**4**

votes

**1**answer

133 views

### Hecke $L$-series exercise in Silverman's Advanced Topics in Arithmetic of EC

This has been posted on SE, but I haven't gotten a reply, so I thought I'll try my luck here.
I would like to refer you to 2.30 & 2.32 in Silverman's book Advanced Topics in the Arithmetic of ...

**4**

votes

**1**answer

208 views

### Elliptic Curves isogenous only over an extension?

Let $l$ be a prime $\geq 5$. Does there exist a pair $E,E'$ of elliptic curves, both defined over the same number field $K$, which are not $l$-isogenous over $K$, but are $l$-isogenous over a ...

**8**

votes

**0**answers

294 views

### A property of supersingular $j$-invariants (reference request)

Edit 2: For those who understandably don't want to read such a long post, I think Voloch's suggestion reduces the problem to asking whether $j$-invariants of supersingular curves are 3rd powers in ...

**4**

votes

**1**answer

102 views

### A certain property of elliptic curves in a paper by Rees

In the paper "On a problem of Zariski", David Rees presents a counterexample to the following problem of Zariski.
Let $F/k$ be a f.g. field extension, $S$ a f.g. normal integral domain over $k$ ...

**3**

votes

**1**answer

289 views

### a question on CM elliptic curves

Let $E$ be an elliptic curve over $\overline{\mathbb{Q}}$ defined by an equation
$y^2=4x^3-g_2x-g_3$
and let $\omega=\int_\gamma \frac{dx}{y}$ be the integral of the regular differential form ...

**1**

vote

**1**answer

114 views

### Hecke Character vs Grossencharakter

I would like to know if there is any difference between
(1) an algebraic Hecke character
(2) a Hecke character
(3) a Grössencharakter
All of the above in the setting of ellitpic curves with complex ...

**5**

votes

**1**answer

249 views

### Canonical differential on Tate curve

I am starting studying the theory of (algebraic) modular forms, and I have some trouble in understanding completely the construction of the Tate curve. My problem is the following: as far as I know ...

**4**

votes

**1**answer

243 views

### Rational points on $X_0(15)$

The modular curve $X_0(15)$ has a canonical model over $\mathbf{Q}$, and it has genus $1$. As the cusp $\infty$ is rational, it is an elliptic curve. Roughly, my question is whether we can find all ...

**1**

vote

**1**answer

380 views

### Hecke Characters

My question today concerns Hecke characters or Größencharaktere. I'm doing a project on complex multiplication of elliptic curves and need some help understanding Hecke characters. My main ...

**19**

votes

**1**answer

792 views

### What is $Aut(Ell)$?

Consider the stack $Ell$ (of groupoids) of elliptic curves. I'm interested in the autoequivalence 2-group of $Ell$, the objects of which consists of transformations $Ell \Rightarrow Ell: Ring \to Gpd$ ...

**4**

votes

**1**answer

233 views

### Abelian image of l-adic representation

For a number field F, let E/F be an elliptic curve with CM by a quadratic field K. Let $\rho_\ell: \text{Gal}_F \to \text{Aut}(T_{\ell}E)$ be the $\ell$-adic representation associated to E for some ...

**2**

votes

**0**answers

142 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

164 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 = ...

**0**

votes

**2**answers

315 views

### A question on degeneration of elliptic curves with actions.

Let $E$ be an elliptic curve. I want to consider its degeneration to the union of two projective lines $C:=\mathbb{P}^1 \cup_{x,y} \mathbb{P}^1$ attaching at two points $x,y$. The involution $-1$ on ...

**3**

votes

**1**answer

189 views

### Affine neighborhood of an $S$-valued point

How can we understand an affine neighborhood of an $S$-valued point on a scheme, and when does it exist?
I am looking at page 111 of Haruzo Hida's Geometric Modular Forms and Elliptic Curves, and he ...

**1**

vote

**0**answers

133 views

### Full $n$-torsion of elliptic curves and the cyclotomic field of order $n$

Hi, overflowers.
I have a question concerning the torsion of elliptic curves over number fields.
Let us consider an elliptic curve $E$ defined over ${\mathbb Q}$. From the Weil pairing one can ...

**5**

votes

**3**answers

379 views

### Surfaces ruled over elliptic curves

Ground field $\Bbb{C}$. Algebraic category. Elliptic surfaces are those surfaces endowed with a morphism onto some smooth curve, with generic fiber an elliptic curve.
Suppose $E$ is an elliptic curve ...

**1**

vote

**1**answer

208 views

### Weiestrass Form

How to convert this to weiestrass form?
$x^{2}y^{2}-2\left( 1+2\rho \right) xy^{2}+y^{2}-x^{2}-2\left( 1+2\rho
\right) x-1=0$

**0**

votes

**0**answers

139 views

### r-torsion points on elliptic curve on finite field

Let $E(\mathbb{F}_q)$ - elliptic curve, $r$ is prime, $|E(\mathbb{F}_q)[r]| > 1$.
Let $r | q-1$. Is it true that $|E(\mathbb{F}_q)[r]| = r^2$?

**9**

votes

**1**answer

340 views

### examples of “exotic” moduli problems for elliptic curves?

Let $\textbf{Ell}$ be the category of elliptic curves over various base schemes, and where a morphism between $E\rightarrow S$ and $E'\rightarrow S'$ is a cartesian diagram with those two maps as ...

**3**

votes

**1**answer

326 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

140 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 ...

**5**

votes

**1**answer

310 views

### Where do the product expansions of modular forms come from?

It is well-known that many modular forms can be expressed as infinite products. For instance, the most famous one is probably the expansion
$$\Delta(q) = q \prod_{n=1}^\infty (1-q^n)^{24}$$
for the ...

**5**

votes

**2**answers

299 views

### elliptic curve with a degree 2 isogeny to itself?

I've come across the following question, which I think must be easy for experts: is there a complex elliptic curve $E$ with an isogeny of degree 2 to itself?
Of course one can ask the same question ...

**9**

votes

**2**answers

531 views

### Best bounds toward Serre's uniformity conjecture

If $E$ is a non-CM elliptic curve over $Q$, then it is a famous theorem of Serre
that there is some integer $M(E)$ such that for any prime $\ell > M(E)$, the image of the
Galois representations ...

**6**

votes

**1**answer

273 views

### Order of torsion group

What can one say about the order of the torsion group of an elliptic curve defined over the compositum of all quadratic extensions of $\mathbb{Q}$ ?

**2**

votes

**0**answers

119 views

### A nice rigid analytic model for local systems over an elliptic curve?

For $E$ an elliptic curve, let $LS(E)$ be the group of line bundles on $E$ with a flat connection. This is an $\mathbb{A}^1$-torsor over $E^\vee$. By Riemann-Hilbert (since the gauge group acts ...

**4**

votes

**1**answer

480 views

### equivalence between katz and classical modular forms

$\newcommand{\CC}{\mathbb{C}}$
$\newcommand{\ZZ}{\mathbb{Z}}$
$\newcommand{\PP}{\mathbb{P}}$
$\newcommand{\QQ}{\mathbb{Q}}$
$\newcommand{\hH}{\mathcal{H}}$
$\newcommand{\eE}{\mathcal{E}}$
...

**0**

votes

**1**answer

148 views

### Can we find a set of elliptic curves over rationals associated with $f$?.

We know that for each elliptic curve over rationals, we can define the Dirichlet series of the Hasse–Weil $L$-function, i.e., the function associated with an elliptic curve over rationals.
Then my ...

**3**

votes

**1**answer

314 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

713 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 ...

**3**

votes

**1**answer

352 views

### Is there an elliptic surface over $Y(1)$?

Actually I have a few related questions.
Here, by $Y(1)$ I mean the affine $j$-line $\text{SL}_2(\mathbb{Z})\backslash\mathcal{H}$.
I know $Y(1)$ is only a coarse moduli space, so there isn't a ...

**1**

vote

**0**answers

216 views

### lifts of maps to $\mathcal{M}_{1,1}$

Hi,
here's there's a construction about elliptic curves that I do not completely understand. Suppose I consider the two following families of elliptic curves over $\mathbb{C}^*$.
The first, which I ...

**3**

votes

**0**answers

122 views

### P-adic Weierstrass Lemma for several variables

The p-adic Weiestrass lemma asserts that a power series $f(z)$ with coefficients in the ring of integers of a local field can be factored as $π^n·u(z)·p(z)$ where u(z) is a unit in the ring of power ...

**5**

votes

**1**answer

261 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

218 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 ...

**15**

votes

**3**answers

760 views

### Elliptic curve over a scheme is a group scheme?

In Katz's article p-adic properties of modular schemes and modular forms in the Antwerp proceedings, the following definition of an elliptic curve over a base scheme $S$ is given:
By an elliptic ...

**2**

votes

**2**answers

152 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

180 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 ...

**4**

votes

**3**answers

534 views

### Division by 3 on elliptic curve

Dear MO,
There is a classical theorem (cf. Theorem 4.1 on Husemoller "Elliptic Curve" book) that states conditions on the coordinates of a point P in an elliptic curve to be twice another point. The ...

**3**

votes

**2**answers

501 views

### About equivalent statements of the Birch and Swinnerton-Dyer Conjecture [closed]

The Birch and Swinnerton-Dyer Conjecture is well known in the current literature
http://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture
My question is about the possible equivalent ...

**0**

votes

**1**answer

688 views

### Pairing on elliptic curve

Let $E(\mathbb{F_q})$ - elliptic curve.
$G_1 = E(\mathbb{F_q})[r]$. $|G_1| = r$.
$k$ is minimal such $r | q^k - 1$.
$\pi_q$ - $q$-power Frobenius endomorphism.
$G_2 = E(\mathbb{F_{q^k}})[r] \cap ...

**2**

votes

**0**answers

232 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.
...

**0**

votes

**0**answers

234 views

### Analytic rank of an elliptic curve with algebraic rank 0

Let $E$ be an elliptic curve over $\mathbb{Q}$ with algebraic rank 0. Is there any way, one can argue that the analytic rank must be even? Of course this would follow from the standard conjectures, ...

**5**

votes

**1**answer

555 views

### A Universal Elliptic Curve

I'm working through Deligne's "Formes modulaires et representations l-adiques" paper and I find one of his constructions particularily ambigious. I'm hoping someone can give me a bit of clarification ...

**3**

votes

**2**answers

280 views

### Inequality relating rank and analytic rank

Let $E$ be an elliptic curve over $\mathbb{Q}$. It is known from Gross and Zagier that if $\textrm{rank}_{\textrm{an}}(E) \leq 1$, then
$$\textrm{rank}(E) \geq \textrm{rank}_{\textrm{an}}(E).$$
...

**3**

votes

**1**answer

257 views

### The elliptic Lehmer problem for several independent algebraic points

The higher dimensional Lehmer problem asserts that if $\alpha_1,\ldots,\alpha_r$ are multiplicatively independent non-zero algebraic numbers generating an extension of $\mathbb{Q}$ of degree $d$, then ...

**8**

votes

**1**answer

187 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 ...

**8**

votes

**1**answer

291 views

### Injective morphism from an elliptic curve to $\mathbb CP^2$.

Let $E$ be the elliptic curve $x^3+y^3+z^3=0$.
Question. Are there injective morphisms $E\to \mathbb CP^2$ of arbitrary high degree?
Comments. 1) There are injective morphisms $E\to \mathbb CP^2$ ...

**8**

votes

**1**answer

412 views

### Best results regarding the Lang-Trotter conjecture

Let $E$ be an elliptic curve over $\mathbb Q$, which does not have complex multiplication over the algebraic closure of $\mathbb Q$. For $x>0$, let $P(x)$ be the number of primes $p < x$ such ...