**5**

votes

**2**answers

241 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 ...

**-5**

votes

**0**answers

32 views

### How to find secret key and public key for ECC cryptosystem? [on hold]

Develop an ECC cryptosystem based on E31(1;1),
point G = (0,1) which has order 32.
nA value of 6.
What is the secret key?
What is the public key?

**-3**

votes

**0**answers

39 views

### How to compute 3P from elliptic curve where P is (28, 8) [on hold]

Consider the elliptic curve E31(1,1):
Calculate 3P, where P = (28,8).

**1**

vote

**0**answers

196 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

111 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

38 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

72 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

121 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" ...

**4**

votes

**3**answers

305 views

### Is there an integral point in the group generated by an rational point?

Let $E$ be an elliptic curve over rational field. Let $P=(a/d^2,b/d^3)\in E(\mathbb{Q})$ and
$$G=\{P,2P,3P,4P,\cdots\}.$$
Is there an integral point $Q\in G?$

**4**

votes

**1**answer

226 views

### What is the complexity of finding an integral point on an elliptic curve?

Let $E$ be an elliptic curve over rational numbers. We know that the set of integral points on $E$ is finite. What is the complexity of finding a point $P\in E(\mathbb{Z})$?
Indeed I'm trying to find ...

**5**

votes

**1**answer

298 views

### Are the Szpiro ratios of 37b1 over certain number fields {33,39,42,48,51,66}?

Related to this question.
According to Hindry p.7 Conj 3.1
and Stein's notes Szpiro's conjecture over number fields states that the Szpiro ratio is:
$$ ...

**4**

votes

**1**answer

270 views

### Why the Szpiro conjecture over number fields doesn't depend on the discriminant of the number field?

According to Hindry p.7 Conj 3.1
and Stein Szpiro's conjecture states that the Szpiro ratio is:
$$ \sigma_{E/K}=\frac{\log{|N_{K/Q}\Delta_{E/K}|}}{\log{|N_{K/Q} f_{E/K}}|}$$
Given $ \varepsilon ...

**3**

votes

**2**answers

194 views

### Sheaf of isogenies representable?

It is well-known that "the" stack of elliptic curves (allow me to be vague as to singular curves, compactifications etc) has a presentation by a groupoid in schemes. One of the things that needs to be ...

**3**

votes

**1**answer

237 views

### Theorem 7b of Serre's “Propriétés galoisiennes des points d'ordre fini des courbes elliptiques”

Could someone please point me towards a proof of the statement in the second paragraph, in the proof of Theorem 7b of Serre's Propriétés galoisiennes...? The statement is as follows:
Let $F$ and $F'$ ...

**1**

vote

**0**answers

151 views

### which sections of elliptic curves are conjugate?

Suppose you have a relative elliptic curves $f : E\rightarrow S$ (say $S$ is connected). Then suppose you have two sections $g,g' : S\rightarrow E$, corresponding to two sections $g_*,g'_*$ to the map ...

**6**

votes

**2**answers

356 views

### How did height in algeb. number theory/elliptic curves started?

Maybe this is obvious but it isn't to me yet. What is the history of heights used in say points of the project plane over a number field or of elliptic curve over a number field? I would guess people ...

**6**

votes

**0**answers

209 views

### Invariant obstructions to gluing Galois representations on elliptic curves

Let $E$ be an elliptic curve over $\overline{\mathbb F_p}$, or another separably closed field of characteristic $p$. Let $K$ be the function field of $E$, and let $K_p$ be the local field at a point.
...

**1**

vote

**0**answers

248 views

### How to find generators to Mordell weil groups of elliptic curves?

I am new to branch of elliptic curve and algebraic number theory .I want to find generators to Mordell Weil group of the Elliptic Curve $y^2=x^3-6321363052$ and class number of $\mathbb ...

**9**

votes

**2**answers

379 views

### Elliptic curves with trace of Frobenius values always congruent to 0 modulo 2

Let $E/\mathbb{Q}$ be an elliptic curve and suppose that the trace of Frobenius values are such that $a_{p}(E) \equiv 0 \pmod{2}$ for all odd primes avoiding the conductor. Is it the case that $E$ ...

**2**

votes

**2**answers

234 views

### Reference for Skinner-Urban on the Iwasawa main conjecture for $GL_2$

Does anyone know the existence of an expository paper or a report discussing the work of Skinner-Urban
"The Iwasawa main conjecture for $GL_2$"?
I am interested in partucular in the case of elliptic ...

**0**

votes

**0**answers

182 views

### Which of the Mochizuki's works are the most closely related to elliptic curves?

I'm very much interest about algebraic geometry and number theory along with cryptography, but I have a special interest about the elliptic curves. I have heard a lot of interesting things about ...

**8**

votes

**1**answer

312 views

### Separation of lattice points on the Mordell elliptic curve

Consider the Mordell equation x^3 – y^2 = k, where x is a non-square positive integer and y^2 is the perfect square nearest to x^3. Noam Elkies (see http://www.math.harvard.edu/~elkies/hall.html) ...

**0**

votes

**0**answers

75 views

### Superelliptic Curves [duplicate]

I'm trying to find information on superelliptic curves and how to solve them over the integers. The equation is $$y^k = f(x)$$ where $k=3$ and $f$ has degree $d=3$. Does anyone know any ...

**4**

votes

**0**answers

212 views

### The Modularity Theorem and Serre's/Faltings's Isogeny Theorem

Earlier this year I completed my Masters dissertation on Andrew Wiles's proof of The Modularity Theorem for semistable elliptic curves as a precursor to the accepted proof of Fermat's Last Theorem. ...

**1**

vote

**1**answer

132 views

### Divisors on an abelian surface

Let $A$ be an abelian surface given by the quotient of a product of two generic elliptic curves $E_1 \times E_2$ by the product $T_1 \times T_2$ of two translations by $2$-torsion points. Then $A$ ...

**2**

votes

**0**answers

99 views

### padic BSD vs. BSD for algorithm to compute rank

Just to be specific, I deal only with the elliptic curves $E$ over $\mathbb{Q}$, and most of the explaination here are obtained from the paper: Algorithms for the Arithmetic of Elliptic Curves using ...

**1**

vote

**1**answer

116 views

### $p$-adic Regulators

Is there some relationship between the $p$-adic regulators of isogenous curves over $\mathbb{Q}$? I've done some computations and their ratio seems to be related (equivalent in all calculations so ...

**4**

votes

**1**answer

159 views

### Explicit formula for the Poincare dual of a CM endomorphism of an elliptic curve

Let $E/\mathbf{C}$ be an elliptic curve with CM by the maximal order $\mathcal{O}_K$ of $K=\mathbf{Q}(\sqrt{-D})$ where $D$ is positive and square-free integer. To make it even more precise, let us ...

**8**

votes

**1**answer

448 views

### Rank of Elliptic Curves

Recently, I have heard of some heuristics that would suggest that the rank of elliptic curves are bounded (specifically in the congruent number family). I always though that the best way to prove ...

**5**

votes

**1**answer

141 views

### Linear systems on bielliptic surfaces

A bielliptic surface is a surface of type $S=E_1 \times E_2/G$ where $E_1, E_2$ are elliptic curves and $G$ is a finite group of translations of $E_1$ acting on $E_2$ such that $E_2/G=\mathbf{P}^1$.
...

**10**

votes

**2**answers

345 views

### Existence of a family of elliptic curves with large torsion subgroup

Andrej Dujella's excellent web-site "High rank elliptic curves with prescribed torsion", at http://web.math.pmf.unizg.hr/~duje/tors/tors.html
gives example of the (current) largest known rank of an ...

**10**

votes

**1**answer

286 views

### K3 surfaces that correspond to rational points of elliptic curves

In his work on mirror symmetry (http://arxiv.org/pdf/alg-geom/9502005v2.pdf) Igor Dolgachev has considered families of K3 surfaces of Picard rank at least 19 with the base given by $X_0(n)^+$, the ...

**2**

votes

**2**answers

194 views

### BSD leading-term coefficient in terms of places without distinction

After reading this blog post, I learned the BSD conjectural formula for the coefficient of the leading term $a_0$ of the L-function of an elliptic curve $E$, namely
$$
a_0 \stackrel{?}{=} ...

**0**

votes

**0**answers

55 views

### If the $L$-series does not vanish

I refer to this paper http://wstein.org/papers/shark/shark.pdf
At the top of page 24, we are dealing with the issue where the $L$-series does not vanish for the case where $p$ is good and ordinary. ...

**2**

votes

**1**answer

243 views

### Integer points on $y^2=x^2-x^3+x^4$

Does the Diophantine equation $y^2=x^2-x^3+x^4$ have solutions other than
$x=1,y=1$? Interestingly, the Diophantine equation $y^2=x^2-x^3+x^5$ has such solutions: $x=3,y=15$, $x=5,y=55$, ...

**2**

votes

**1**answer

91 views

### How the equality in the first case is equivalent to the inequality in the last case?

The motivation to this question can be found in:
About equivalent statements of the Birch and Swinnerton-Dyer Conjecture
My question is about the last equivalences:
$\mathrm{ord}_{s=1} L(E/K,s) = ...

**0**

votes

**0**answers

38 views

### Can we deduce something about the nature of those solutions?

To describe the problem, we note that we can find an affine model for any elliptic curve $C$ over $ℚ$ in Weierstrass form
$$C:y^2=x^3+ax+b$$
with $a,b∈ℤ$. The full L-series of $C$ is given by
...

**4**

votes

**0**answers

84 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

146 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

**0**answers

266 views

### Proportion of rational elliptic curves of a given rank

This morning appeared on Arxiv the following article by Manjul Bhargava et al: http://arxiv.org/pdf/1407.1826.pdf, in which the authors give a lower bound for th proportion of rational elliptic curves ...

**2**

votes

**2**answers

368 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

170 views

### Is the leading Taylor coefficient at $s = 1$ of the $L$-series of an elliptic curve over $\mathbb{Q}$ positive, as predicted by BSD?

Let $E$ be an elliptic curve over $\mathbb{Q}$. As proved by Wiles et al., its $L$-series $L(E, s)$ is entire. Set $r := \mathrm{ord}_{s = 1} L(E, s)$, a value conjecturally equal to ...

**6**

votes

**1**answer

365 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) = ...

**1**

vote

**1**answer

182 views

### Some questions related to Iwasawa invariants of elliptic curves

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at an odd prime $p$.
Let $\mathbb{Z}_{p}$ denote the ring of $p$-adic integers, and $\mathbb{Q}^{cyc}$ be the ...

**8**

votes

**2**answers

427 views

### Elliptic Curves with equal trace of Frobenius Values

Suppose we have two elliptic curves over $\mathbb{Q}$ with trivial rational torsion. Is there some density $\delta$ such that if the trace of Frobenius values of the two elliptic curves are equal on a ...

**3**

votes

**0**answers

88 views

### Lang's height conjecture over $\mathbb{F}_q(T)$?

Is the canonical height of a non-torsion $\mathbb{F}_q(T)$-rational point of an elliptic curve over $\mathbb{F}_q(T)$ known or supposed to be bounded from below by an absolute positive number (or ...

**1**

vote

**1**answer

109 views

### Can we use this formula to construct rational points on the curve $C$?

One of the techniques used to quantifying the size of a point on an elliptic curve is the so called canonical height defined as follow: Let $R=(x,y)∈C(ℚ)$ where $x=(p/d),p,d∈ℤ$. Define the naive or ...

**0**

votes

**1**answer

186 views

### Is there is a known relation or expression containing the algebraic rank $r$?

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 ...

**6**

votes

**3**answers

330 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 ...

**3**

votes

**1**answer

158 views

### Galois representation attached to $3$-torsion points of an elliptic curve

Let
$ E $ - Elliptic curve defined over $ {\mathbb{Q}} $.
$G_{\mathbb{Q}}$ - The absolute Galois group, $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) $ of $\mathbb{Q}$.
$ E[3] $ - $3$-torsion points ...