# Tagged Questions

**4**

votes

**0**answers

72 views

### minimal conductors among elliptic curves with a fixed CM type

Let $K$ be a quadratic imaginary field. To simplify my life, let us assume
that $K$ has class number one.
Consider the following infinite set:
$S_1:=$ $\{$ $E\subseteq\mathbf{P}^2(\mathbf{C})$ is an ...

**1**

vote

**2**answers

130 views

### On the conductor of the Groessencharacter of a CM elliptic curve

Let $K$ be a quadratic imaginary field. Let $L$ be a number field which contains $K$ and let $E/L$ be an elliptic curve defined over $L$ with complex multiplication by $K$, i.e. such that ...

**2**

votes

**2**answers

335 views

### Elliptic curve E and Galois representation

Assume that an elliptic curve $E$ over $\Bbb Q$ has a reducible mod $p$ representation. Then
Q: Why is the semi-simplification of $E[p]$ the direct sum of ${\Bbb Z}/p{\Bbb Z}$ and $\mu_p$?
Next
...

**6**

votes

**1**answer

340 views

### Elliptic curve and Galois representation

For an elliptic curve $E$ over ${\Bbb{Q}}$, let us consider Serre's mod $l$ representation by
$\rho_{E,l} \colon {\mathrm{Gal}}({\overline{\Bbb{Q}}}/{\Bbb{Q}}) \to {\mathrm{Aut}}(\phantom{}_lE) = ...

**6**

votes

**3**answers

316 views

### Torsion group of the following elliptic curve

Let $p_1=2, p_2 = 3,\ldots,$ be the prime numbers, and define $n_i = \prod_{j=1}^i p_j$. Moreover, let $E_i $ be the elliptic curve defined by $y^2 = x^3 + n_i$.
Can one compute the torsion group ...

**7**

votes

**1**answer

354 views

### Explicit calculation of Weil Deligne representations

According to Grothendieck monodromy theorem, l-adic galois representations of a local field corresponds to Weil-Deligne representations.
However, given a galois representation, it is usually difficult ...

**6**

votes

**2**answers

324 views

### Is there a largest prime p such that J_0(p) completely splits into elliptic curves

The question in the title is related to a more general question. Namely does there exist an integer $N$ such that for all curves $C/\mathbb C$ of genus $> N$ one has that not all simple isogeny ...

**5**

votes

**0**answers

144 views

### When are Galois representations with open image attached to elliptic curves?

Let $K$ be a number field with absolute Galois group $G_K$.
Let $\rho:G_K \rightarrow GL_2(\hat{\mathbb{Z}})$ be a Galois representation such that the image of $\rho$ is open in ...

**3**

votes

**1**answer

236 views

### Are elliptic Kummer extensions big?

Loosely speaking, are elliptic Kummer extensions big? More concretely:
Let $E$ be an elliptic curve over $\mathbb{Q}$, let $p$ be a prime, and
let $F$ be a subfield of $\overline{\mathbb{Q}}$ ...

**2**

votes

**1**answer

216 views

### Elliptic units and Euler system

Maybe this question is quite obscure and ambiguous. I am really sorry for such ambiguity.
My question is, what is the good thing we get from defining elliptic units and Euler system? There are lots ...

**1**

vote

**0**answers

126 views

### What does Hodge theory tell us about simply connected surfaces of general type

Let $X$ be a smooth complex projective variety. We know that $\Omega^1_X$ has a non-zero section if and only if the abelianization of the fundamental group of X is infinite. This follows from Hodge ...

**2**

votes

**1**answer

147 views

### Specialization of sections in an elliptic fibration

Let $\pi: X \rightarrow S$ be the Neron model of an elliptic curve over a dedekind domain (but probably any minimal elliptic fibration will suffice).
Let $\eta$ be the generic point of $S$, $K = ...

**3**

votes

**1**answer

297 views

### Shafarevich's theorem for elliptic curves defined over function field of algebraic curve over algebraically closed field

Let $K$ be a number field and $S$ a finite set of places of $K$. Then Shafarevich's theorem states that there are only finitely many isomorphism classes of elliptic curves $E$ over $K$ with good ...

**3**

votes

**0**answers

134 views

### Is Hasse-witt map isomorphism?

Fix a level $N \geq 3$ and denote by $\Gamma(N)\subset SL(2,\mathbb{Z})$ the subgroup of matrices which are congruent to the identity modulo $N$. The open modular curve $Y(N)$ corresponding to ...

**3**

votes

**1**answer

269 views

### Heegner Points on $X_0(N)$ when some primes dividing $N$ are inert in the imaginary quadratic field

If $K = \mathbb{Q}(\sqrt{-D})$ is a imaginary quadratic field with discriminant $-D$, then we get Heegner points on $X_0(N)$ as long as there exists $\mathfrak{n} \subset \mathcal{O}_K$ such that ...

**3**

votes

**3**answers

529 views

### reduction types of elliptic curves

Let $E/K$ be an elliptic curve, where $K$ is a complete local field with residue field $k$ and char$(k) = p$. I'm trying to make sense of Kodaira symbols and Tate's algorithm.
My current ...

**5**

votes

**1**answer

253 views

### Inertia subgroup in the ordinary reduction case when $p=2$

Dear MO,
Let $K/\mathbb{Q}_2$ be a finite extension, and let $E/K$ be an elliptic curve with good ordinary reduction, and such that $\mathbb{Q}_2(j(E))=K$. Let ...

**5**

votes

**0**answers

197 views

### Isogenous elliptic curves have same conductor

Let $E/K, E'/K$ be elliptic curves defined over a number field $K$. Let $\phi: E \to E'$ be a non-constant isogeny defined over $K$. Why must the conductors be equal?
I know that this is an ...

**2**

votes

**2**answers

144 views

### Possible reasons why the image of 0 in the modular parametrization would always be O for a family of quadratic twists of elliptic curves

Let $E$ be an elliptic curve and $E_n$ be its quadratic twist by $n$. Let $\phi_n: X_0(N_n) \to E_n$ be the normalized modular parametrization of $E_n$ ($\infty \to O$)
For some particular curves ...

**4**

votes

**2**answers

169 views

### computing the order of the image of 0 under the modular parametrization map for an elliptic curve

Let $E/\mathbb{Q}$ be an elliptic curve of rank 0, with modular parametrization $\phi: X_0(N) \to E$. Let $\Omega_0$ be the least positive real period. A paper I'm reading (Yoshida, Some variants of ...

**2**

votes

**0**answers

225 views

### On the expilicite example of Parshin Construction.

In the Proof of Mordell Conjecture by Gerd Faltings, it is famous that Parshin constructed curve C_P for each Q-rational point P on the given curve C over Q such that genus of C satisfies g(C) > 0.
...

**8**

votes

**1**answer

171 views

### Which level structures on elliptic curves are twist-invariant?

Let $N \geq 5$ be a prime, and $H$ a subgroup of $GL_2(\mathbb{F}_N)$. As shown in Chapter IV of [DeRap], there is a curve $X_H(N)$, defined over $K_N := \mathbb{Q}(\zeta_N)^{\det H}$, which is a ...

**3**

votes

**1**answer

217 views

### Torsion of elliptic curves is finite

Let $S$ be an integral 1-dimensional scheme with function field $K$.
Let $E$ be an elliptic curve over $K$. The torsion of $E$ over $K$ is not necessarily finite. As an example consider an elliptic ...

**21**

votes

**1**answer

861 views

### Do all curves have Néron models

Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Assume that $g\geq 2$.
Does there exist a Néron model $\mathcal X$ for $X$ over $O_K$?
By a Néron model, I mean ...

**1**

vote

**1**answer

239 views

### What is the reduction of this hyperelliptic curve

Let $K$ be a number field and $E/K$ an elliptic curve with equation $Y^2Z = X^3 +AXZ^2+BZ^3$ in $\mathbf{P}^2_K$, where $A,B\in K$.
Let $S$ be non-empty finite set of finite places of $K$ and suppose ...

**9**

votes

**5**answers

1k views

### The significance of modularity for all Galois representations

On pg. 1 of the slides of a talk, Henri Darmon wrote:
Question: What is an interesting Diophantine equation?
A “working definition”. A Diophantine equation is interesting
if it reveals or ...

**18**

votes

**3**answers

1k views

### Over which fields does the Mordell-Weil theorem hold?

According to a well-known theorem of Mordell, the group of rational points $E(\mathbf{Q})$ of an elliptic curve $E/\mathbf{Q}$ is finitely generated. Weil generalized this theorem to abelian varieties ...

**1**

vote

**0**answers

134 views

### $K$-groups and dual graphs of special fibers

Let $p$ be a prime number, let $E$ be an elliptic curve defined over $\mathbb{Q}_p$. Let $\mathcal{E}_p$ be the special fiber of the Néron model of $E$ over $\mathbb{Z}_p$ and let ...

**6**

votes

**2**answers

979 views

### Isogeny classes of elliptic curves

Let $E \subset \mathbb{P}_\mathbb{C}^2$ be an elliptic curve. If $E$ has complex multiplication (by anything) then the theory of complex multiplication in particular tells us that if $\sigma \in ...

**5**

votes

**1**answer

541 views

### Special value of $L$-function

Let $p$ be a prime number. Let $f$ be a newform of weight 2 on $Γ_0(p)$, and $E_f$ denote the associated newform quotient of $J_0 (N)$ over $\mathbb{Q}$. Is there a way to express the
algebraic part ...

**2**

votes

**1**answer

217 views

### Are abelian varieties degree two covers of some projective space

Let $A$ be an abelian variety over a field $k$ of dimension $g\geq 2$.
There exists a finite morphism $A\to \mathbf{P}^g_k$. Here's the question.
Does there exist a finite morphism $A\to ...

**20**

votes

**6**answers

2k views

### Does the moduli space of smooth curves of genus g contain an elliptic curve

Let $M_g$ be the moduli space of smooth projective geometrically connected curves over a field $k$ with $g\geq 2$. Note that $M_g$ is not complete.
Does $M_g$ contain an elliptic curve?
The answer ...

**1**

vote

**1**answer

361 views

### Elliptic subfields of a function field

Let $C$ be a curve and $K(C)$ be its function field of genus 2, where $K$ = $\mathbb{C}$.
The number of essential elliptic subfields of $K(C)$ is 0 or 2 or $\infty$.
Edit: I am looking for a proof. ...

**2**

votes

**1**answer

431 views

### Conductor of an elliptic curve

Given any elliptic curve over $\mathbb{Q}$ of conductor $N$, by modularity of elliptic curves,
there exists a surjective morphism from $X_0(N)$ $\rightarrow$ $E$.There may be several such 'N' and ...

**7**

votes

**1**answer

2k views

### Order of Ш (Sha)

To prove the BSD conjecture, one has to know about 'the finiteness of the Shafarevich Tate group'. But, an example of an elliptic curve of rank 2 (whose Sha group $Ш(E/\mathbb{Q})$ is finite) is not ...

**4**

votes

**1**answer

727 views

### understanding the main theorem of complex multiplication (of elliptic curves)

I want to understand the meaning of the main theorem of complex multiplication (of elliptic curves) as given in Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, II.8, or Shimura, ...

**3**

votes

**1**answer

1k views

### Trace of Frobenius over $F_q$

Let $q_0$ be a prime and $q$ = $q_0^n$.
Let $a(F_q/F_{q_0})$ denote any integer which is trace of Frobenius over the field $F_q$ for some elliptic curve which can be defined over $F_{q_0}$.
It is ...

**4**

votes

**1**answer

927 views

### Bound for the number of rational points on the modular curve

By Mazur's theorems (Reference:- Modular curves and the Eisenstein ideal, 1977),
we know that the only rational points of X_0(N) for N any prime > 163 are
the two cusps (o) and (oo) (|X_0(N)(Q)| = 2 ...

**2**

votes

**2**answers

370 views

### branch points of modular parametrization of an elliptic curve

Let $E$ be an elliptic curve over a number field K. Then there is a morphism $\phi:X_0(n) \to E$. Consider composition $f:X_0(n)\to \mathbf{P}^1_K$, where we compose with degree 2 cover $E\to ...

**3**

votes

**0**answers

170 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 ...

**5**

votes

**0**answers

406 views

### Formal groups in the supersingular reduction case

Dear MO,
Let $E/\mathbb{Q}$ be an elliptic curve with potential good supersingular reduction at $p$. Thus, there is a finite extension $K/\mathbb{Q}_p$ such that $E/K$ has good supersingular ...

**10**

votes

**1**answer

526 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 ...

**13**

votes

**3**answers

918 views

### rational points on algebraic curves over $Q^{ab}$

Motivation:
Let $Q_{\infty,p}$ be the field obtained by adjoining to $Q$ all $p$-power roots of unity for a prime $p$. The union of these fields for all primes is the maximal cyclotomic extension ...

**8**

votes

**3**answers

1k views

### Elliptic Curves over Global Function Fields

I am currently thinking about a problem, and I feel that by knowing more about elliptic curves over extensions of $\mathbb{F}_q(T)$, for $q$ a power of $p$ say, might lead to insight. I am also ...

**10**

votes

**1**answer

694 views

### Which primes can divide orders of Tate-Shafarevich groups?

Heuristic arguments due to (I believe) Delauney predict that every prime divides the order of the Tate-Shafarevich group of infinitely many elliptic curves over $\mathbf{Q}$. However, is it even ...

**14**

votes

**3**answers

2k views

### Are there any rational solutions to this equation?

I am not sure if this is an appropriate question, but I was asked this by a colleague today and do not know how to answer it.
1) Are there any rational solutions to the following equation:
...

**18**

votes

**2**answers

1k views

### unboundedness of number of integral points on elliptic curves?

If $E/\mathbf{Q}$ is an elliptic curve and we put it into minimal Weierstrass form, we can count how many integral points it has. A theorem of Siegel tells us that this number $n(E)$ is finite, and ...

**5**

votes

**2**answers

843 views

### Direct proof of special case of Hasse's theorem for elliptic curves

Consider the elliptic curve $y^2 = x^3 + x$ over $\mathbb{F}_p$, where $p \equiv 1 \pmod 4$.
If memory serves correctly, the number of points (excluding the point at infinity) is $p - a$ where $a$ is ...

**6**

votes

**0**answers

624 views

### Can you get Siegel's theorem “for free” from modularity and Mazur's Eisenstein Ideal paper?

There is a well-known theorem of Shafarevich that given a finite set $S$ of primes the number of isomorphism classes of elliptic curves over $\Bbb Q$ with everywhere good reduction outside $S$ is ...

**2**

votes

**1**answer

1k views

### How is the period of an elliptic curve defined exactly?

I sometimes read $\int_{E(\mathbf{R})} \frac{dx}{2y + a_1x + a_3}$ and sometimes $\int_{E(\mathbf{R})} |\frac{dx}{2y + a_1x + a_3}|$. Furthermore, one has to choose an orientation on $E(\mathbf{R})$.
...