All Questions
1,978 questions
5
votes
3
answers
2k
views
Elliptic curves on abelian surface
Let $Y$ be an abelian surface. Is it true that for every general point $P \in Y$, there exists an elliptic curve passing through $P$?
CommunityBot
- 1
5
votes
2
answers
977
views
Elliptic curve group law, Sum of intersection points
If a plane curve of degree n intersects an elliptic curve in 3n points, then do those points always sum to zero when added using the group law on the points of an elliptic curve ?
- 3,338
14
votes
1
answer
986
views
P-adic L-functions of nonabelian twists of elliptic curves
Let $E$ be an elliptic curve and $\rho$ an Artin representation of $\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$. Then there is a "twisted L-function" $L(E, \rho, s)$, corresponding to the ...
CommunityBot
- 1
0
votes
2
answers
386
views
Zariski closures of one parameter additive maps in positive characteristic
Suppose we are in characteristic $p$, and that the field, $K$, that we are working over is imperfect. We have a map $\Theta: K \to K^{\delta}$ where each coordinate function $\Theta_i$ is an additive ...
- 30.5k
5
votes
1
answer
710
views
Log resolutions on surfaces and 3-folds in characteristic p
If $X$ is a normal projective variety and $D$ a divisor in it, we say that $\pi\colon (\widetilde X,\widetilde D)\rightarrow (X,D)$ is a log resolution if $\widetilde X$ is a resolution of $X$, the ...
- 10.5k
22
votes
4
answers
2k
views
Does p-adic $L$- function determine the $L$ function
Let $E_1$ and $E_2$ be two Elliptic curve defined over $\mathbb Q$ . Let $p$ be an fixed given odd prime of $\mathbb Q$ at which both the curves have good ordinary reduction. Moreover p-adic $L$-...
CommunityBot
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8
votes
0
answers
873
views
Resolution of singularities in positive characteristic
I am currently trying to make some small parts of the minimal model program work for some very explicit varities in positive characteristic. I have such a variety $X_1$ and I know that there is a ...
- 783
7
votes
2
answers
536
views
What are the polynomial relations between these characteristic 2 "thetas" ?
Suppose $\ell=2m+1$, $m>0$. Define $[i]$ in $\mathbb{Z}/2\mathbb{Z}[[x]]$ to be $$\sum_{n\equiv i\mod l} x^{n^2}.$$ Note that $[0]=1$, and that $[i]=[j]$ whenever $\ell$ divides $i+j$ or $i-j$.
...
- 5,422
9
votes
0
answers
560
views
Function fields of characteristic p modular curves, and mod p reductions of the classical modular equation
Let l and p be distinct primes, l>2. There are "characteristic p modular curves" X_0(l) and X(l), defined over an algebraic closure, K, of Z/p, solving moduli problems for elliptic curves with some ...
- 5,422
5
votes
0
answers
642
views
Can the Galois representation on the $p$-adic Tate module of $E/\mathbf{Q}_p$ be recovered from the $p$-divisible group associated to the mod $p$ good reduction of $E$?
Let $p$ be a prime number, and $E/\mathbf{Q}_p$ and elliptic curve with good reduction $\bar E_p$. Can the Galois representation of $\mathbf{Q}_p$ given by the $p$-adic Tate module of $E$ be recovered ...
- 2,406
1
vote
0
answers
298
views
Can we make a useful ring on an Elliptic curve? [closed]
I know that given an elliptic curve $E$ we can define an addition $+$ over the set of points on the curve to make in an abelian group. However
Can we define multiplication on $E$ in a natural way so ...
- 333
11
votes
1
answer
858
views
Can local duality for elliptic curves be proven with "big rings"?
From Exercise 5.14, Ch. V of Silverman's "Advanced Topics in the Arithmetic of Elliptic Curves", I learned that the local duality for elliptic curves over $p$-adic fields can be proven for Tate curves ...
- 4,807
5
votes
1
answer
948
views
Elliptic curves over the complex numbers: everything "well known"?
This may be a very naive question. We always hear about elliptic curves over the rational numbers, or over other arithmetically significant fields or rings.
But, are there open problems or recent ...
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3
votes
0
answers
204
views
Computing Elliptic Curves of Conductor Divisible by a Large Prime Factor
A little while ago, I came across a paper (or slides from a talk or something) that seemed to suggest that the modular symbol method for computing elliptic curves over $\mathbf{Q}$ of prescribed ...
- 309
2
votes
2
answers
494
views
Fields obtained by adjoining x coordinates of torsion points on elliptic curves
Let $E$ and $E'$ be elliptic curves over a field $K$ of characteristic zero such that $E$ and $E'$ are non-isogenous over $\bar{K}$. Let $l$ be a large prime and suppose that $K(x(E[l]))=K(x(E'[l]))$ (...
- 1,905
3
votes
0
answers
350
views
Cyclotomic fields and singular moduli
Let $\mu$ be the roots of unity and $S$ be the image under the modular $j$-function of all imaginary quadratic $\tau$. Then what is $\mathbb{Q}(\mu)\cap\mathbb{Q}(S)$?
- 1,905
2
votes
1
answer
772
views
Serre's open image theorem for products of elliptic curves over function fields via specialization
In Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15, 259--331 (1972), Serre proved the following (Theorem 6
′′, p. 325):
Let $K$ be a number field and let
$K^...
- 1,905
2
votes
1
answer
406
views
Isomorphism between pull-backs of an F-crystal by different liftings of Frobenius
This might be a naive question. But since I haven't seen this in any reference, I'll try to ask it here. Let $T$ be a smooth scheme over the algebraically closed field $k$ of characteristic $p>0$ (...
- 3,032
5
votes
1
answer
446
views
More questions involving characteristic 2 theta series identities
In my answer to my earlier question, "Existence of certain identities involving characteristic 2 thetas", I established some curious identities when the thetas have prime "level" congruent to 1 mod 4 ...
- 5,422
20
votes
2
answers
2k
views
Frobenius splitting and derived Cartier isomorphism
Let $X$ be a smooth algebraic variety over an algebraically closed field $k$ of characteristic $p>\dim X$. The motivation for my question comes from the following results.
1. If $X$ is Frobenius ...
CommunityBot
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6
votes
1
answer
804
views
Del pezzo surfaces in positive characteristic
For me a Del Pezzo surface $X$ over an algebraically closed field of characteristic $p$ is an algebraic surface where the anticanonical bundle $\omega^{-1}_X$ or $-K_X$ is ample. (I prefer the second ...
- 2,215
17
votes
1
answer
3k
views
How do you calculate the group scheme of E[p] for a an elliptic curve E in characteristic p?
I know that the answer is $\mu_p \times \mathbb{Z}/p\mathbb{Z}$ if $E$ is ordinary, and $\alpha_p$ if $E$ is supersingular, where $\mu_p$ and $\alpha_p$ are the kernels of Frobenius on $\mathbb{G}_m$ ...
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21
votes
1
answer
2k
views
When does the relative differential $df=0$ imply that $f$ comes from the base?
Let $A \to B$ be a map of commutative rings, and $d : B \to I/I^2$ be
defined by $df = f\otimes 1 - 1\otimes f$, where $I$ is the kernel of
$B \otimes_A B \to B$, as in [Hartshorne II.8].
If $df=0$,...
- 11.1k
1
vote
1
answer
453
views
Isogenies from hyperelliptic to elliptic curves
Is it always possible to find an isogeny from a hyperelliptic curve of genus 4, to a 'normal' elliptic curve (genus 1), or a product of elliptic curves?
Are such isogenies easy to compute?
This ...
- 3,338
15
votes
0
answers
779
views
Lifting varieties from char. $p$ to char. 0 after alterations
The question is related to this MO question:
Lifting varieties to characteristic zero.
Let $X$ be a projective smooth variety over $k$ alg. closed field of char. $p.$ Does there always exist an ...
CommunityBot
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4
votes
2
answers
336
views
Analogue of Shafarevich-Ogg's theorem over complex numbers
Let $f:E\to D^*$ be a family of complex elliptic curves parametrized by the punctured open disk $D^*.$ Assume that the monodromy on $H^1$ is trivial (i.e. $R^1f_*\mathbb Z$ is a constant sheaf on $D^*$...
- 66.3k
6
votes
2
answers
590
views
Intersection of field extensions of torsion points of non-isogenous elliptic curves
Let $E$ and $E'$ be non-isogenous elliptic curves over a field $k$ (characteristic 0) such that $Gal(k(E[p^{\infty}])/k)=Gal(k(E'[p^{\infty}])/k) = SL_2(\mathbb{Z}_p)$ with $p \geq 5$ (where $E[p^{\...
- 1,905
2
votes
1
answer
247
views
On pseudo rational modular forms of weight 2 and level N
So consider the $\mathbb{Q}$-vector space $V$ of functions which satisfy the following conditions
(1) $f:\mathbb{H}\rightarrow\mathbb{C}$ is holomorphic. Here $\mathbb{H}$ stands for the
upper half ...
- 176
11
votes
1
answer
615
views
Do Richardson varieties have rational singularities in arbitrary characteristic?
The title basically asks the question. I'll review the relevant terminology and explain what I have and haven't found in the literature.
Let $G$ be a reductive group. Let $v \leq w$ be elements of ...
- 156k
1
vote
1
answer
179
views
What can we say if E^sigma is isogeneous to a twist of E?
Let $K$ be a quadratic number field, and let $E_1$ and $E_2$ be two isogeneous elliptic curves over $K$. Assume we know that $j(E_1)^\sigma=j(E_2)$ where $\sigma$ is the generator of the Galois group ...
- 19.2k
2
votes
1
answer
304
views
Connected extensions of finite by connected algebraic groups
Let $1\to H\to E\to G\to 1$ be a short exact sequence of algebraic groups defined over an algebraically closed field $k$ of characteristic $p$. Suppose $H$ is a finite group, and $G$ and $E$ are ...
- 22.6k
5
votes
1
answer
461
views
Given a branched cover with branch cycle description $(g_1,...,g_r)$, does $g_i$ generate some decomposition group?
Classically:
Let $a_1,...,a_r$ be points in $\mathbb{P}^1_{\mathbb{C}}$, and let $\alpha_1,...,\alpha_r$ be simple loops around the $a_i$, all counterclockwise, and none touching (so $\alpha_1...\...
CommunityBot
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4
votes
2
answers
3k
views
Rank 2 flat bundles on an elliptic curve, via extensions
I have some hopefully elementary questions about rank 2 flat bundles on an elliptic curve $E$.
Take $p\in E$, and consider the exact sequence
$$0\to \mathcal{O}(-p) \to V \to \mathcal{O}(p)\to 0$$
...
- 1,129
2
votes
0
answers
688
views
Elliptic Curves and cryptography. Recommended Reading [closed]
I have been studying RSA cryptography and want to extend this to ECC. I am interested in any books on the topic, that start off with basic principles of elliptic curves as I have almost zero knowledge ...
CommunityBot
- 1
3
votes
1
answer
1k
views
Automorphism groups of Elliptic curves as Galois module
Let $E/k$ be an elliptic curve over a field of characteristic $\neq$ 2, 3. Then we have an isomorphism $ [ \ \ ] :\mu_n \rightarrow\mathrm{Aut}_{\overline{k}}(E)$, $[ \zeta ] : (x,y) \rightarrow (\...
- 47.4k
3
votes
0
answers
281
views
What would be a characteristic-$p$ analogue for $C^{\infty}$-fiber bundles?
I'd like to know a notion for a morphism between algebraic varieties in characteristic $p$ that plays the role of a $C^{\infty}$-fiber bundle. It should be, in particular, flat. I'm not assuming the ...
- 4,265
13
votes
2
answers
935
views
Sums of two cubes
In his solution of the equation $x^3 + dy^3 = 1$, Nagell
comes across the equation
$$ u^3 + 6u^2v + 3uv^2 - v^3 = w^3. $$
He then observes that
$$ (u^3 + 6u^2v + 3uv^2 - v^3) U^3 = V^3 + W^3 $$
for
$$...
CommunityBot
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2
votes
2
answers
787
views
Picard sheaves for elliptic curves
Hi,
my question concerns the Picard bundles in the special case of an elliptic curve $E$, say over the complex numbers. Let $p$ and $q$ be the projections of $E \times E$.
One defines the $n$-th ...
- 21k
13
votes
1
answer
690
views
Obstructions to formally integrating vector fields in characteristic p?
Let $M$ be a smooth scheme over some field $k$ of characteristic $p$, and $\vec X$ a vector field on it. Equivalently, $\vec X$ gives a map $Spec\ k[\epsilon]/\langle \epsilon^2 \rangle \times M \to M$...
CommunityBot
- 1
8
votes
2
answers
1k
views
Distance functions on elliptic curves over number fields
My question originates from the book of Silverman "The Aritmetic of Elliptic Curves", 2nd edition (call it [S]). On p. 273 of [S] the author is considering an elliptic curve $E/K$ defined over a ...
- 11.1k
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 ...
- 13k
12
votes
1
answer
2k
views
Replacement for derivations in characteristic p?
Let $k$ be a field.
If $f \in k[x]$ is a polynomial, and $d/dx\ f = 0$, then either
$f$ is constant, or
$char\ k = p$ and $f \in k[x^p]$.
So "annihilated by all derivations" is perhaps not the right ...
CommunityBot
- 1
1
vote
0
answers
527
views
Galois groups of quadratic number fields
Is there an algorithm that finds,
given a subgroup $G$ of a finite permutation group $S_n$,
a quadratic number field $K=Q(\sqrt a), a \in Q$ and a Polynom $f \in K[x]$ such that the Galois Group of ...
- 79.9k
21
votes
4
answers
2k
views
Simplest example of jumping of cohomology of structure sheaf in smooth families?
Using Hodge theory (and the ill-defined Lefschetz principle), one can show that in characteristic 0, given a proper smooth family $X \rightarrow B$, the cohomology groups of the structure sheaf of the ...
CommunityBot
- 1
9
votes
2
answers
1k
views
Bounding the modular discriminant of an elliptic curve in the j-invariant
Consider an elliptic curve $X=\mathbf{C}/ (\mathbf{Z}+\tau \mathbf{Z})$, where $\tau$ is an element in the complex upper half plane. We define $$\Vert \Delta\Vert(X) = (\Im \tau)^6 \vert q\prod_{k=1}^\...
- 79.9k
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 ...
CommunityBot
- 1
11
votes
1
answer
1k
views
Extensions obtained adding torsion points of an elliptic curve
When adding to the rational the $p$-torsion points $E[p]$ of an elliptic curve we obtain an extension containing the $p$-th roots of the unity, and whose Galois group can be embedded in $GL(2, \mathbb{...
- 57.6k
4
votes
1
answer
627
views
Can the projection (tensor algebra) -> (symmetric algebra) be forced to split in char. p by factoring out p-th powers?
Question 1 (the weak and simple statement, which, I think, already is wrong): Let $p$ be a prime. Let $k$ be a field with characteristic $p$.
For any $k$-vector space $V$, consider the canonical ...
CommunityBot
- 1
12
votes
1
answer
1k
views
Ramification in p-division fields associated to elliptic curves with good ordinary reduction
Dear MO,
Let $p$ be a prime and let $E/\mathbb{Q}_p$ be an elliptic curve. Suppose that $E/\mathbb{Q}_p$ has good ordinary reduction at $p$. In his 1972 paper ``Propriétés galoisiennes des points d'...
- 1,694
32
votes
10
answers
3k
views
Which 'well-known' algebraic geometric results do not hold in characteristic 2?
A smooth curve $X$ in $\mathbb{P}^n$ is strange if there is a point $p$ which lies on all the tangent lines of $X$.
Examples are $\mathbb{P}^1$ is strange and so is $y=x^2$ in characteristic $2$. ...
- 22.6k