**-4**

votes

**0**answers

42 views

### finding points on elliptic curve over finite field [on hold]

Find the points on the elliptic curve y^2 = x^3 + 2x + 2 in F17 (field of prime 17).
Do I have to guess a first point and then use an algorithm to spit out all other points?

**4**

votes

**0**answers

113 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

309 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

144 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? ...

**-4**

votes

**0**answers

105 views

### reduction of elliptic curves to finite field [closed]

Let $E$ be an elliptic curve which is defined over $\mathbb{Q}$ and $p$ be a prime number.
I know we can reduced $E(\mathbb{Q})$ to $E(\mathbb{F}_p)$, is there an algorithm to reduce $E(\mathbb{Q})$ ...

**-1**

votes

**0**answers

125 views

### elliptic curves and tower of finite fields [closed]

Let $E$ be an elliptic curve which is defined over $\mathbb{F}_{p^n}$ and $m< n$. Can we reduce $E(\mathbb{F}_{p^n})$ to $E(\mathbb{F}_{p^m})$?
Specially in the case where $m=1$?
I mean, let $A$ ...

**2**

votes

**1**answer

124 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?

**9**

votes

**1**answer

526 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

41 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

**3**answers

1k 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

445 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

**4**answers

386 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 ...

**15**

votes

**0**answers

218 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 ...

**6**

votes

**3**answers

402 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 ...

**1**

vote

**0**answers

202 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

148 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

41 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

78 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

138 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

317 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

233 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

303 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

278 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

197 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

248 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

157 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 ...

**7**

votes

**2**answers

366 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

211 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

257 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

387 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

250 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

187 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

318 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

221 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

135 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

103 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

120 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

161 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

461 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

143 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

348 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

300 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

56 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

253 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

94 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

39 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

85 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

149 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 ...