**5**

votes

**2**answers

466 views

### Langlands program vs Shimura-Taniyama-Weil conjecture

Edward Frenkel said that "we can see Langlands program as a generalization of Shimura-Taniyama-Weil conjecture in the case of elliptic curves"
I hope I'm not distorting his phrase, can someone ...

**5**

votes

**0**answers

188 views

### Average size of $p$-part of the Tate-Shafarevich group for elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve defined over $\mathbb{Q}$. For a given prime $p$, the $p$-Selmer group $\operatorname{Sel}_p(E)$ of $E$ and the $p$-part of the Tate-Shafarevich $Ш_E[p]$ group ...

**0**

votes

**0**answers

62 views

### Fastest algorithm to compute isogeny

Let $E/GF(p)$ and $E'/GF(p')$ are two isogenous elliptic curves($\#E=\#E'$). We know that there exist the map
$$\psi : E \to E'$$
Suppose that we haven't any information about degree of $\psi$.
...

**11**

votes

**0**answers

201 views

### Computing endomorphism rings of supersingular elliptic curves

I would like to know what algorithms there are to compute the linearly independent generators $(1,i,j,k)$ for quaternion algebra containing the endomorphism ring of a supersingular curve. The curve in ...

**2**

votes

**0**answers

81 views

### what is the structure of the group of isogenies between two ordinary elliptic curve?

Let $E_1$ and $E_2$ be two ordinary elliptic curves. It is well known that the group $Home(E_1,E_2)$ is a $\mathbb{Z}$ module of rank at most four. In the case where $E_1=E_2$, this module has rank ...

**3**

votes

**0**answers

94 views

### Equivalent definitions of the Hasse invariant

As probably many others before me, I got stuck in verifying all the nice properties of the Hasse invariant.
Let me start by recalling one definition:
Let $E\to S$ be an elliptic curve in ...

**4**

votes

**1**answer

125 views

### Congruence Primes and Modular Degrees

Let $\mathcal{S}=S_2(\Gamma_0(N) \cap \mathbf{Z} [[ q ]]$ be the set of cusp forms of weight $2$ on $\Gamma_0(N)$ with integral coefficients.
Let $f \in \mathcal{S}$ be a normalized newform, so it ...

**5**

votes

**0**answers

74 views

### Reference request: "effective'' semistable reduction

I am looking for the origin of the following idea: suppose $m$ and $n$ are relatively prime integers $\geq 3$. Let $E$ be an elliptic curve over a number field $K$. Let $L/K$ be a finite extension ...

**3**

votes

**0**answers

103 views

### Complex multiplication and ray class fields

This question is mainly referring to the proof of Theorem 5.6, Chapter 2 of Silverman's "Advanced Topics in the AEC". Basically, let $K$ be an imaginary quadratic field, and $E$ be an elliptic curve ...

**7**

votes

**0**answers

324 views

### Field of definition of a point in $[p]^{-1}E(K)$

Let $E$ be an ordinary elliptic curve defined over a non-perfect field $K$ of characteristic $p$. If $P \in E(K)$ satisfies $P \not\in [p]E(K)$, is it true that its $p^m$-division points of $P$ are ...

**2**

votes

**0**answers

43 views

### What is the complexity of finding a distortion map on a supersingular elliptic curve?

Let $E$ be a supersingular elliptic curve which is defined over $\mathbb{F}_q$ and $P\in E$. Then there exist a distortion map with respect to $P$. I am looking for an algorithm which finds the map ...

**1**

vote

**0**answers

40 views

### How can I find the specific endomorphism in a supersingular elliptic curve?

Let $E$ be a supersingular elliptic curve. As we know the endomorphism of $E$ is an order in a quaternion algebra. Suppose that $End(E)=\mathcal{O}$ and $a\in \mathcal{O}$. How can I find the ...

**21**

votes

**2**answers

500 views

### CM $j$-invariants in $p$-adic fields

I'm trying to understand the $p$-adic distribution of $j$-invariants for elliptic curves with complex multiplication.
Specifically, suppose $\sigma$ is some embedding $\sigma:\overline{\mathbb Q}\to ...

**0**

votes

**0**answers

67 views

### Elliptic curve : determine size of group $E/E_0$

My curve is given by E : $y^2 = x^3-3267x+45630$. Bad primes are 2,3,17. I want to find the size of group $E/E_0$. I know that $E_0(Q_2)$ are points on $E(Q_2)$ that do not reduce to a singular point. ...

**8**

votes

**0**answers

136 views

### Elliptic curves and the $\ell$-adic image of the decomposition group

Let $E$ be an elliptic curve over $\mathbb{Q}_\ell$ and consider the image of the $\ell$-adic representation. Is there a description of this image similar to Serre's description of the image of the ...

**0**

votes

**1**answer

222 views

### On elliptic curves, $\sqrt{x^2-101y^2} ,\sqrt{x^2+101y^2}$, and their ilk

I. Elliptic curves
Given integers $a,b,m_k$. Let,
$$x^2+a = m_1u_1^2\\x^2+b = m_1u_2^2\tag1$$
If there is a rational point $x_i$, then the pair (after a transformation) is birationally ...

**3**

votes

**1**answer

69 views

### Finiteness of the $p$-primary subgroup of an elliptic curve over the cyclotomic $\mathbb{Z}_p$-extension

Let $E$ be an elliptic curve defined over a number field $F$ and $F_\infty$ be the cyclotomic $\mathbb{Z}_p$-extension of $F$. Is it true that the $p$-primary subgroup of $E$ over $F_\infty$ i.e. ...

**3**

votes

**1**answer

309 views

### Cohomology of elliptic curves

Assume $K$ is an imaginary quadratic extension of $\mathbb{Q}$, and $E$ an elliptic curve defined over $\mathbb{Q}$.
Let $p\neq l$ be primes in $\mathbb{Q}$ where $E$ has good reduction. Assume $p$ ...

**3**

votes

**1**answer

139 views

### Identifying the canonical principal polarization of a Jacobian

Let $X$ be a curve over an algebraically closed field $k$ (even over $k = \mathbb{C}$ if you want), let $J = Pic^0_{X/k}$ be its Jacobian, let $P \in X(k)$ be a point, and let $i \colon X ...

**5**

votes

**3**answers

420 views

### Extending rational Diophantine triples to sextuples

(This is a follow-up to a previous post.) A rational Diophantine $m$-tuple is a set of rationals {$a_1,a_2,\dots a_m$} such that (with $i\neq j$), all $a_i a_j+1$ is a square. Problem: Find a class of ...

**4**

votes

**1**answer

135 views

### Lifting of Frobenius on torsors over abelian varieties

This is related to my previous question Assume that $A$ is an abelian variety over a field $k$ of characteristic $p$, $\mathcal{L}$ is a line bundle on $A$. Assume that $A$ is ordinary and ...

**3**

votes

**1**answer

162 views

### Lifting of Frobenius on semi-abelian varieties

Let $A$ be a semi-abelian variety over a field $k$($char\, k=p$). Namely, there is an exact sequence of group schemes $$0\to T\to A\to B\to 0$$ where $T$ is a torus, $B$ an abelian variety. Assume ...

**8**

votes

**0**answers

140 views

### Hilbert scheme of projectively normal elliptic curves

Consider the Hilbert scheme of degree $n$, genus $1$ curves in $\mathbb P^{n-1}$. It contains the locus of smooth curves embedded by the complete linear system of a degree $n$ divisor. Let $X_n$ be ...

**6**

votes

**0**answers

121 views

### Is there a prime degree endomorphism on supersingular elliptic curves?

Let $E$ be a supersingular elliptic curve which is defined over field $\mathbb{F}_{p^2}$ and $l$ be a prime such that $gcd(l,p)=1$.
Is there an endomorphism $\phi\in End(E)$ such that $deg(\phi)=l$?
...

**12**

votes

**1**answer

332 views

### Weak Mordell-Weil over number fields

I have a question regarding the Mordell Weil theorem a number field $K$.
I read the proof of the Mordell Weil theorem in "rational points on elliptic curves" by Tate and Silverman.
They presented a ...

**1**

vote

**0**answers

64 views

### Complex Structure Moduli of Elliptic Fibrations

Given an elliptically fibered Calabi-Yau threefold in Weierstrass form I want to compute the number of complex structure moduli of the fibration.
I know how it is done for the generic Weierstrass ...

**0**

votes

**1**answer

132 views

### Find special elliptic curves from j-invariant

Let $E$ be the elliptic curve defined over $GF(p)$ and $j$ be j-invariant of $E$, where $p$ is a big prime number. Also suppose $l$ be small prime number (for example $l<5000$) and $\#E$ denote ...

**5**

votes

**0**answers

216 views

### An example of a coarse moduli space of elliptic curves

Let $\mathcal{S}$ be the moduli stack (over $\text{Spec }\mathbb{Z}$) of elliptic curves endowed with a trivialisation of its Hodge bundle.
Classically, we know that $\mathcal{S}\otimes ...

**0**

votes

**0**answers

66 views

### Algebraic operations with memory hardness properties

In cryptography, there are password hash functions like scrypt and argon2 for which the fastest known algorithms employ large ...

**6**

votes

**3**answers

246 views

### Uniform bounds on the number of integer points on a family of elliptic curves

Let $P(x,y)$ be a binary cubic polynomial with integer coefficient. Let $n$ be an integer. Suppose the (complex) curve $P(x,y)=n$ is nonsingular, so is an elliptic curve. Is there any bound on the ...

**1**

vote

**0**answers

341 views

### The Sato-Tate conjecture (Frobenius eigenvalues)

Another question crossed my mind.
In the statement of the Sato-Tate conjecture, one usually assumes that
the elliptic curve has no CM. But, I read the Morita's paper for the BSD in the CM
case and ...

**2**

votes

**2**answers

487 views

### Curves of higher genus

I saw the question:
Abelian varieties with CM
and though I know that there are rare CM elliptic curves, I wonder
what kind of curves with higher genus have the CM Jacobians?

**1**

vote

**1**answer

169 views

### Finding cyclic subgroups of points on elliptic curves for isogeny based cryptography

Isogeny based cryptography is one of the newest post-quantum cryptography. Hardness of this system is based on finding isogeny between two elliptic curves. Also this is a theorem:
Elliptic curves ...

**0**

votes

**0**answers

42 views

### lower bound for solve ECDLP

Let $P$ and $Q$ are two points of NIST elliptic curve $E$ (defined over $F_{2^m}$ with prime $m$) and $k$ is a private key such that $k.P=Q$. Suppose $\ell$ be the number of bits in $k$, and let $k_i$ ...

**3**

votes

**1**answer

715 views

### Abelian varieties with CM

In this site, I looked at a paper of Kazuma Morita claiming the BSD conjecture for the CM case
posted on his homepage (he made a mistake three years ago for full BSD).
But, I am interested in this ...

**3**

votes

**0**answers

282 views

### Lifting a real quadratic twist of an Elliptic Curve to the modular curve

Let $E$ be an elliptic curve of conductor $N\cdot p^2$ over $\mathbb{Q}$, defined by the equation
$$y^2=x^3+p^2b\cdot x + p^3\cdot c$$
and parametrized by a map
$$X_{0}(N\cdot {p}^{2})\rightarrow E$$
...

**17**

votes

**1**answer

485 views

### Why do the $2$-Selmer ranks of $y^2 = x^3 + p^3 $ and $y^2 = x^3 - p^3 $ agree?

I was playing around with sage, when I found that the ranks (over $\mathbf{Q}$) of the elliptic curves $y^2=x^3+p^3$ and $y^2=x^3-p^3 $ almost always agree, for $p$ prime. The first few exceptions ...

**30**

votes

**2**answers

1k views

### Estimating the size of solutions of a diophantine equation

A. Is there natural numbers $a,b,c$ such that $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b}$ is equal to an odd natural number ?
(I do not know any such numbers).
B. Suppose that $\frac{a}{b+c} + ...

**5**

votes

**1**answer

179 views

### Is there $t\in\operatorname{Gal}(\overline{K}/K)$ s.t. $\operatorname{rank}_{\mathbf{Z}_p}((t-1)E_{p^\infty}(\overline{K}))=1$?

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, let $$K:=\varinjlim_{k\in\mathbb{Q}[\mu_{p^\infty}]} \mathbb{Q}\left[\mu_{p^\infty},k^{1/p^\infty}\right]$$
and ...

**5**

votes

**1**answer

281 views

### Division by $n$ in elliptic curves

Let $E/\mathbb F_{p^m}$ be an arbitrary elliptic curve over the Galois field $\mathbb F_{p^m}$, and let $$[n]^{-1}(P)\cap E(\mathbb F_{p^m})=\{Q\in E(\mathbb F_{p^m})\mid nQ=P\}.$$ Also let ...

**5**

votes

**1**answer

524 views

### Elliptic curves and prime numbers

Let $p_n$ be the $n^{th}$ prime number. Suppose $E(F_{p_n})$ denotes an elliptic curve over the Galois field $GF(p_n)$ which is defined by $y^2=x^3+ax+b$. Is the below claim true?
For each integer ...

**18**

votes

**2**answers

862 views

### elliptic curves and group cohomology

Recently, I've been trying to understand Jacob Lurie's 2-equivariant elliptic cohomology a bit better than I had in the past.
From what I can tell, the fragment of the story that only deals with ...

**8**

votes

**2**answers

403 views

### What do we know about the structure of $J_{0}(N)$ over $\mathbb{Q}[{\mu}_{{p}^{\infty}},{{k}}^{\frac{{1}}{{p}^{n}}}])$?

What is known about the structure of $J_{0}(N)$ over $\mathbb{Q}[\mu_{p^{\infty}}]$?
More generally, what do we know about $J_{0}(N)$ over
$\mathbb{Q}[\mu_{p^{\infty}},k^{1/p^{n}}]$, where ...

**19**

votes

**0**answers

212 views

### “High-concept” explanation for proof of a theorem of Ochanine?

See Akhil Mathew's notes on Ochanine's theorem for elliptic genera here and here.
Let $\phi: \Omega_{SO} \to \Lambda$ be a genus. We might ask when $\phi$ satisfies the following multiplicative ...

**10**

votes

**1**answer

416 views

### Galois representations for the curve $y^{2} = x^{3} - x$

Let $E / \mathbb{Q}$ be the elliptic curve given by $y^{2} = x^{3} - x$. I would like to know explicitly what the field of all $2$-power torsion looks like, as well as the image in ...

**6**

votes

**1**answer

245 views

### Why are some solutions of these diophantine equations off the usual patterns?

This is inspired by a recent question about complete multipartite integral graphs. I am wondering if more can be said about tripartite integral graphs with block sizes $a<b<c$. It is easy to see ...

**7**

votes

**1**answer

297 views

### What are some open problems regarding elliptic curves in finite fields?

I accept that my question seems so vague and broad, and I already looked into some similar questions in MO. But I would like to learn specifically about some open problems and conjectures regarding ...

**4**

votes

**0**answers

121 views

### Ranks of elliptic curves over Q(t)

I have an elliptic curve $E/\mathbb{Q}(t)$, and I want to compute its rank. Does knowing the rank over $\mathbb{F}_p(t)$ for some prime of good reduction give a bound on the rank over ...

**5**

votes

**1**answer

368 views

### The Weil numbers and modulus of an elliptic curve

I have an ignorant question about elliptic curves which I'll be slightly imprecise about. If I have an elliptic curve $X$ defined over $\mathbb Z$, I can base change to $\mathbb C$, and then ...

**0**

votes

**1**answer

161 views

### Rational maps between elliptic curves [closed]

I am studying Silverman's "The Arithmetic of Elliptic Curves" and I got the following question:
In the first chapters he defines rational between projective varieties (see the first definition in ...