# Tagged Questions

**1**

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**2**answers

141 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**

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**1**answer

250 views

### Mordell-Weil and finiteness of rational points

Let $E$ be a CM elliptic curve defined over a quadratic imaginary field $K$ with maximal order, that is, $\mathrm{End}_K(E)\cong \mathcal{O}_K$. Suppose the class number of $K$ is equal to $1$. Let ...

**3**

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**1**answer

188 views

### The existence of elliptic curves with prescribed supersingular primes

For a given infinite set of primes, not too big, eg, satisfying Lang-Trotter conjecture, can we always find an E.C. with supersingular reduction (at least) at these primes? How about E.C. without CM?
...

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297 views

### The existence of infinitely many supersingular primes for every elliptic curve over Q

Elkies proved The existence of infinitely many supersingular primes for every elliptic curve over Q. I read his paper, but found the supersingular primes he constructed are all 3(mod 4) type. So, how ...

**2**

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**2**answers

192 views

### What is the exact meaning of the real period in the $p$-adic formulation of BSD?

Let $E$ be an elliptic curve over $\mathbf{Q}$ which has split multiplicative reduction at $p$ (a prime). If one chooses a global Neron model of $E$ over $\mathbf{Z}$ (unique up to unique isomorphism ...

**3**

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**0**answers

118 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**

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**1**answer

255 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**

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**1**answer

473 views

### Heegner Points and Binary Quadratic Forms

I've been trying to read Gross' paper on Heegner points on $X_0(N)$ and I am stuck on a few details. The definition he is working with is that a heegner points is a pair $y=(E,E')$, where $E$ and $E'$ ...

**5**

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**1**answer

543 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**

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**1**answer

432 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**

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**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 ...

**2**

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**2**answers

476 views

### Special values of a doubly periodic meromorphic function

Consider the following function: $G(z) = \prod_{n \in \mathbb{Z}} {1 \over{\tanh^2\left(\pi\left(z-n\right)\right)}}$.
By constuction, it has poles at $z=m+in$ with $m,n \in \mathbb{Z}^2$.
...

**3**

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**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 ...

**7**

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**0**answers

301 views

### Tameness criterion in the reducible case

Dear MO,
This is a follow up to a previous question here in MO, but I will make this question self-contained for convenience. Those already familiar with the following paper [G] by Gross can safely ...

**4**

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**1**answer

932 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 ...

**6**

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**1**answer

450 views

### Can elliptic integral singular values generate cubic polynomials with integer coefficients?

For the elliptic integral of first kind, $K(m)=\int_0^{\pi/2}\frac{d\theta}{\sqrt{1-m^2sin^2\theta}} $, it is well-known that $K(m)$ can be expressed in what Chowla and Selberg call "finite terms" ...

**4**

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**0**answers

218 views

### A natural mod-n representation of the automorphisms of an elliptic curve

Let $E$ be an elliptic curve over a field of characteristic $p$, and let $n$ be an integer coprime to $p$. Then $E[n]$, the kernel of multiplication by $n$ on $E$, is (etale-locally) isomorphic to ...

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**2**answers

1k views

### Image of a Galois representation

Notation:
$E$ is a non-CM Elliptic curve over $\mathbb{Q}$.
$p$ is an ordinary prime.
$f$ - cuspidal eigenform of weight $k$ = 2 attached with $E$.
$\rho_f$ - the global 2-dimensional $p$-adic ...

**3**

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**0**answers

336 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 $\mathbb{Q}(\mu)\cap\mathbb{Q}(S)$?

**2**

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**1**answer

556 views

### Q_p*/(Q_p*)^2 and descent for elliptic curves

Is there a simple description of the group Q_p*/(Q_p*)^2 where Q_p denotes the p-adic integers?
I am doing descent calculations for elliptic curves, and so am most interested in the case p = 2. ...

**14**

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**1**answer

1k views

### If p is a prime congruent to 9 mod 16, can 4 divide the class number of Q(p^(1/4))?

When $p$ is a prime $\equiv9\bmod16$, the class number, $h$, of $\mathbb Q(p^{1/4})$ is known to be even. In
[Charles J. Parry,
A genus theory for quartic fields.
Crelle's Journal 314 (1980), ...

**2**

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**2**answers

489 views

### When is the period of elliptic curve over the rationals transcendental?

Given an elliptic curve $E/\mathbf{Q}$, when is its period transcendental/algebraic?

**13**

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**1**answer

479 views

### components of E[p], E universal in char p.

I have just realised that a group scheme I've known and loved for years is probably a bit wackier than I'd realised.
In this question, in Charles Rezk's answer, I erroneously claim that his ...

**13**

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**4**answers

988 views

### The ring of algebraic integers of the number field generated by torsion points on an elliptic curve

(Warning: a student asking)
Let $E$ be an elliptic curve over $\mathbf Q$. Let $P(a,b)$ be a (nontrivial) torsion point on $E$. Is there an easy description of the ring of algebraic integers of ...

**16**

votes

**3**answers

2k views

### Conceptual understanding of the Gross-Zagier theorem.

The Gross-Zagier paper "Heegner points and derivatives of $L$-series", is really computational and hard to plow through. It seems it is futile to read it as such and one must look for a more ...

**15**

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**3**answers

1k views

### Regulators of Number fields and Elliptic Curves

There is supposed to be a strong analogy between the arithmetic of number fields and the arithmetic of elliptic curves. One facet of this analogy is given by the class number formula for the leading ...