All Questions
1,978 questions
4
votes
1
answer
448
views
Quantitative lifting for mod-p elliptic curves to characteristic zero CM elliptic curves
Let $A$ be a supersingular elliptic curve over $\mathbb{Z}/p\mathbb{Z}$ and $\mathcal{O}$ an order in an imaginary quadratic field contained in the quaternion algebra $\operatorname{End}(A)$, then by ...
- 685
2
votes
0
answers
241
views
Inertia group representation from $p^{n}$-torsion of ordinary elliptic curve
Let $K$ be a complete local field. Suppose that $K$ is an unramified extension of $\mathbb{Q}_{p}$ and let $E$ be an elliptic curve over $K$ with good ordinary reduction. Let $G_{K}=\text{Gal}(\...
- 305
1
vote
0
answers
237
views
Tate uniformization and reduction of elliptic curves
Let $E$ be an elliptic curve over $K$ (nonarchimedean) with $j$-invariant satisfying $|j(E)|>1$.
Tate uniformization theorem says that we have an isomorphism : $E \simeq \mathbf G_m/q^{\mathbf Z}$.
...
- 11
3
votes
0
answers
105
views
Points with residue fields having big inseparability degree cannot be contained in smooth hypersurfaces
Let $X$ be a $k$-scheme over imperfect field $k$ and $x \in X $ some (reduced) point with residue field $\kappa(x) = \mathcal{O}_{X,x}/ \mathfrak{m}_x$. How to check that if $\kappa(x)$ has "big ...
- 619
7
votes
2
answers
636
views
Mordell curves with large rank
An elliptic curve is (for the purpose of this question) a cubic algebraic curve defined by an equation (short Weierstrass equation) of the form
$$\displaystyle E_{a,b} : y^2 = x^3 + ax + b, a, b \in \...
- 27k
2
votes
1
answer
370
views
Elliptic curve with CM and image of Galois representation in normalizer of nonsplit Cartan
I am trying to understand the following. Let $E/\mathbb{Q}$ be an elliptic curve with complex multiplication given by the ring of integers $\mathcal{O}_K$. We are given a fixed rational prime $p$ ...
- 637
2
votes
0
answers
137
views
Tangential basepoint of a log singular local system
Consider the Legendre family $f: X\longrightarrow Y = \mathbb{P}^1\setminus\{0,1,\infty\}$, defined over $K = \mathbb{Q}/\mathbb{Q}_p$.
having fibre above $t\in Y$, the elliptic curve $E_t := y^2 = x(...
- 2,907
3
votes
0
answers
303
views
Analytic continuation of $L$-functions of base changed elliptic curves
Suppose that $E$ is an elliptic curve over $\mathbf{Q}$. Let $K$ be a number field and let $L(E/K, s)$ be the Hasse-Weil $L$-function of $E$ base-changed to $K$. The modularity theorem tells us that $...
- 1,949
6
votes
0
answers
163
views
Explicit computations of Serre duality for elliptic curves
I have an elliptic curve $E$ defined over a ring $R$, I want to compute the pairing
$$
H^1(E,\mathcal{O}_E)\times H^0(E, \Omega_E^1){\rightarrow}R.
$$
Clearly we have that $H^0(E, \Omega_E^1)=R \...
- 109
2
votes
0
answers
254
views
When is a prime considered to be ramified, split or inert in a non-maximal order of an imaginary quadratic number field?
I am reading this paper on "Averages of Elliptic curve constants" here and in section 2.2 page no. 693 the formula for the conjectural constant in the asymptotics of the Lang-Trotter ...
- 309
0
votes
1
answer
324
views
An identity for Weierstrass elliptic functions evaluation
Let $\wp(z), \zeta(z)$ and $\sigma(z)$ be the Weierstrass $\wp$, zeta and sigma functions associated to the ODE:
$$\wp'(z)^2 = 4(\wp(z)-e_1)(\wp(z)-e_2)(\wp(z)-e_3)$$
and we assume $e_1=\frac{2-c}3>...
- 43.2k
1
vote
0
answers
211
views
Period calculation for elliptic curve
In the paper "Hodge cycles on Abelian Varieties" (Proposition 1.5), Deligne proves the following theorem:
Let $X$ be a smooth projective variety over $\overline{\mathbf{Q}}$ of dimension $n$...
- 1,949
1
vote
1
answer
263
views
On the equation $a^4+b^4+c^4=2d^4$ in natural numbers with $a<b<c<d$
I asked a simillar question with the weaker restriction:
On the equation $a^4+b^4+c^4=2d^4$ in positive integers $a\lt b\lt c$ such that $a+b\ne c$
.
I couldn't find any solution to this equation. ...
user178594
3
votes
0
answers
96
views
Descent obstruction of an open curve in an elliptic curve
Let $E$ be an elliptic curve over a number field $k$, and for an extension $K/k$ we denote by $E_K$ the base change $E \times_k K$. By fixing an embedding $k \hookrightarrow \mathbb{C}$, the etale ...
- 895
4
votes
1
answer
515
views
Recent works on the Hardy-Littlewood conjecture on primes represented by quadratic polynomials
I have been working on my master's thesis which is about the equivalence of the Hardy-Littlewood conjecture on primes represented by quadratic polynomials and the Lang-Trotter conjecture for CM ...
- 309
1
vote
1
answer
185
views
Deduce kernel of isogeny from action on torsion points
I'm stuck with the following problem:
In Petit's work "Faster Algorithms for Isogeny Problems using Torsion Point Images", p. 8, he says that we can deduce $\ker \psi_{N_2}$ knowing the ...
- 73
2
votes
0
answers
119
views
Resolution of singularities of the resultant locus
We consider projective space of dimension $n$ as the parameter space of degree $n$ polynomials in one variable. Then, I am interested in resolving the singularities of the "resultant locus" $...
- 7,746
6
votes
1
answer
391
views
Tate–Shafarevich group of Jacobian of Selmer curve $3X^3 + 4Y^3 + 5Z^3 = 0$
$C/ \Bbb{Q}: 3X^3 + 4Y^3 + 5Z^3 = 0$ is known to be a nontrivial element of the Tate–Shafarevich group of the elliptic curve $E/\Bbb{Q}:X^3 + Y^3 + 60Z^3 = 0$. It is also an example of an abelian ...
- 1,541
2
votes
0
answers
119
views
Finding a Hodge theoretic condition to measure the rank of isogeny of product abelian surfaces
Let $A$ be an abelian surface over $\mathbb{C}$, then there is a condition on $H^0\left(\Omega^1_A\right)$
to determine if $A$ contains an elliptic curve $E$ as a subvariety. If $A$ were to contain
...
- 91
26
votes
0
answers
567
views
Elliptic analogue of primes of the form $x^2 + 1$
I have a project in mind for an undergraduate to investigate next quarter -- a curiosity really, but I'm surprised I can't find it in the literature. I do not want a detailed analysis here... but ...
- 13.3k
4
votes
1
answer
363
views
What is meant by a meet-in-the-middle approach?
I'm studing C. Petit's work "Faster algorithms for isogeny problems using torsion point images" (link) and he talks about meet-in-the-middle approach/strategy for solve some isogenies ...
- 73
5
votes
1
answer
371
views
How the automorphism group of an elliptic curve acts at the localization of the stack $\mathcal{M}_{1, 1, k}$ at the corresponding point
I am studying the enlightening article "The Picard Group of $\mathcal{M}_{1, 1, S}$", written by Fulton and Olsson, but I have some problems with a proof.
Setting
Let $\mathcal{M}_{1, 1, k}$ ...
- 53
7
votes
2
answers
615
views
Reference request for recurrence relation of division polynomials
The recurrence relations for division polynomials of elliptic curves are well known:
$$\Psi_{2n} = \Psi_n \left( \Psi_{n+2} \Psi_{n-1}^2 - \Psi_{n-2} \Psi_{n+1}^2 \right) / \ 2y$$
$$\Psi_{2n+1} = \...
- 265
0
votes
0
answers
184
views
Why Lubin Tate character acts on torsion points of CM elliptic curve implies the group of torsion points is infinite?
Let $F$ be quadratic imaginary field, and $R_F$ be its ring of integers.
Let $E /\Bbb{Q} $ be an elliptic curve which has CM by $F$. Suppose $E$ has good reduction at $P$,which is prime ideal of $R_F$....
- 1,541
4
votes
0
answers
129
views
Weak version of (elliptic analog) Artin's primitive roots conjecture
Let $E/\mathbb{Q}$ be an elliptic curve, and $P\in E(\mathbb{Q})$ be any non-torsion point. Given any $\varepsilon>0,$ how often it is true that $\mathrm{ord}(P \pmod p)>p^{1-\varepsilon},~p~\...
- 521
1
vote
0
answers
88
views
Pure, residual, 2-dimensional, semisimple $G_{\mathbb{Q}}$ representations
Let $G_{\mathbb{Q}}$ be the absolute Galois group of the rationals. I want to know how many isomorphism classes of 2-dimensional, irreducible, pure Galois representations are there over a finite field ...
- 2,907
1
vote
0
answers
109
views
Why is the kernel cyclic if and only if the walk does not backtrack?
I'm reading Mathematics of Isogeny Based Cryptography by Luca De Feo. At some point (pg. 32), he says
"A walk of length $e_A$ in the $l_A$-isogeny graph corresponds to a kernel of size $l_A^{e_A};...
- 73
5
votes
1
answer
289
views
Transforming the Kondo quintic $5T2$ into the Lehmer quintic $5T1$?
I. Kondo-Brumer quintic
The deceptively simple solvable quintic,
$$x^5 + (a - 3)x^4 + (-a + b + 3)x^3 + (a^2 - a - 1 - 2b)x^2 + b x + a=0$$
is quite important for imaginary quadratic fields. For ...
- 12.6k
4
votes
0
answers
77
views
Conjugacy of cocharacters from conjugacy of labelled diagrams
Everything to follow is over some fixed algebraically closed field $k$. Although all the definitions make sense regardless of characteristic, the meat of the question is about small positive ...
- 12.9k
1
vote
0
answers
132
views
Is there an analog of Weil pairing for modular forms?
Given a newform $f(z)$ (of weight $k$) and a prime $p,$ consider the classical Galois representation
$$\rho_{f,p}: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \text{GL}_2(\mathbb{Z}/p\mathbb{Z}).$...
- 521
4
votes
0
answers
214
views
Effect of the surjectivity of Galois representation
Let $K$ be any arbitrary number field and $E$ be any elliptic curve over it. For any integer $m,$ consider the well-known Galois representation
$$\rho:\text{Gal}(\overline{K}/K)\to \text{GL}_2(\mathbb{...
- 521
2
votes
0
answers
132
views
How to compute torsion subgroup $E[24]$ over $\overline{\mathbb{Q}}$
If I have an elliptic curve $E: y^2=x^3-15x+22$ over $\mathbb{Q}$ with CM from the imaginary quadratic field $\mathbb{Q}(\sqrt{-3})$ then how do I compute the $24$-torsion subgroup $E[24]$ over $\...
- 309
0
votes
0
answers
80
views
Projection map $\pi:\left(\mathcal{O}/n\mathcal{O}\right)^\times \to\left(\mathcal{O}/\gcd(n,m)\mathcal{O}\right)^{\times}$ of a CM elliptic curve
In this paper the author has mentioned in page $693$ under section $2.2$ that for an Elliptic curve $E/\mathbb{Q}$ with CM by an order $\mathcal{O}$ of an imaginary quadratic field $K$ there is a ...
- 309
5
votes
2
answers
314
views
Generalization of $j(E) \in \overline { \Bbb{Z}}$ to abelian varieties of arbitrary dimension
Let $E/ \Bbb{C}$ be an elliptic curve which has complex multiplication over a number field $K$.
Then it is widely known that $j(E) \in \overline { \Bbb{Z}}$.
What is the known generalization of this ...
- 1,541
1
vote
1
answer
88
views
Generic finite subgroups, associated to small finite fields, of reductive algebraic groups
Theorem 1 of [LS] Liebeck and Seitz - On the subgroup structure of exceptional groups says:
Let $X = X(q)$ be a quasisimple group of Lie type in characteristic $p$, and suppose that $X < G$, where ...
- 12.9k
2
votes
0
answers
145
views
How to compute the character of the Steinberg module for the group $\mathrm{SL}_n$ over a field of characteristic $p$?
It is known that the Steinberg representation $V$ of the group $\mathrm{SL}_n$ over a field $k$ of characteristic $p$ (maybe one needs to assume that $k$ is perfect, I am not sure) is the irreducible ...
3
votes
0
answers
127
views
Isogeny of elliptic curve over positive characteristic $p$ which does not come from characteristic $0$
Let $K$ be quadratic imaginary field. Let $E$ be an elliptic curve which has CM over $R_K$
($R_K$ is ring of integers of $K$).
According to SIlverman's ''ADvanced topics in the arithmetic of elliptic ...
- 1,541
2
votes
0
answers
47
views
Characters of simple $\mathfrak{sl}_n$-modules in positive characteristic with subregular nilpotent central character
Consider representations of $\mathfrak{sl}_n$ in positive characteristic with a subregular nilpotent central character $\chi$ (i.e. $\chi$ is a nilpotent matrix whose Jordan normal form has two blocks ...
2
votes
1
answer
159
views
Decomposition of the Galois group of the $m$-th division field of an elliptic curve with CM into a direct product of Galois groups
Let $E/\mathbb{Q}$ be an elliptic curve with CM from an imaginary quadratic field $K$. Let $K(E[m])$ denote $m$-th division field (number field obtained by adjoining the coordinates of the $m$-torsion ...
- 309
3
votes
0
answers
262
views
Reverse engineering an elliptic curve from its modular form?
Does there exist an algorithm or something of the sort to reverse-engineer a curve from its modular form (weight two eigenform with complex coefficients)? I am aware that sometimes there isn’t a ...
- 565
2
votes
0
answers
142
views
Upper bound for the torsion subgroup of an elliptic curve over arbitrary number fields
Let $K$ be a finite extension of $\mathbb{Q}$ of degree $d$ and let $E(K)$ be an elliptic curve over the field $K$ with coefficients in $K$. Let us fix $d$ and vary over all the possible $K$, in turn ...
- 565
4
votes
0
answers
64
views
An analog of a BGG resolution in subregular case in positive characteristic
Consider representations of $\mathfrak{sl}_n$ in positive characteristic with a subregular nilpotent central character $\chi$. For every regular weight $\lambda$ of $\mathfrak{sl}_n$, we have the ...
1
vote
0
answers
180
views
How to link the rank of Elliptic Surfaces and Elliptic Curves over function fields?
I have been investigating certain elliptic surfaces for my research, and when giving a presentation, I was asked why when given an elliptic surface $E$, say given in Weierstrass form
$$E\colon y^2+a_1(...
- 11
4
votes
0
answers
157
views
Can you determine the least degree of a morphism between algebraic curves?
I have several questions regarding the degrees of morphisms between algebraic curves.
If we have algebraic curves $X$ and $Y$ defined over some perfect field $k$, can we determine the least degree of ...
- 41
6
votes
2
answers
310
views
Does the $p$-adic regulator depend on Weierstrass model?
I am a little confused on the $p$-adic regulator on elliptic curves and what happens when you switch to different Weierstrass models. Restrict to ell. curves over $\mathbb Q$ for simplicity.
From my ...
- 207
3
votes
1
answer
179
views
Why an isogeny induces a surjection between points over maximal unramified extension?
Let $E$ and $E'$ be elliptic curves over $\mathbb Q$, and let $\phi\colon E\to E'$ be an isogeny defined over $\mathbb Q$. Let $p$ be a prime relatively prime to the degree of $\phi$. Let $\mathbb Q_p^...
- 2,375
3
votes
0
answers
120
views
Resolving the "wild" singularities of $\mathbb A^n/C_n$
Let the cyclic group on $n$ elements, $C_n$, act on $\mathbb A^n$ by permuting the co-ordinates (over a field $k$). If $n \neq 0 \in k$, we can resolve the singularities of $X = \mathbb A^n/C_n$ by ...
- 7,746
1
vote
0
answers
95
views
An elliptic threefold and the Mordell–Weil lattices of its reductions
Let $T\!: y^2 = x^3 + a(t, s)x + b(t, s)$ be an elliptic threefold over a finite field $\mathbb{F}_q$ of characteristic $p > 3$. In other words, we have an elliptic curve over the function field $\...
- 1,833
0
votes
0
answers
101
views
Identity component of $\mathrm{Ker}(E^n→E^m)$ in the advanced topics in the arithmetic of elliptic curves
$\DeclareMathOperator\Ker{Ker}$Silverman's "Advanced topics in the arithmetic of elliptic curves", p.115 reads $$0\to\mathfrak{a}^{-1}Λ\to\Bbb{C}\to\Ker(E^n\to E^m)\to Λ^n/A^tΛ^m\quad (1)$$ ...
- 1,541
2
votes
0
answers
177
views
How do characters of representations in cohomology depend on the (positive-characteristic) field?
The following sentence appears in Jantzen - Representations of algebraic groups, 2nd edition, p. x, where $G$ is a reductive group over an algebraically closed field $k$, $B$ is a Borel subgroup, $T$ ...
- 12.9k