An elliptic curve is an algebraic curve of genus one with some additional properties. Questions with this tag will often have the top-level tags nt.number-theory or ag.algebraic-geometry in addition; note also the tag arithmetic-geometry as well as some related tags such as rational-points, ...

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Least supersingular prime

Given an elliptic curve over the rationals, what can one say about the size of the smallest supersingular prime?
11
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2answers
399 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?
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1answer
127 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 ...
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0answers
114 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
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3answers
272 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 ...
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68 views

How do we usually calculate the size of the 2-division field of an elliptic curve over local fields

Let $E$ be an elliptic curve over $\mathbb Q$, with rank $0$ and $E(\mathbb Q)[2]=0$. Let $p$ be a prime number. How do we usually calculate $\# E(\mathbb Q_p)[2]$ when $E$ has good, bad reduction at ...
11
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0answers
218 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
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1answer
189 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 ...
3
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1answer
165 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
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1answer
295 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
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1answer
330 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
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1answer
153 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
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1answer
133 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
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1answer
548 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
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0answers
47 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
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3answers
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
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3answers
455 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
4answers
397 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
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0answers
226 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
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3answers
414 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
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0answers
214 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
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0answers
151 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 $\alpha = \left[\begin{matrix} 1&0 \\ 10&1 \end{matrix}\right] \in SL_2(\mathbb{Z})$. ...
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0answers
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 ...
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0answers
79 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
1answer
141 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
3answers
323 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
1answer
242 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
1answer
304 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
1answer
282 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
2answers
199 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
1answer
253 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
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0answers
160 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
2answers
374 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
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0answers
214 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. ...
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0answers
265 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
2answers
389 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
2answers
253 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
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0answers
188 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
1answer
331 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) ...
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0answers
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
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0answers
225 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
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1answer
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 $A$ ...
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0answers
104 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
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1answer
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
1answer
163 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
1answer
472 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
1answer
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
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2answers
350 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
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1answer
310 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
2answers
205 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{?}{=} ...