**3**

votes

**0**answers

76 views

### How can I describe explicitely a nonsingular model of the elliptic surface?

Consider the surface $\mathcal{E} = Q_1 \cap Q_2 \subset \mathbb{P}_k^3\times \mathbb{P}_k^1$ with homogenous coordinates $x$, $x^\prime$, $y$, $z$ and $t$, $s$ respectively and a field $k$ of even ...

**0**

votes

**0**answers

94 views

### Selmer and free rank of Elliptic Curves

If I am not mistaken, the equality of the $p$-Selmer rank and the free rank of an elliptic curve are conjectured to be equal.
This is one of the many equivalent formulations of the Birch and ...

**3**

votes

**0**answers

146 views

### Fields generated by torsion points of CM elliptic curves

I'm using the same setup as Corollary 1.7 on p. 44 of de Shalit manuscript (Iwasawa theory of elliptic curves with complex multiplication).
I think there is a mistake in his Corollary 1.7 and I'm ...

**4**

votes

**1**answer

148 views

### Does complex multiplication for higher dimensional abelian varieties give some generalization of class field theory?

I am currently learning some aspects of the theory of complex multiplication for elliptic curves, and the relationship with class field theory.
As I understand it, there is a very special class of ...

**9**

votes

**2**answers

514 views

### congruent number problem [closed]

I am studying the congruent number problem
and I heard that there is a paper by Kazuma Morita
which claims to solve this problem from my colleague.
I saw the paper on his homepage but it is very ...

**2**

votes

**0**answers

64 views

### Rank growth in ray class fields of primes that are inert in an imaginary quadratic extension

If $K=\mathbb{Q}[\sqrt{-d}]$, $E$ is an elliptic curve and $\operatorname{rank}(E(K))=\operatorname{rank}(E(\mathbb{Q}))$, are there infinitely many primes $l$ of $\mathbb{Q}$ so that:
(1.) $\...

**3**

votes

**1**answer

185 views

### Is there an infinite family of primes $q_{1},q_{2},…$ so that the rank of $E(\mathbb{Q}(\sqrt{-q_{i}}))$ equals that of $E(\mathbb{Q})$?

It is known that the group of $K$-rational points of an elliptic curve $E$ is finitely generated if $K$ is a number field of finite degree over $\mathbb{Q}$.
Much less is known if $K$ is infinite-...

**4**

votes

**2**answers

283 views

### Mordell-Weil rank of an elliptic curve over $\mathbb{Q}(\sqrt{-1},\sqrt{2},\sqrt{3},\sqrt{5},…)$?

It is known that the group of $K$-rational points of an elliptic curve $E$ is finitely generated if $K$ is a number field of finite degree over $\mathbb{Q}$.
The picture is less clear if $K$ is ...

**10**

votes

**1**answer

239 views

### Why is there a factor $p$ in the definition of $T_p$ via Hecke correspondences on modular curves?

Fix $N\ge4$. Let $Y_1(N)$ and $X_1(N)$ be the usual modular curves. I want to view them as schemes over $\mathbb Z$ representing the moduli functors of (usual or generalized) elliptic curves with (...

**2**

votes

**1**answer

315 views

### An elliptic curve for Ramanujan-type cubic identities?

Given the roots $x_i$ of the depressed cubic,
$$x^3+px+q=0$$
with rational coefficients. It can be shown that, in general, one can find rational $u,v$ such that,
$$(u-x_1)^{1/3}+ (u-x_2)^{1/3}+ (u-...

**1**

vote

**0**answers

70 views

### Explicit construction of a bielliptic curve

Let $C$ be a (projective smooth complex) curve such that $K_C=2(D+p)$, with $D+p$ defining a $g_7^2$; $p$ is a base point and $D$ defines a 2-to-1 map $\varphi:C\rightarrow E\subset\mathbb{P}^2$ onto ...

**0**

votes

**1**answer

133 views

### An example of ``all non-torsion rational points on an elliptic curve are integral points''?

For an elliptic curve $E$ over $\mathbb{Q}$, it is well-known that the torsion points on $E$ are integral points.
Then, is it possible that there exists an example whose all of non-torsion rational ...

**5**

votes

**1**answer

632 views

### Ordinary primes vs supersingular primes

Let $E$ be an elliptic curve over $\mathbb{Q}$ without CM. As shown by Serre, the set of supersingular primes for $E$ has density zero.
Is the analytic rank of $L(E,1)$ determined only by the ...

**1**

vote

**2**answers

166 views

### How to prove that $A$ is supersingular iff the Picard number $\rho(A)$ is equal to the second $l$-adic Betti number $b_2(A) = 6$?

Let $A$ be an abelian surface over algebraically closed field $k$ of characteristic $p > 2$. How to prove that $A$ is supersingular (in other words, there is an isogeny between $A$ and $E^2$, where ...

**6**

votes

**1**answer

67 views

### Bilinearity of the Cassels-Tate pairing

Let $K$ be a number field and let $A$ be an abelian variety over $K$ (I'm mostly interested in the case that $A$ is an elliptic curve). We use $v$ to denote places of $K$ and we write $H^i(k, A)$ for ...

**2**

votes

**0**answers

121 views

### Reduction “modulo $p$” of $\mathfrak{p}$-torsion points of CM elliptic curves

Let $E/L$ be an elliptic curve defined over a number field $L$. Assume moreover that $E$ has complex multiplication by an imaginary quadratic field $K$. Let $\mathfrak{p}$ be a prime ideal of $K$. ...

**3**

votes

**1**answer

188 views

### Tate modules of elliptic curves with complex multiplications

Let $E/K$ be an elliptic curve with complex multiplication
over an imaginary quadratic field $K$. Then, I heard that
it is well-known that the Tate module $V_{p}(E)$ over
$\mathbb{Q}_{p}$ ...

**0**

votes

**1**answer

71 views

### Is $H^{1}_{Sel}\left(K,E_{p^{n}}\right)\rightarrow\prod_{q \nmid \infty} H^1\left(K_{q},E_{p^{n}}\right)$ an injection?

If $K$ is a number field, $E$ an elliptic curve and $p$ a prime, does the Selmer group
$$H^{1}_{\operatorname{Sel}}\left(K,E_{p^{n}}\right)$$
always inject into
$$\prod_{q \text{ a nonarchimedean ...

**0**

votes

**1**answer

111 views

### Finding out $p$-torsion elements of an elliptic curve $E$ over $\mathbb{Q}_p$

Let $E$ be an elliptic curve over $\mathbb{Q}$.
Then how to compute the $p$-torsion elements of $E$ over the $p$-adic field $\mathbb{Q}_p$ using SAGE or any other means ?
At least can we say whether ...

**1**

vote

**0**answers

58 views

### What is the relation between roots of classical and Atkin modular polynomial?

Modular polynomials are a good tools in elliptic curves. I need to find the roots of $l$-th classical modular polynomial $\Phi_l(X,Y)$ over the prime field. Unfortunately the coefficients of these ...

**7**

votes

**4**answers

844 views

### Clarification on the weak BSD conjecture

It is usually told that Birch and Swinnerton-Dyer developped their famous conjecture after studying the growth of the function
$$
f_E(x) = \prod_{p \le x}\frac{|E(\mathbb{F}_p)|}{p}
$$
as $x$ tends to ...

**3**

votes

**1**answer

284 views

### Frobenius at ramified primes

Let $E$ be an elliptic curve defined over $\mathbf{Q}$, fix an odd prime $p>3$, let $T_p$ denote the $p$-adic Tate module of $E$, and let $V_p = T_p \otimes \mathbf{Q}_p$.
If the action of $G_\...

**11**

votes

**1**answer

426 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 ...

**17**

votes

**2**answers

920 views

### What is the smallest positive integer for which the congruent number problem is unsolved?

The congruent number problem is the problem of figuring out whether a given positive integer $N$ is the area of a right-angled triangle with all side lengths rational. According to Dickson's "History ...

**2**

votes

**0**answers

128 views

### Uniform bound on the Mordell-Weil rank of elliptic curves

I want to know that if there is an uniform upper bound for the rank of elliptic curves over $\mathbb{C}(t)$, the rational function field over complex numbers, and generaly over the function field of ...

**5**

votes

**1**answer

236 views

### j-invariants for isogenous elliptic curves

Let $E$ be a smooth complex elliptic curve, and $\sigma$ translation of $E$ given by a point $p$ on $E$ of finite order, with respect to some fixed origin.
What are the $j$-invariants related with $E$...

**1**

vote

**0**answers

72 views

### “Algebrazing” canonical subgroups of elliptic curves

I'm puzzled by a part of the construction of the canonical subgroup of a "not too supersingular" elliptic curve. In Katz's paper, one produces a subgroup of the formal group of the elliptic curve but ...

**9**

votes

**2**answers

311 views

### Neron models and ramification

I encountered this result while reading a few things, and it was stated without reference. I am having a hard time finding a reference for it (or a simple proof), so maybe you can help me:
let $E$ be ...

**3**

votes

**1**answer

189 views

### Connected-étale sequence for ordinary CM elliptic curves

Let $E/k$ be an elliptic curve over algebraically closed field of characteristic $p$ with CM, for simplicity, by the maximal order of a quadratic imaginary field $K/\mathbb{Q}$.
Suppose that $p$ is ...

**5**

votes

**3**answers

172 views

### Birationally transforming a quartic elliptic curve

Consider the elliptic curve
$$y^2=ax^4+cx^2+dx+f$$
I am aware that there are algorithmic methods for birationally transforming a nondegenerate cubic curve into the Weierstrass canonical form (...

**0**

votes

**1**answer

73 views

### Heights of multiples of rational points on elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve, given by some minimal Weierstrass equation (say $Y^2 = X^3 + aX + b$ for some integer $a$ and $b$), and let $P$ be a rational point on $E$ which is not the ...

**5**

votes

**2**answers

394 views

### BSD and generalisation of Gross-Zagier formula

The classical Gross-Zagier formula and the modularity theorem leads to a proof of half-BSD (i.e. an inequality and not equality) for elliptic curves of analytic rank 0.
The Gross-Zagier formula gives ...

**5**

votes

**2**answers

624 views

### Langlands program vs Shimura-Taniyama-Weil conjecture

Edward Frenkel said that "we can see Langlands program as a generalization of Shimura-Taniyama-Weil conjecture in the case of elliptic curves"
I hope I'm not distorting his phrase, can someone ...

**5**

votes

**1**answer

245 views

### Average size of $p$-part of the Tate-Shafarevich group for elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve defined over $\mathbb{Q}$. For a given prime $p$, the $p$-Selmer group $\operatorname{Sel}_p(E)$ of $E$ and the $p$-part of the Tate-Shafarevich $Ш_E[p]$ group ...

**0**

votes

**0**answers

80 views

### Fastest algorithm to compute isogeny

Let $E/GF(p)$ and $E'/GF(p')$ are two isogenous elliptic curves($\#E=\#E'$). We know that there exist the map
$$\psi : E \to E'$$
Suppose that we haven't any information about degree of $\psi$.
...

**11**

votes

**1**answer

285 views

### Computing endomorphism rings of supersingular elliptic curves

I would like to know what algorithms there are to compute the linearly independent generators $(1,i,j,k)$ for quaternion algebra containing the endomorphism ring of a supersingular curve. The curve in ...

**2**

votes

**0**answers

83 views

### what is the structure of the group of isogenies between two ordinary elliptic curve?

Let $E_1$ and $E_2$ be two ordinary elliptic curves. It is well known that the group $Home(E_1,E_2)$ is a $\mathbb{Z}$ module of rank at most four. In the case where $E_1=E_2$, this module has rank ...

**3**

votes

**0**answers

171 views

### Equivalent definitions of the Hasse invariant

As probably many others before me, I got stuck in verifying all the nice properties of the Hasse invariant.
Let me start by recalling one definition:
Let $E\to S$ be an elliptic curve in ...

**4**

votes

**1**answer

137 views

### Congruence Primes and Modular Degrees

Let $\mathcal{S}=S_2(\Gamma_0(N) \cap \mathbf{Z} [[ q ]]$ be the set of cusp forms of weight $2$ on $\Gamma_0(N)$ with integral coefficients.
Let $f \in \mathcal{S}$ be a normalized newform, so it ...

**5**

votes

**0**answers

81 views

### Reference request: "effective'' semistable reduction

I am looking for the origin of the following idea: suppose $m$ and $n$ are relatively prime integers $\geq 3$. Let $E$ be an elliptic curve over a number field $K$. Let $L/K$ be a finite extension ...

**3**

votes

**0**answers

115 views

### Complex multiplication and ray class fields

This question is mainly referring to the proof of Theorem 5.6, Chapter 2 of Silverman's "Advanced Topics in the AEC". Basically, let $K$ be an imaginary quadratic field, and $E$ be an elliptic curve ...

**7**

votes

**0**answers

375 views

### Field of definition of a point in $[p]^{-1}E(K)$

Let $E$ be an ordinary elliptic curve defined over a non-perfect field $K$ of characteristic $p$. If $P \in E(K)$ satisfies $P \not\in [p]E(K)$, is it true that its $p^m$-division points of $P$ are ...

**2**

votes

**0**answers

45 views

### What is the complexity of finding a distortion map on a supersingular elliptic curve?

Let $E$ be a supersingular elliptic curve which is defined over $\mathbb{F}_q$ and $P\in E$. Then there exist a distortion map with respect to $P$. I am looking for an algorithm which finds the map ...

**1**

vote

**0**answers

44 views

### How can I find the specific endomorphism in a supersingular elliptic curve?

Let $E$ be a supersingular elliptic curve. As we know the endomorphism of $E$ is an order in a quaternion algebra. Suppose that $End(E)=\mathcal{O}$ and $a\in \mathcal{O}$. How can I find the ...

**21**

votes

**2**answers

538 views

### CM $j$-invariants in $p$-adic fields

I'm trying to understand the $p$-adic distribution of $j$-invariants for elliptic curves with complex multiplication.
Specifically, suppose $\sigma$ is some embedding $\sigma:\overline{\mathbb Q}\to \...

**0**

votes

**0**answers

69 views

### Elliptic curve : determine size of group $E/E_0$

My curve is given by E : $y^2 = x^3-3267x+45630$. Bad primes are 2,3,17. I want to find the size of group $E/E_0$. I know that $E_0(Q_2)$ are points on $E(Q_2)$ that do not reduce to a singular point. ...

**8**

votes

**0**answers

150 views

### Elliptic curves and the $\ell$-adic image of the decomposition group

Let $E$ be an elliptic curve over $\mathbb{Q}_\ell$ and consider the image of the $\ell$-adic representation. Is there a description of this image similar to Serre's description of the image of the ...

**0**

votes

**1**answer

244 views

### On elliptic curves, $\sqrt{x^2-101y^2} ,\sqrt{x^2+101y^2}$, and their ilk

I. Elliptic curves
Given integers $a,b,m_k$. Let,
$$x^2+a = m_1u_1^2\\x^2+b = m_1u_2^2\tag1$$
If there is a rational point $x_i$, then the pair (after a transformation) is birationally equivalent ...

**3**

votes

**1**answer

75 views

### Finiteness of the $p$-primary subgroup of an elliptic curve over the cyclotomic $\mathbb{Z}_p$-extension

Let $E$ be an elliptic curve defined over a number field $F$ and $F_\infty$ be the cyclotomic $\mathbb{Z}_p$-extension of $F$. Is it true that the $p$-primary subgroup of $E$ over $F_\infty$ i.e. $E[p^...

**3**

votes

**1**answer

324 views

### Cohomology of elliptic curves

Assume $K$ is an imaginary quadratic extension of $\mathbb{Q}$, and $E$ an elliptic curve defined over $\mathbb{Q}$.
Let $p\neq l$ be primes in $\mathbb{Q}$ where $E$ has good reduction. Assume $p$ ...