**35**

votes

**0**answers

691 views

### The exponent of Ш of y^2 = x^3 + px, where p is a Fermat prime

For $d$ a non-zero integer, let $E_d$ be the elliptic curve
$$
E_d : y^2 = x^3+dx.
$$
When we let $d$ be $p = 2^{2^k}+1$, for $k \in \{1,2,3,4\}$, sage tells us that, conditionally on BSD,
$$
\# ...

**34**

votes

**0**answers

2k views

### Constructing non-torsion rational points (over Q) on elliptic curves of rank > 1.

Consider an elliptic curve E defined over Q. Assume that the rank of E(Q) is >=2. (Assume the Birch-Swinnerton-Dyer conjecture if needed, so that analytic rank = algebraic rank.) How do you construct ...

**20**

votes

**0**answers

1k views

### What are the possible singular fibers of an elliptic fibration over a higher dimensional base?

An elliptic fibration is a proper morphism $Y\rightarrow B$ between varieties such that the fiber over a general point of the base $B$ is a smooth curve of genus one.
It is often required for the ...

**17**

votes

**0**answers

739 views

### Fake CM elliptic curves

Suppose one has an elliptic curve $E$ over $\mathbb{Q}$ with conductor $N < k^3$ for some (large) positive $k$, with the property that its Fourier coefficients satisfy
$$
a_p=0, \; \mbox{ for all ...

**16**

votes

**0**answers

231 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 ...

**15**

votes

**0**answers

399 views

### Cohomological characterization of CM curves

In his 1976 classical Annals paper on $p$-adic interpolation, N. Katz uses the fact that if $E_{/K}$ is an elliptic curve with complex multiplications in the quadratic field $F$, up to a suitable ...

**12**

votes

**0**answers

274 views

### For how many primes does an elliptic curve over a totally imaginary field have supersingular reduction?

An elliptic curve over a finite field, $k$, of characteristic p is called supersingular if it has no $p$-torsion over $k^{\mathrm{alg}}$, or equivalently, if $\mathrm{End}(E)$ is an order in a ...

**12**

votes

**0**answers

1k views

### Why doesn't functoriality immediately imply the modularity theorem?

Let $E/\mathbb{Q}$ be an elliptic curve. By the modularity theorem, the prime indexed coefficients of its $L$-function agree with those of a weight $2$ cusp eigenform $f$ with integer coefficients. ...

**12**

votes

**0**answers

534 views

### Endomorphisms rings of elliptic curves and congruences of $j$

Let $p$ be a prime number, $K/\mathbf{Q}_p$ a finite extension, with integers $O_K$, valuation ideal $\mathfrak{p}$, and residue field $k_\mathfrak{p}$. Let $E$ be an elliptic curve over $K$ with good ...

**11**

votes

**0**answers

229 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 ...

**10**

votes

**0**answers

374 views

### divisibility of Tamagawa numbers

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$. Let $p\ge11$ be a prime of good ordinary reduction for $E$ and assume that $p$ does not divide the degree of a minimal modular parametrization ...

**9**

votes

**0**answers

235 views

### Preimage of torsion points under modular parametrizations

Let $E$ be an elliptic curve defined over $\mathbf{Q}$. It is known that $E$ admits a modular parametrization $\phi : X_0(N) \to E$, where $N$ is the conductor of $E$.
We may normalize $\phi$ such ...

**9**

votes

**0**answers

487 views

### Isogenies between supersingular elliptic curves

Suppose we are given two non-isomorphic supersingular elliptic curves $C$ and $C'$ (in characteristic $p$). Is there an isogeny $C\to C'$ of a given degree (say, power of a prime $l$ different from ...

**8**

votes

**0**answers

289 views

### A property of supersingular $j$-invariants (reference request)

Edit 2: For those who understandably don't want to read such a long post, I think Voloch's suggestion reduces the problem to asking whether $j$-invariants of supersingular curves are 3rd powers in ...

**8**

votes

**0**answers

753 views

### Torsion points of CM elliptic curves

Let $K$ be an imaginary quadratic field, and $\mathfrak{f}$ an integral ideal of $K$ which is stable under complex conjugation. Assume that $(1 + \mathfrak{f} ) \cap \mathcal{O}_K^\times = \{1\}$.
...

**7**

votes

**0**answers

245 views

### Modular interpretation of Ramanujan theta operator?

I'm a beginner to the theory of modular forms trying to understand a certain construction from the point of view of elliptic curves. Let $f(q) = \sum a_n q^n$ be a formal power series. Define $\theta ...

**7**

votes

**0**answers

312 views

### Tameness criterion in the reducible case

Dear MO,
This is a follow up to a previous question here in MO, but I will make this question self-contained for convenience. Those already familiar with the following paper [G] by Gross can safely ...

**7**

votes

**0**answers

588 views

### Semistable Elliptic Curves and irreducible Galois representations

I am interested in the set of number fields $K$ having the property that for \emph{any semistable} elliptic curve $E$ defined over $K$, there exists a constant $c(E,K)>0$ such that
...

**7**

votes

**0**answers

335 views

### Binary quadratic forms attached to supersingular elliptic curves over F_p?

The question that I have is a more precise version of an earlier one (1), posted by myself on MO a little bit ago. Sorry for repeating myself.
Let $p$ be prime number which for simplicity shall be ...

**6**

votes

**0**answers

215 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.
...

**6**

votes

**0**answers

295 views

### elliptic curves over function fields

Let $K$ be a number field, $E$ an elliptic curve over $K$, and $p$ an odd prime. If $v$ is a place of $K$, we know by the Kummer injection that $E(K_v)/pE(K_v) \hookrightarrow H^1 (K_v, E[p])$ for ...

**6**

votes

**0**answers

213 views

### Hilbert space from the Tate pairing

Fix an elliptic curve $E$ over ${\Bbb Q}$ (or if you prefer, something more general over something more general). For each extension $F$ of ${\Bbb Q}$, the Néron-Tate height pairing gives
an inner ...

**6**

votes

**0**answers

427 views

### Constructing large rank elliptic curves by multiplying quadratic imaginaries by cubes so that all have same imaginary part

I've been thinking of the relation between elliptic curves of large rank and quadratic imaginary fields with large 3-rank class groups. There are quite a few papers constructing infinitely many ...

**6**

votes

**0**answers

637 views

### Can you get Siegel's theorem “for free” from modularity and Mazur's Eisenstein Ideal paper?

There is a well-known theorem of Shafarevich that given a finite set $S$ of primes the number of isomorphism classes of elliptic curves over $\Bbb Q$ with everywhere good reduction outside $S$ is ...

**5**

votes

**0**answers

241 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 ...

**5**

votes

**0**answers

153 views

### When are Galois representations with open image attached to elliptic curves?

Let $K$ be a number field with absolute Galois group $G_K$.
Let $\rho:G_K \rightarrow GL_2(\hat{\mathbb{Z}})$ be a Galois representation such that the image of $\rho$ is open in ...

**5**

votes

**0**answers

214 views

### Isogenous elliptic curves have same conductor

Let $E/K, E'/K$ be elliptic curves defined over a number field $K$. Let $\phi: E \to E'$ be a non-constant isogeny defined over $K$. Why must the conductors be equal?
I know that this is an ...

**5**

votes

**0**answers

472 views

### Elliptic curve with no points in a number field

The following is probably well-known (I'd appreciate a link):
for a field $K$ that is a finite extension of the field of rational numbers, give a polynomial $f(x,y) ∈ Q[x,y]$ of the form $y^2 − x^3 ...

**5**

votes

**0**answers

386 views

### Can the Galois representation on the $p$-adic Tate module of $E/\mathbf{Q}_p$ be recovered from the $p$-divisible group associated to the mod $p$ good reduction of $E$?

Let $p$ be a prime number, and $E/\mathbf{Q}_p$ and elliptic curve with good reduction $\bar E_p$. Can the Galois representation of $\mathbf{Q}_p$ given by the $p$-adic Tate module of $E$ be recovered ...

**5**

votes

**0**answers

417 views

### Formal groups in the supersingular reduction case

Dear MO,
Let $E/\mathbb{Q}$ be an elliptic curve with potential good supersingular reduction at $p$. Thus, there is a finite extension $K/\mathbb{Q}_p$ such that $E/K$ has good supersingular ...

**5**

votes

**0**answers

653 views

### Elliptic curves over finite fields and 2x2 matrices

Let $k$ be a finite field of order $p^a$ and characteristic $p$, and $\mathcal{C}$ a $k$ isogeny class of elliptic curves over $k$. Let $w$ be the (Galois conjugacy class of the) Weil $k$-number ...

**5**

votes

**0**answers

585 views

### Curious propositon in “Les schemas de modules de courbes elliptiques”

Currently I am reading "Les schemas de modules de courbes elliptiques" by Deligne and Rapoport and I got myself seriously confused about the following proposition (in English translation):
(II ...

**4**

votes

**0**answers

229 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. ...

**4**

votes

**0**answers

88 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 ...

**4**

votes

**0**answers

297 views

### Explicit family of generalized elliptic curves with level n structure

Let $\pi:\mathcal{E}\rightarrow U$ be a family of elliptic curves with level $n$ structure (in the sense of Deligne-Rapoport) where $U\subseteq C$ is some (non-empty) Zariski open set of a smooth ...

**4**

votes

**0**answers

188 views

### Evaluation of $E_{\ell,2}$ on supersingular curves over $\mathbb{F}_{p^2}$

As mentioned in an answer to Modularity of $E_2$ on congruence subgroups, there exist modular forms $E_{\ell,2}$ of level $\Gamma_{0}(\ell)$ and weight 2, with $q$-expansion ...

**4**

votes

**0**answers

256 views

### Abelian cubic extensions of Q[i],

Recently I was considering cubic extensions $K/Q$ that have discriminant negative of a perfect square. Classifying these curves reduces to solving a Diophatine equation of the form $4a^3+27b^2=c^2$ ...

**4**

votes

**0**answers

284 views

### What is the status of the equidistribution root numbers of elliptic curves' L-functions

In Section 7 of Alice Silverberg's Rank "Cheat Sheet", Silverberg stated
The Bhargava Conjecture: For each $n >
> 1$ the average size of
$S_{n}(E/\mathbb{Q})$ is
...

**4**

votes

**0**answers

222 views

### A natural mod-n representation of the automorphisms of an elliptic curve

Let $E$ be an elliptic curve over a field of characteristic $p$, and let $n$ be an integer coprime to $p$. Then $E[n]$, the kernel of multiplication by $n$ on $E$, is (etale-locally) isomorphic to ...

**3**

votes

**0**answers

208 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 ...

**3**

votes

**0**answers

157 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 $\alpha = \left[\begin{matrix} 1&0 \\ 10&1 \end{matrix}\right] \in SL_2(\mathbb{Z})$. ...

**3**

votes

**0**answers

90 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 ...

**3**

votes

**0**answers

274 views

### Integer solutions of $ z^3 y^2 = x(x-1)(x+1)$

According to a conjecture there are no three
consecutive powerful numbers.
Necessary condition for this is integer solution of
$$ z^3 y^2 = x(x-1)(x+1) \qquad (1) $$
What are integer solutions ...

**3**

votes

**0**answers

119 views

### Elliptic curves and quasi-self-reciprocal polynomials

I am reading Shoichi Kihara's On the rank of the elliptic curve $y^2=x^3+k, II$ [Proc. Japan Acad. Ser. A Math. Sci. Volume 72, Number 10 (1996), 228-229] which is available here ...

**3**

votes

**0**answers

205 views

### Is it expected that every natural number is the rank of some elliptic curve over the rationals?

It is a well-known problem on the theory of elliptic curves that the rank of an elliptic curve (the number of generators of the free part of the Mordell group of the elliptic curve) cannot be ...

**3**

votes

**0**answers

286 views

### Birch/Swinnerton-Dyer “Notes on Elliptic Curves II”

I would like to know if any of you know if there is a more general treatment to what Birch and Swinnerton-Dyer did in "Notes on Elliptic Curves II" (http://www.ams.org/mathscinet-getitem?mr=179168).
...

**3**

votes

**0**answers

140 views

### Is Hasse-witt map isomorphism?

Fix a level $N \geq 3$ and denote by $\Gamma(N)\subset SL(2,\mathbb{Z})$ the subgroup of matrices which are congruent to the identity modulo $N$. The open modular curve $Y(N)$ corresponding to ...

**3**

votes

**0**answers

122 views

### P-adic Weierstrass Lemma for several variables

The p-adic Weiestrass lemma asserts that a power series $f(z)$ with coefficients in the ring of integers of a local field can be factored as $π^n·u(z)·p(z)$ where u(z) is a unit in the ring of power ...

**3**

votes

**0**answers

285 views

### Rational points and Tesla cards

I'm rapidly approaching 300,000 curves in my ongoing search for Mordell curves of rank >=8.
Currently I'm finding that I have a bottleneck in the code that locates rational points on these curves.
...

**3**

votes

**0**answers

102 views

### Extending cohomology classes to compactifications of Kuga varieties

I am trying to understand the proof of lemma 3 in the paper "Algebraic cycles and the Hodge structure of a Kuga fiber variety" by B. Brent Gordon,
available at ...