q})$. I am wondering whether the cohomology group $H^1(G, E[p])$ can be ...
- 429
13
votes
2
answers
1k
views
Best bounds toward Serre's uniformity conjecture
If $E$ is a non-CM elliptic curve over $Q$, then it is a famous theorem of Serre
that there is some integer $M(E)$ such that for any prime $\ell > M(E)$, the image of the
Galois representations $\...
- 26.1k
13
votes
2
answers
1k
views
Are Kato's zeta elements integral?
Let $E$ be an elliptic curve over $\mathbb{Q}$ and $T$ the $p$-adic Tate module of $E$. Kato's Euler system, constructed in the paper "P-adic Hodge theory and values of zeta functions of modular forms"...
- 37k
13
votes
2
answers
928
views
A reference for $\mathbb{A}^1_R$ being a coarse moduli space of the stack of elliptic curves
Let $R$ be a ring and let $\mathcal{M}_{1, R}$ be the algebraic $R$-stack of elliptic curves (over $R$-schemes as bases). One knows that the coarse moduli space of $\mathcal{M}_{1, R}$ is supposed to ...
- 1,103
12
votes
3
answers
1k
views
The order of the discriminant of a good-reduction elliptic curve
Notation. Let $p$ be a prime number, $K$ a finite extension of
$\mathbb{Q}_p$ and $E|K$ an elliptic curve which has good reduction.
The discriminant $d_{E|K}$ of $E|K$ is an element of the
...
- 18.7k
12
votes
0
answers
676
views
Kihara-like Z/6Z elliptic curve families
Shoichi Kihara constructed a family of elliptic curves with Mordell–Weil group $\mathbb{Z}/6\mathbb{Z}\times\mathbb{Z}^3$ (generic rank at least 3) in 2006. Kihara's family produces a number of rank 8 ...
- 913