**3**

votes

**1**answer

182 views

### Some clarifications regarding Deligne's paper on $\ell$-adic representations arising from modular forms

I've posted this question few days ago on math.stackexchange because it seems quite superficial. However, since I've got no responses at all, I'm posting it here. If the question is not suitable, ...

**0**

votes

**0**answers

43 views

### Equivalent of Lauricella $F_D$ on an elliptic curve?

Lauricella's hypergeometric function $F_D$ is related to (weighted) configurations of points on $\mathbb{P}^1$. I am looking for generalizations to weighted point configurations on an elliptic curve. ...

**4**

votes

**1**answer

581 views

### How good can we approximate with algebraic curve the egg shaped vanishing of $\Re \zeta(s)$ near the origin?

Related to this question
where degree $2$ algebraic curve is good approximation to vanishing of
the real part of expression involving zeta.
Near the origin, $\Re \zeta(s)$ vanishes in egg shaped ...

**2**

votes

**1**answer

196 views

### An explicit formula for Weil pairing on a complex torus

I begin by defining the Weil pairing in general (as in Oda's 1969 paper). My question is about an explicit formula for this pairing in the case of an elliptic curve over complex numbers.
Let ...

**0**

votes

**0**answers

69 views

### Is it possible the division polynomials evaluated at fixed point to be perfect powers unbounded number of times?

Let $E$ be elliptic curve over the rationals and $P=(X_P,Y_P)$ point on $E$.
$\psi_n$ are the division polynomials.
Define $a_n=\psi_n(X_P,Y_P)$.
Is it possible $a_n$ to be perfect power ...

**4**

votes

**2**answers

545 views

### Unable to find any information regarding this fact (Frey, elliptic curves)

Frey states in 'Links between stable elliptic curves and certain Diophantine equations' the following
"The most important fact about elliptic curves with reduction of muItipIicative type is due to ...

**0**

votes

**0**answers

22 views

### Hardness assumptions on composite order bilinear groups

I am not at all knowledgeable in elliptic curve cryptography. So, here lies a couple of questions that I failed to find answers for to my satisfaction.
Is there any known Type-III bilinear pairing ...

**3**

votes

**0**answers

127 views

### Galois descent for a non-Galois extension

Suppose that $k$ is an algebraically closed field of characteristic $p > 0$ and $E/k$ is a supersingular elliptic curve equipped with a full level $N$ structure $\phi$ for some $N \ge 3$ that is ...

**1**

vote

**1**answer

196 views

### Heegner points on elliptic curves

I want to know about Heegner point computations for a CM elliptic curve. What is the best reference book/paper for reading?

**1**

vote

**0**answers

171 views

### Fourier expansions at the cusps of $\Gamma_0(N)$

My question may be basic but I can't find any answer. Let $N$ be a positive integer. I need to find the constant term (of the Fourier series) at each cusps of a modular form
...

**4**

votes

**1**answer

204 views

### Proof of a Proposition regarding the reduction of N-torsion groups on elliptic curves

In Diamond-Shurman A first course in Modular forms p.334 Prop. 8.4.4. It is stated,
For E elliptic curve over $\bar{\mathbb{Q}}$ with good reduction at the prime ideal $\mathfrak{p}$ the reduction ...

**2**

votes

**0**answers

151 views

### A morphism of elliptic schemes that preserves the identity is a homomorphism

I am trying to understand the proof of the fact that any morphism $f \colon E_1 \rightarrow E_2$ of elliptic curves over an arbitrary base scheme $S$ satisfying $f(0) = 0$ must respect the group ...

**2**

votes

**1**answer

176 views

### Difference between Frobenii on Tate modules of special and generic fibre

Let $E$ be elliptic curve over $\mathbb Q$ and $p$ a prime of good reduction for $E$. Fix $\ell \neq p$.
If $E_p$ is ordinary then we have Frobenius $F_p$ on $E_p$. Assume $F_p$ lifts to ...

**2**

votes

**1**answer

165 views

### rational point of a curve [closed]

Let $X$ be a smooth projective curve over $\mathbb{Q}$. I heard (if I did not misunderstood) that the geometry of the complex points $X(\mathbb{C})$ (flat, hyperbolic case) dicts the shape (group ...

**0**

votes

**0**answers

33 views

### Reading roots of the supersingular polynomial from an explicit formula

I wish to obtain an explicit expression for the supersingular polynomial such that the roots of the polynomial can be easily read from it. I am aware of the work of Brillhart and Morton -- ...

**5**

votes

**1**answer

550 views

### The elliptic curve for $x_1^9+x_2^9+\dots+x_6^9 = y_1^9+y_2^9+\dots+y_6^9$

I. If there are $a,b,c,d,e,f$ such that,
$$a+b+c = d+e+f\tag1$$
$$a^2+b^2+c^2 = d^2+e^2+f^2\tag2$$
$$3u^3-3uv+w=-def\tag3$$
where $u=a+b+c,\; v = ab+ac+bc,\;w = abc$, then,
$$(a + u)^k + (b + ...

**1**

vote

**0**answers

91 views

### Need information about particular kind of quotients of semisimple algebraic groups by free abelian discrete subgroups

Let me start with the simplest version of the question since already there I don't know anything.
For a complex number $q$, consider the quotient space $X_q:=\mathrm{SL}_2(\mathbb ...

**0**

votes

**0**answers

105 views

### Asymptotic Expansion of Double integral

Crosspost from math.stackexchange. Have a look at the great answers there, even though they do not quite answer the question completely.
Define
$$G(\theta) = \int\limits_0^\infty \int\limits_0^{2\pi} ...

**5**

votes

**0**answers

312 views

### is the modular curve X(N) defined over Q?

In most sources, the field of definition of the modular curve $X(n)_\mathbb{C}$ (quotient of the upper half plane by the subgroup $\Gamma(n)$ of $SL_2(\mathbb{Z})$ congruent to $I\mod n$) is ...

**2**

votes

**2**answers

537 views

### Is it possible on an elliptic curve both $x,y$ to be arbitrary large powers infinitely often?

Let $f(x,y)=0$ be irreducible elliptic curve over the rationals.
Are there $f$ for which:
Both $x,y$ are arbitrary large powers infinitely often,
i.e. infinitely many rational points $(u^k,v^m)$ ...

**3**

votes

**0**answers

243 views

### On 7th and 8th powers for $x_1^k+x_2^k+x_3^k+x_4^k = y_1^k+y_2^k+y_3^k+y_4^k$

The Diophantine equation,
$$x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k\tag1$$
for either $k=5$ or $6$ is quite well explored, and it has long been known that it has an infinite number of primitive ...

**24**

votes

**1**answer

1k views

### $x^4+y^4$ powerful for relatively prime $x,y$

I asked this question on the NMBRTHRY mailing list on
17 February 2014, but it remains unsolved as far as I know.
Recall that a "powerful
number" is a positive integer whose prime factorizations
$m = ...

**5**

votes

**1**answer

298 views

### On the elliptic curve $x(x+a^2)(x+b^2) = y^2$

Ajai Choudhry showed that special cases of the elliptic curve,
$$x(x+a^2)(x+b^2)=y^2\tag1$$
can be used to prove that,
$$u_1^7+u_2^7+\dots + u_9^7 = 0\tag2$$
has an infinite number of primitive ...

**13**

votes

**2**answers

1k views

### More elliptic curves for $a^4+b^4+c^4+d^4 = (a+b+c+d)^4$?

(Note: See also the $a^4+b^4+c^4 = 1$ version in this old MSE post.)
The equation discussed in a paper by Jacobi and Madden,
$$a^4+b^4+c^4+d^4 = (a+b+c+d)^4 = z^4\tag1$$
or equivalently,
$$(p-2q + ...

**3**

votes

**2**answers

147 views

### Can the pre-image of the real points in the complex upper-half plane of a modular elliptic curve under the modular parametrization be identified?

Consider an elliptic curve $E/\Bbb Q$ and let $\Phi:\Gamma_0(N)\backslash\overline{\mathfrak{H}}\rightarrow E(\Bbb C)$ be the analytical description of its modular parametrization. We know that this ...

**2**

votes

**1**answer

104 views

### Simple Isogeny Question

I'm looking for a reference of an isogeny fact that I've used many times but am having a hard time proving formally.
One can define the degree of an isogeny as the degree of extension fields of the ...

**7**

votes

**2**answers

348 views

### Least supersingular prime

Given an elliptic curve over the rationals, what can one say about the size of the smallest supersingular prime?

**12**

votes

**3**answers

654 views

### Are there nonisotrivial elliptic curves over $\mathbb{G}_m$?

Is there an elliptic curve over $\mathbb{C}[t, t^{-1}]$ that has a nonconstant $j$-invariant? What is an equation for such a curve, if it exists?

**1**

vote

**1**answer

180 views

### Canonical form of cubic curves over general fields

Given a field of characteristic not 2 or 3 containing a primitive third root of unity, is it true that every nonsingular cubic curve, i.e. a curve defined by one homogeneous form of degree 3 in 3 ...

**0**

votes

**0**answers

151 views

### Is the Jacobi theta function invertible?

Let $\theta$ denote the Jacobi theta function:
$$\theta=\sum_{k=0}^{\infty}{(-1)^kq^{k(k+1)}sin((2k+1)\frac{2\pi}{\omega_1}Re(z))},$$
and we have a complex number $t$. Suppose that we know there ...

**0**

votes

**3**answers

328 views

### Why there are two point at infinity on certain elliptic curve [closed]

In article Adams, W. W., & Razar, M. J. (1980). Multiples of points on elliptic curves and continued fractions. Proc. London Math. Soc, 41, 481-498. is said on ...

**11**

votes

**0**answers

260 views

### Am I missing something about this notion of Mirror Symmetry for abelian varieties?

This is a continuation of my recent question: Mirror symmetry for polarized abelian surfaces and Shioda-Inose K3s.
In the comments of the question, I was directed to the paper ...

**1**

vote

**1**answer

221 views

### computing height on elliptic curve of the form $y^2=x^3-nx$

Let $E$ be the elliptic curve
$$y^2 =x^3 - 19*67 x$$
and $P=[26011/625,2159616/15625]$, I want to compute $\hat{h}(P)$ using formula given in
Fujita, Y., & Terai, N. (2011). Generators for the ...

**4**

votes

**1**answer

190 views

### Elliptic curves with maximal order in an imaginary field

Let $K/\mathbb{Q}$ be an imaginary quadratic extension with discriminant $-D$. Then there is an elliptic curve $E$ over $\overline{\mathbb{Q}}$ such that End$(E)^{0}: =$ End$(E) \otimes Q = K$.
Now ...

**5**

votes

**1**answer

339 views

### Many integral points on quartic models of elliptic curve via differences of squares

Pick fourth power free integer $n$ ($p^4$ doesn't divide $n$).
Represent $n$ as difference of possibly negative integer squares
$n=v_i^2-u_i^2$.
The goal is to find quadratic polynomial with integer ...

**9**

votes

**1**answer

381 views

### Eichler-Shimura congruence

I'm trying to understand the Eichler-Shimura congruence which relates the Hecke operator $T_p$ to Frobenius at $p$ in characteristic $p$.
Two possible ways to compute $T_p$ mod $p$ seem to be:
A) ...

**0**

votes

**1**answer

193 views

### Is elliptic curve point division defined over the field of real numbers?

An elliptic curve is defined over the field of real numbers:
$y^2=x^3 + ax + b$
A point P and scalar n can be multiplied using a combination of point doubling and adding.
What about point division? ...

**2**

votes

**1**answer

164 views

### Trivial Weil-Châtelet group

Does there exist an elliptic curve over a number field $K$ such that $WC(E/K)\cong H^1(G_K, E)$ is trivial?

**10**

votes

**1**answer

651 views

### Questions about the “universal elliptic curve” over the affine $j$-line punctured at 0 and 1728

So my question refers to families of elliptic curves over the $\mathbb{A}^1_\mathbb{C}\setminus\{0,1728\}$ whose fiber above a point $j$ has $j$-invariant equal to $j$ (I understand it's not ...

**0**

votes

**0**answers

91 views

### Geometric meaning of the group law on Jacobi quartic

Is there any geometric "face" of the group law on the Jacobi quartic similar to Weierstraß cubic? (Reminder: given any two distinct points on the cubic, let's draw a line containing both of them and ...

**11**

votes

**2**answers

2k views

### How many Pythagorean triples are there in which every member is triangular?

How many Pythagorean triples $(a,b,c)$ are there such that $a, b$ and $c$ are triangular?
Any two solutions with only $a$ and $b$ interchanged are considered equivalent.
The question of existence ...

**12**

votes

**3**answers

498 views

### For an elliptic curve $E/\mathbb{Q}$ can the cohomology group $H^1(\text{Gal}(\mathbb{Q}(E[p])/\mathbb{Q}), E[p])$ be nontrivial?

Suppose that $E$ an elliptic curve defined over $\mathbb{Q}$ and $p$ an odd prime. Let $G=\text{Gal}(\mathbb{Q}(E[p])/\mathbb{Q})$. I am wondering whether the cohomology group $H^1(G, E[p])$ can be ...

**3**

votes

**3**answers

412 views

### Pairs of quadratic polynomials taking values pairs of consecutive squares

Let $f,g \in \mathbb{Z}[x]$ be quadratic and neither square.
For $x,y,z \in \mathbb{Z}$ what is the maximal number
of solutions to $f(x)=z^2,g(y)=(z+1)^2$?
Solutions are integral points on the genus ...

**16**

votes

**0**answers

264 views

### Elliptic $\infty$-line bundles over $B G$

Theorem 5.2 in Jacob Lurie's "Survey of Elliptic Cohomology" (pdf) states the equivalence of two maps
$$
B G \longrightarrow B \mathrm{GL}_1(A)
$$
for $A$ an $E_\infty$-ring carrying an oriented ...

**8**

votes

**3**answers

621 views

### $j$-invariants of elliptic curves over finite fields

Let $K$ be a finite field, and $\overline{K}$ its algebraic closure. It is well known that two curves are isomorphic over $\overline{K}$ if and only if they have the same $j$-invariant. If two such ...

**2**

votes

**1**answer

358 views

### Is the upper half plane an algebraic stack?

Here by algebraic stack I mean an algebraic stack over the etale site $\textbf{Sch}/\mathbb{C}$.
So I've read from various nonrigorous sources that the upper half plane $\mathcal{H}$ is a fine moduli ...

**3**

votes

**0**answers

172 views

### Fourier expansions of newforms at width-1 cusps

Let $f_E$ be the newform attached to the Elliptic Curve $E$ with cremona label
$\textbf{100a1}$ and let $\alpha = \left[\begin{matrix} 1&0 \\ 10&1 \end{matrix}\right] \in SL_2(\mathbb{Z})$. ...

**0**

votes

**0**answers

50 views

### Some of the real zeros of those $k^{th}$ derivatives are also simple?

Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. The modularity theorem implies that $L(C,s)$ is the ...

**0**

votes

**0**answers

97 views

### How can information about morphisms between modular curves be read from the morphism of their corresponding stacks?

$\newcommand{\yY}{\mathcal{Y}}$
$\newcommand{\PSL}{\text{PSL}}$
$\newcommand{\SL}{\text{SL}}$
$\newcommand{\ZZ}{\mathbb{Z}}$
$\newcommand{\CC}{\mathbb{C}}$
$\newcommand{\mM}{\mathcal{M}}$
Let $Y(1)$ ...

**5**

votes

**1**answer

232 views

### Modular polynomials for elliptic curves point counting

The Schoof-Elkies-Atkin (SEA) algorithm (for counting points on elliptic curves over a finite field) performs computations over polynomials modulo some modular polynomials. Originally the "classical" ...