All Questions
1,978 questions
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
7
votes
1
answer
799
views
Liftability of Enriques Surfaces (from char. p to zero)
Let $k$ be an algebraically closed field of characteristic $p > 0$, $X$ a variety over $k$.
We say $X$ lifts to characteristic zero, if there exists a local ring $R$ containing $\mathbb Z$ with ...
- 839
1
vote
3
answers
338
views
Homomorphism of Legendre curve
Let E be an elliptic curve over a finite field k (char(k) is not 2) be given by y^2 = (x-a)(x-b)(x-c) where a,b and c are distinct and are in k. Then why is (c,0) is in [2]E(k) iff c-a and c-b is a ...
- 71
8
votes
1
answer
471
views
Is there an R=T type result for modular forms with additive reduction?
Let E be an elliptic curve over the rationals with conductor $Mp^2$ with p>5 and M and p coprime, and let $\rho$ be the Galois representation attached to the p-torsion points of E. Is there a way to ...
- 818
3
votes
1
answer
868
views
isogeny of elliptic curves
Let $E$ and $F$ be two abelian varieties of dimension 1 over $\mathbb{C}$. Let $f : E \to F$ be a surjective homomorphism of abelian varieties ($f(0) = 0$). If $\ker (f) \cong \mathbb{Z}/2\mathbb{Z} ...
- 363
5
votes
3
answers
2k
views
Additive reduction of elliptic curves
Suppose $E/ \mathbf{Q}$ is an elliptic curve with additive reduction at a prime $p$. Is there an easy way to tell if $E$ is a quadratic twist of an elliptic curve $E'/\mathbf{Q}$ with good reduction ...
- 13.1k
3
votes
2
answers
661
views
Repeated digits of squares in different bases
Hello, I am Mahima. I would like to ask the following clarifications. If any one answered, I am so thankful to you.
In which bases is 1111 a square?
b^3 + b^2 + b + 1 = n^2.
(b + 1)(b^2 + 1) = n^2.
...
- 31
9
votes
1
answer
1k
views
Visualizing a complex plane cubic together with the real plane
In Alain Roberts "Elliptic curves: notes from postgraduate lectures given in Lausanne 1971/72" page 11 (available on google books unless you already tried to read another chapter), there is a hand ...
- 4,394
12
votes
3
answers
815
views
Decomposition of Tate-Shafarevich groups in field extensions
Suppose $E/\mathbb{Q}$ is an elliptic curve with rank zero. According to the conjecture of Birch and Swinnerton-Dyer, the special value $L(1,E_{/\mathbb{Q}})$ should be equal (up to some harmless ...
- 13.1k
16
votes
5
answers
8k
views
Is the ABC conjecture known to imply the Riemann hypothesis?
I once heard from a graduate student that the ABC conjecture implies the Riemann hypothesis. I can't find a reference for this, but given the department the student is from I tend to believe he might ...
- 3,268
9
votes
2
answers
657
views
Is it possible for the repeated doubling of a non torsion point of an elliptic curve stays bounded in the affine plane?
Let $P=(x_1,y_1)$ be a non torsion point on an elliptic curve $y^2=x^3+Ax+B$.
Let $(x_n,y_n)=P^{2^n}. x_n,y_n$ are rationals with heights growing rapidly. Can ${x_n} {y_n}$ stay bounded?
- 111
12
votes
3
answers
2k
views
What is the etymology for the term conductor?
This is related to the previous question of how to define a conductor of an elliptic curve or a Galois representation.
What motivated the use of the word "conductor" in the first place?
A friend ...
- 3,268
12
votes
0
answers
716
views
Lifting abelian varieties in (the closed fiber of) a fixed Neron model
Suppose that $R$ is a dvr with field of fractions $K$ and residue field $k$ and that $A_K$ is an abelian variety over $K$ with Neron model $A$ over $R$. Then the closed fiber $A_k$ is a smooth ...
- 1,609
6
votes
2
answers
2k
views
j-invariant of a supersingular elliptic curve
Let E be a supersingular curve over a finite field. Why is the j-invariant always in F_p^2?
- 71
9
votes
4
answers
3k
views
reduction of CM elliptic curves
Can someone indicate how to prove the following equivalences for a CM elliptic curve $E$:
(i) $p$ is inert in End($E$)
(ii) $E_p$ is supersingular
(iii) The trace of the Frobenius at $p$ is $0$ [...
user19475
24
votes
6
answers
15k
views
What are the recommended books for an introductory study of elliptic curves?
I am currently doing a self study on algebraic geometry but my ultimate goal is to study more on elliptic curves. Which are the most recommended textbooks I can use to study? I need something not so ...
- 513
5
votes
2
answers
794
views
Is there a presentation of the cohomology of the moduli stack of torsion sheaves on an elliptic curve?
Let $E$ be your favorite elliptic curve, and let $Tor^m$ be the moduli stack of torsion sheaves of degree $m$ on $E$. This sounds horrible, but it's not so bad; it's a global quotient of a smooth ...
- 44.7k
4
votes
2
answers
402
views
lower bound for torsion of abelian varieties
Let $A$ be an abelian variety defined over a field $K$ of characteristic $p>0$. Let $A[\ell]$ be the group of $\ell$-torsion points, $\ell\neq p$ a prime. Are there positive constants $C, \eta$ ...
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
4
answers
1k
views
The ring of algebraic integers of the number field generated by torsion points on an elliptic curve
(Warning: a student asking)
Let $E$ be an elliptic curve over $\mathbf Q$. Let $P(a,b)$ be a (nontrivial) torsion point on $E$. Is there an easy description of the ring of algebraic integers of $\...
- 889
3
votes
2
answers
732
views
If the morphism of root data induced by an isogeny of a reductive group is a Frobenius, is then the isogeny itself a Frobenius?
Let $G$ be a reductive (or just semisimple) algebraic group over an algebraically closed field $k$ of characteristic $p > 0$, let $T$ be a maximal Torus and let $f:G \rightarrow G$ be an isogeny. ...
- 5,243
15
votes
3
answers
3k
views
How do you explicitly compute the p-torsion points on a general elliptic curve in Weierstrass form?
Consider the Weierstrass cubic
$$y^2z = x^3 + A\, xz^2+B\,z^3.$$
This defines a curve $E$ in $\mathbb{P}^2$, which if smooth is an elliptic curve with basepoint at $[0,1,0]$.
I'm interested in having ...
- 27.2k
32
votes
4
answers
4k
views
Modular curves of genus zero and normal forms for elliptic curves
This is maybe the first question I actually need to know the answer to!
Let $N$ be a positive integer such that $\mathbb{H}/\Gamma(N)$ has genus zero. Then the function field of $\mathbb{H}/\Gamma(N)...
- 118k
7
votes
2
answers
1k
views
Hochschild Kostant Rosenberg theorem for varieties in positive characteristic?
Is there is a known version of the HKR theorem as proved in say Swan's paper "Hochschild Cohomology of Quasiprojective Varieties" in positive characteristic? I assume something is known about this as ...
- 3,758
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
5
votes
3
answers
739
views
Smoothness of hyperplane sections
Suppose $X\subset \mathbb{P}^n$ is a smooth hypersurface defined over $\mathbb{Q}$. For a "generic" prime $p$, what can be said about the set of hyperplanes $H$ in $\mathbb{P}^n(\mathbb{F}_p)$ for ...
- 13.1k
15
votes
6
answers
6k
views
bad reduction for elliptic curves
Why do elliptic curves have bad reduction at some point if they are defined over Q, but not necessarily over arbitrary number fields?
- 1,022
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
13
votes
2
answers
4k
views
Example of connected-etale sequence for group schemes over a Henselian field?
Can someone give a really concrete example of such a sequence? I am looking at several notes related with such things, but haven't seen any well-calculated example. And I'm really confused at this ...
- 1,503
7
votes
4
answers
736
views
Simply connected quasi-projective varieties in positive characteristic
I am looking for examples of non-projective (quasi-projective) varieties $X$ defined over a field of positive characteristic, which have trivial étale fundamental group.
It is well known that the ...
- 4,450
18
votes
1
answer
2k
views
What's the Hilbert class field of an elliptic curve?
My question points in a direction similar to Qiaochu's, but it's not the same (or so I think). Let me provide you with a little bit of background first.
Let E be an elliptic curve defined over some ...
- 32.6k
0
votes
1
answer
962
views
Quadratic Twist of Legendre Form
What is the quadratic twist of an elliptic curve in Legendre Form?
How do you show an elliptic curve and its quadratic twist is isomorphic when they are in Legendre Form?
- 11
22
votes
3
answers
2k
views
One dimensional (phi,Gamma)-modules in char p
I would like to better understand the simplest case of the correspondence between Galois representations and (phi,Gamma)-modules. Namely, consider 1-dimensional Galois representations of $G_{Q_p}$ ...
- 1,680
9
votes
1
answer
763
views
Restriction theorems over finite fields
A short while ago, Dvir proved the Kakeya conjecture over finite fields. Does this have any implications for restriction theorems over finite fields? I am aware only of implications going in the ...
- 20.2k
19
votes
1
answer
2k
views
The Infinitesimal topos in positive characteristic
This question was inspired by and is somewhat related to this question.
In his article "Crystals and the de Rham cohomology of schemes" in the collection "Dix exposes sur la cohomologie ...
- 4,450
6
votes
2
answers
507
views
Concerning the dimension of a complex variety modulo a prime
Let V be a complex affine variety given as the vanishing set of a set of polynomials with integral coefficients. I have 3 questions.
1)
Under what assumption will the dimension of V over C remain ...
- 63
2
votes
2
answers
1k
views
Isomorphic elliptic curves
If we have an elliptic curve E over a field k and we pick a non-square d in k-{0}. Suppose
E is isomorphic to E^(d). (E^(d) is the quadratic twist) Why must j(E) = 1728 and why is k(sqrt(d)) = k(...
- 21
4
votes
2
answers
923
views
What is the correct formulation of the CDE triangle?
The CDE triangle with C Cartan matrix and D decomposition matrix is well-known for finite groups. I have not seen a full account of this for finite dimensional algebras which I find surprising as ...
- 9,132
10
votes
3
answers
2k
views
Is $Sym^n (V^*) \cong Sym^n (V)^\ast$ naturally in positive characteristic?
Background/motivation
It is a classical fact that we have a natural isomorphism $Sym^n (V^*) \cong Sym^n (V) ^\ast$ for vector spaces $V$ over a field $k$ of characteristic 0. One way to see this is ...
- 14.8k
25
votes
3
answers
2k
views
product of all F_p, p prime
Let $R$ be the ring $$R = \prod_{p\ \text{prime}} \mathbb{F}_p$$ where $\mathbb{F}_p$ is the field having $p$ elements.
Is it true that $R$ has a quotient by a maximal ideal which is a field of ...
- 5,163
27
votes
4
answers
3k
views
Have people successfully worked with the full ring of differential operators in characteristic p?
This question is inspired by an earlier one about the possibility of using the full ring of differential operators on a flag variety to develop a theory of localization in characteristic $p$. (Here ...
- 57.6k
0
votes
1
answer
216
views
Q-isogeny and Q-torsion subgroup
What is meant by a Q-isogeny and the Q-torsion subgroup? (And by Q, I mean rational 'Q')`
`
- 71
13
votes
0
answers
943
views
Beilinson-Bernstein localization in positive characteristic
This is a follow-up to this question; in particular, I'm wondering if anyone can expand upon the interesting answers given by Kevin McGerty and David Ben-Zvi there. (In particular, in this question I'...
- 3,637
10
votes
3
answers
2k
views
Is this naive test to tell whether a complex elliptic curve has complex multiplication effective?
I have a question about a naive test to tell whether a complex elliptic curve $E$ has complex multiplication.
Recall that the endomorphism ring $End(E)$ of $E$ is isomorphic to either $\mathbb{Z}$ or ...
user1073
10
votes
2
answers
1k
views
Integrable dynamical system - relation to elliptic curves
From seminar on kdV equation I know that for integrable dynamical system its trajectory in phase space lays on tori. In wikipedia article You may read (http://en.wikipedia.org/wiki/Integrable_system):
...
- 1,626
34
votes
2
answers
3k
views
The work of E. Artin and F. K. Schmidt on (what are now called) the Weil conjectures.
I was reading Dieudonne's "On the history of the Weil conjectures" and found two things that surprised me. Dieudonne makes some assertions about the work of Artin and Schmidt which are no doubt ...
- 41.4k
7
votes
3
answers
3k
views
congruent to 1 mod p
This is a somewhat vague question: for a prime number p, we often see that various counts come out to be 1 modulo p. What are the possible reasons for this?
Here are some I've encountered:
For some ...
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
2
votes
5
answers
2k
views
CM of elliptic curves
This question is related to this one.
Tate module of CM elliptic curves
There seem to be several versions of "complex multiplication".
Fact 1: We say $E/\mathbb{C}$ has CM if $End_C(E) \supsetneq Z$. ...
- 1,503
7
votes
1
answer
220
views
Cyclic extensions coming from E[p] \equiv F[p],
Let p be a prime and let K be a field containing the p'th roots of unity. Let E be an elliptic curve over K. We consider the the moduli problem $Y_E(p)$, which sends L to set of elliptic curves F/L, ...
- 818