# Tagged Questions

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, abelian-...

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### 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 ...
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### Is the Modularity Theorem (currently) effective?

The Modularity Theorem says every elliptic curve over $\mathbb{Q}$ can be gotten from the classic modular curve $X_0(N)$ by a rational map. Here $N$ is the conductor, easily calculable from a ...
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If $E$ is a non-CM elliptic curve over $Q$, then it is a famous theorem of Serre that there is some integer $M(E)$ such that for any prime $\ell > M(E)$, the image of the Galois representations $\... 1answer 522 views ### Parametric Families for Large Torsion Subgroups The following are two facts about$\mathbb{Z}/9\mathbb{Z}$,$\mathbb{Z}/10\mathbb{Z}$,$\mathbb{Z}/12\mathbb{Z}$,$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/8\mathbb{Z}$. (a) According to Andrej ... 3answers 790 views ### Can the number of solutions$xy(x-y-1)=n$for$x,y,n \in Z[t]$be unbounded as n varies? We've recently seen this question: Can the number of solutions$ab(a+b+1)=n$for$a,b,n \in \mathbb{Z}$be unbounded as$n$varies? It appears initially plausible that the answer is yes, but evidently ... 2answers 345 views ### Root number of a quadratic twist of an elliptic curve Could someone provide a reference for the following fact which is stated without proof in section 4.3 of Alice Silverberg's survey "Open Questions in Arithmetic Algebraic Geometry"? Let E be an ... 3answers 2k views ### Re: Mordell's Equation$y^2 = x^3 + k$and Perfect Numbers I have already tried a somewhat exhaustive search of the literature, but couldn't find anything close to the problem that I am working on. My question is: When does Mordell's Equation $$y^2 = x^3 +... 0answers 541 views ### Automorphisms of the L-function associated to an elliptic \mathbb{Q}-curve Edited after Noam Elkies' comment: From what I understand (very few actually), there exist elliptic curves defined over some number fields \mathbb{K} Galois over \mathbb{Q} which are isogenous to ... 1answer 290 views ### Algebraic equations for modular parameterizations I was wondering if there some place where for some small N I can find explicit modular parameterizations in an algebraic way. One type of model for X_0(N) is just given by a single algebraic ... 1answer 1k views ### Which elliptic curves over totally real fields are modular these days? As the title says. In particular, every elliptic curve over \mathbb{Q} is modular; but what is the current state of the art for general totally real number fields? I assume the answer is ... 1answer 1k views ### Fermat's Bachet-Mordell Equation Fermat once claimed that the only integral solutions to y^2 = x^3 - 2 are (3, \pm 5). Fermat knew Bachet's duplication formulas (more precisely, Bachet had a formula for computing what we call -... 1answer 423 views ### Elements of arbitrary large order in the first Galois cohomology of an elliptic curve Let E be an elliptic curve over k=\mathbb{Q}. Consider H^1(k,E). In this answer Daniel Loughran writes: "I'm pretty sure that this cohomology group has elements of arbitrarily large order". I ... 1answer 348 views ### A reference for \mathbb{A}^1_R being a coarse moduli space of the stack of elliptic curves Let R be a ring and let \mathcal{M}_{1, R} be the algebraic R-stack of elliptic curves (over R-schemes as bases). One knows that the coarse moduli space of \mathcal{M}_{1, R} is supposed to ... 1answer 581 views ### how many consecutive integers x can make ax^2+bx+c square ? The following problem was raised in a Mathlinks thread: If a,b,c\in\mathbb Z such that a\ne0 and b^2-4ac\ne 0, for how many consecutive integers x can ax^2+bx+c ba a perfect square ? The ... 0answers 237 views ### Is the compositum of all quadratic extensions of the rationals an ample field? Let K be the compositum of all quadratic extensions of \mathbb{Q}, that is$$K = \mathbb{Q}(\sqrt{d} \ : \ d \in \mathbb{Q}).$$Is there a (geometrically irreducible) smooth variety V/\mathbb{... 1answer 169 views ### Is an elliptic curve that is isomorphic to its Frobenius conjugate defined over \mathbb{F}_p? Let p be prime and q = p^n. Let E be an elliptic curve over \mathbb{F}_q, and let E^{(p)} be the pullback of E by the p-power Frobenius of \mathbb{F}_q. If E is isomorphic (over \... 1answer 317 views ### Modular polynomials for elliptic curves point counting The Schoof-Elkies-Atkin (SEA) algorithm (for counting points on elliptic curves over a finite field) performs computations over polynomials modulo some modular polynomials. Originally the "classical" ... 1answer 739 views ### Abelian varieties with CM In this site, I looked at a paper of Kazuma Morita claiming the BSD conjecture for the CM case posted on his homepage (he made a mistake three years ago for full BSD). But, I am interested in this ... 1answer 248 views ### The existence of elliptic curves with prescribed supersingular primes For a given infinite set of primes, not too big, eg, satisfying Lang-Trotter conjecture, can we always find an E.C. with supersingular reduction (at least) at these primes? How about E.C. without CM? 0answers 258 views ### Does the property (P) holds true for the derivatives of L? Let$$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve C over ℚ. As s takes on real negative values, there are trivial zeros ... 2answers 862 views ### Legitimacy of reducing mod p a complex multiplication action of an elliptic curve? I scoured Silverman's two books on arithmetic of elliptic curves to find an answer to the following question, and did not find an answer: Given an elliptic curve E defined over H, a number field, ... 2answers 578 views ### Is there a presentation of the cohomology of the moduli stack of torsion sheaves on an elliptic curve? Let E be your favorite elliptic curve, and let Tor^m be the moduli stack of torsion sheaves of degree m on E. This sounds horrible, but it's not so bad; it's a global quotient of a smooth ... 2answers 264 views ### Argument for unboundedness of integral points of elliptic curves over number fields Probably this is well known to those who know it. Got an argument and numerical support that over number fields elliptic curves in minimal models might have unbounded number of integral points, the ... 1answer 142 views ### Lifting of Frobenius on torsors over abelian varieties This is related to my previous question Assume that A is an abelian variety over a field k of characteristic p, \mathcal{L} is a line bundle on A. Assume that A is ordinary and \mathcal{L}... 1answer 171 views ### Lifting of Frobenius on semi-abelian varieties Let A be a semi-abelian variety over a field k(char\, k=p). Namely, there is an exact sequence of group schemes$$0\to T\to A\to B\to 0$$where$T$is a torus,$B$an abelian variety. Assume ... 1answer 333 views ### a question on CM elliptic curves Let$E$be an elliptic curve over$\overline{\mathbb{Q}}$defined by an equation$y^2=4x^3-g_2x-g_3$and let$\omega=\int_\gamma \frac{dx}{y}$be the integral of the regular differential form$\...
Is there an elliptic curve $E/\mathbb{Q}$ such that $E(K)$ is trivial for every finite extension $K/\mathbb{Q}$ with discriminant a power of $7$ ?
Hopefully this is better than what I asked yesterday and Milton solved. Let $C : F(x,y,z)=0$ be a projective genus $1$ curve over $\mathbb{Q}$ with no restriction on the degree. Write a point \$P = (...