0
votes
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 ...
8
votes
1answer
437 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 ...
0
votes
1answer
184 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 ...
1
vote
1answer
223 views

Embedding of an elliptic curve into $\mathbb{P}^2 \times \mathbb{P}^2$

Let $E$ be a smooth elliptic curve over a field $k$. Let $$ i : E \to \mathbb{P}^2 \times \mathbb{P}^2, $$ be an embedding. How one can find an explicit canonical forms of equations cutting $E$ in ...
0
votes
0answers
118 views

Parabolic bundles on elliptic curves

as a warm up for his thesis I would like a student of mine to read something on parabolic bundles. He is reading the famous Atiyah paper on vector bundles on elliptic curves, so I think it would be ...
6
votes
1answer
392 views

What did Shimura say about $y^2 + y = x^3 - x$?

From the introduction of Ribet-Stein: Shimura showed that if we start with the elliptic curve $E$ defined by the equation $y^2 +y = x^3 −x^2$ then for “most” $n$ the image of $\rho$ is all of ...
2
votes
0answers
134 views

Finite Heisenberg groups action on cohomology of line bundles

Let $E$ be a smooth elliptic curve over algebraically closed field $k$ of characteristic zero, $\mathcal{L}$ is a line bundle over $E$, $\operatorname{deg}(\mathcal{L})=n \geq 1$. Then I define the ...
0
votes
1answer
114 views

Does the modified Szpiro conjecture require minimal model?

The modified Szpiro conjecture is described in Wikipedia and here and here. The modified Szpiro conjecture states that: given $\varepsilon > 0$, there exists a constant $C(\varepsilon)$ such ...
2
votes
1answer
143 views

picard group of moduli of elliptic r-prym curves

Let $\overline{\mathcal{M}}_{1,1}$ be the DM compactification of the moduli stack of elliptic curves. Its Picard group is $\mathbb{Z}$. Let us now consider stack of $r$-prym curves ...
4
votes
0answers
186 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 ...
3
votes
0answers
257 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). ...
8
votes
0answers
284 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
1answer
390 views

Best results regarding the Lang-Trotter conjecture

Let $E$ be an elliptic curve over $\mathbb Q$, which does not have complex multiplication over the algebraic closure of $\mathbb Q$. For $x>0$, let $P(x)$ be the number of primes $p < x$ such ...
12
votes
0answers
248 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 ...
18
votes
1answer
2k views

Modern proof of Serre's open image theorem?

Let $E$ be an elliptic curve defined over a number field $K$ without complex multiplication. Serre's open image theorem (which appears in his book 'Abelian $l$-Adic Representations and Elliptic ...
4
votes
2answers
399 views

What does this quotient of the upper half plane parametrize?

Let $G(N)$ be the congruence subgroup $\big\{ \begin{pmatrix} a&b \\ c&d \end{pmatrix} \in SL_2(\mathbb{Z}) \ \ | \ \ a \equiv d \mod N \textrm{ and } b \equiv c\equiv 0 \mod N \big\}$. ...
4
votes
2answers
355 views

Intersection of Hilbert class fields of imaginary quadratic fields

In this question Hilbert class field of Quadratic fields it is mentioned that if $d\equiv 1 \mod 4$ then the Hilbert class field of $\mathbb{Q}(\sqrt{-d})$ contains $\mathbb{Q}(i,\sqrt{d})$. Could ...
5
votes
1answer
555 views

Must the $j$-invariant of an elliptic curve with an isogeny be integral?

Let $K$ be a quadratic field, and $E/K$ a non-CM elliptic curve with a $K$-rational $p$-isogeny, for $p$ a prime. I would like to say the following: For large enough $p$, the $j$-invariant $j(E)$ ...
4
votes
3answers
545 views

Effective way of finding generators on the curve and the rank conjecture

Hello everyone, I have never heard of a polynomial time running algorithm that finds the generators of elliptic curves efficiently. I do know that Nagell-Lutz theorem is useful in computing the ...
4
votes
1answer
234 views

Reference for Rank Distribution Conjecture.

I am currently writing my master's thesis and I was wondering if the rank distribution conjecture was ever formally written down. Recall that it says that: Half of all elliptic curves have rank $0$, ...
9
votes
3answers
2k views

A recommended roadmap to Fermat's Last Theorem

I was inspired to undertake math as a career after watching a documentary on the proof of Fermat's Last Theorem. As such it's been a small goal of mine to understand Wiles et al's proof. In a ...
4
votes
2answers
468 views

Tamagawa Number of Elliptic Curves over $\mathbb{Q}$

I am currently reading a paper by De Weger and one theorem in it proves a bound for the Tamagawa number of any elliptic curve defined over $\mathbb{Q}$. I was wondering if anyone has any good ...
1
vote
1answer
361 views

Elliptic subfields of a function field

Let $C$ be a curve and $K(C)$ be its function field of genus 2, where $K$ = $\mathbb{C}$. The number of essential elliptic subfields of $K(C)$ is 0 or 2 or $\infty$. Edit: I am looking for a proof. ...
10
votes
1answer
598 views

Finiteness of Tate-Shafarevich

Does anyone happen to know who conjectured the finiteness of the Tate-Shafarevich group? We recall the conjecture. Let $E/K$ be an elliptic curve where $K$ is a number field. Then $Ш(E/K)$ is finite. ...
5
votes
1answer
299 views

Modular curve parametrizing two cyclic subgroups of an elliptic curve

The aim of this question is to better understand the following moduli space/modular curve, for which I propose (temporarily) the name $Y_0(M,N)$. We define $Y_0(M,N)$ as the moduli space parametrizing ...
2
votes
1answer
439 views

Generalization of singular moduli

$j$-invariants of CM curves $E$ over (say) the complex numbers are known as singular moduli. As the theory of complex multiplication explains, singular moduli are algebraic integers of great ...
2
votes
0answers
272 views

Moduli space of points of fixed order N on elliptic curves

Let us consider the moduli space $Y_1(N)$, parametrizing an elliptic curve, together with a choice of a point of N-torsion on it. Over the complex numbers, it is easy to see that this moduli space is ...
3
votes
2answers
361 views

What's known about complete split primes in Q(E[p])?

Let E be an elliptic curve over $\mathbf{Q}$, and p be a prime of good reduction for E such that the Galois representation $\bar\rho_p$ of $\mathbf{Q}$ on the p-torsion of E surjects onto Aut(E[p]). ...
2
votes
1answer
499 views

Serre's open image theorem for products of elliptic curves over function fields via specialization

In Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15, 259--331 (1972), Serre proved the following (Theorem 6 ′′, p. 325): Let $K$ be a number field and let ...
6
votes
1answer
486 views

Beilinson's height pairing vs Neron-Tate

In the literature there are several different definitions of what is often referred to as Beilinson's height pairing (see for example section 4.3.8 of Gillet and Soulé's paper Arithmetic intersection ...
8
votes
3answers
1k views

Elliptic Curves over Global Function Fields

I am currently thinking about a problem, and I feel that by knowing more about elliptic curves over extensions of $\mathbb{F}_q(T)$, for $q$ a power of $p$ say, might lead to insight. I am also ...
16
votes
6answers
3k views

Algorithms for finding rational points on an elliptic curve?

I am looking for algorithms on how to find rational points on an elliptic curve $$y^2 = x^3 + a x + b$$ where $a$ and $b$ are integers. Any sort of ideas on how to proceed are welcome. For example, ...
2
votes
2answers
271 views

For which composite $N$ does $X_0(N)$ possess a non-cuspidal rational point?

According the the introduction to Mazur's Rational Isogenies of Prime Degree the following question was open in 1978: Let $N$ be one of the integers 39, 65, 91, 125, or 169. Does the modular ...
19
votes
0answers
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 ...
7
votes
3answers
1k views

Convergence of L-series

I remember to have read that the L-function of an elliptic curve, which a priori only converges for $\Re s > \frac{3}{2}$ also converges at $s=1$ provided that the $L$-function satisfies the ...
9
votes
3answers
1k views

What is the etymology for the term conductor?

This is related to the previous question of how to define a conductor of an elliptic curve or a Galois representation. What motivated the use of the word "conductor" in the first place? A friend ...
16
votes
3answers
2k views

Conceptual understanding of the Gross-Zagier theorem.

The Gross-Zagier paper "Heegner points and derivatives of $L$-series", is really computational and hard to plow through. It seems it is futile to read it as such and one must look for a more ...
6
votes
1answer
1k views

Reference for the `standard' Tate curve argument.

I'd like a reference (e.g. something published somewhere that I can cite in a paper) for the proof of the following: Let $E$ be an elliptic curve over $\mathbb Q$ with minimal discriminant ...