Deriving from an Assumption a semistable elliptic Curve for \(x_{0}^{k} + y_{0}^{l} = z_{0}^{m}\)
Robert J. Betts
备注：
This paper has been withdrawn by the author. There were at three skeptical responses from others by implication, with two from qualified reviewers. Also if revision could have improved the presentation, no suggestions were made to the author as to how to revise the argument
With a simple transformation of the three exponents the generalized Fermat equation can be put into the same form as the Fermat equation. When it is rewritten into this new altered form any real solutions to the altered equation equal a subset of real solutions to the Fermat equation. This result, along with a paradigm previously established by G. Frey and Y. Hellegouarch, can be used to show that the generalized Fermat equation has no solutions in coprime integers. In fact we prove one can derive an elliptic curve over $\mathbb{Q}$ for primes (p \geq 3), from the assumption that (x_{0}^{k} + y_{0}^{l} = z_{0}^{m}), (k, l, m \geq 3), has coprime solutions in integers. The curve then can be used to show the unsolvability of this equation. Our aim is not to arrive at very deep results. Rather the goal in this paper is to apply prior results obtained by G. Frey, K. Ribet and A. Wiles, to show the lack of solutions to the generalized Fermat equation.
If P is an algebraic point on a commutative group scheme A/K, then P is _almost_rational_ if no two non-trivial Galois conjugates sigma(P), tau(P), have sum equal to 2P. In this paper, we classify almost rational torsion points on semistable elliptic curves over Q.
Given an elliptic curve $E$ in Legendre form $y^2 = x(x - 1)(x - λ)$ over the fraction field of a Henselian ring $R$ of mixed characteristic $(0, 2)$, we present an algorithm for determining a semistable model of $E$ over $R$ which depends only on the valuation of $λ$. We provide several examples along with an easy corollary concerning $2$-torsion.
Let $K$ be a totally real field, and let $S$ be a finite set of non-archimedean places of $K$. It follows from the work of Merel, Momose and David that there is a constant $B_{K,S}$ so that if $E$ is an elliptic curve defined over $K$, semistable outside $S$, then for all $p>B_{K,S}$, the representation $\barρ_{E,p}$ is irreducible. We combine this with modularity and level lowering to show the existence of an effectively computable constant $C_{K,S}$, and an effectively computable set of elliptic curves over $K$ with CM $E_1,\dotsc,E_n$ such that the following holds. If $E$ is an elliptic cur…
Let $p$ be an odd prime number, and let $E$ be an elliptic curve defined over a number field $F'$ such that $E$ has semistable reduction at every prime of $F'$ above $p$ and is supersingular at at least one prime above $p$. Under appropriate hypotheses, we compute the Akashi series of the signed Selmer groups of $E$ over a $\mathbb{Z}_p^d$-extension over a finite extension $F$ of $F'$. As a by-product, we also compute the Euler characteristics of these Selmer groups.
Let $p$ be an odd prime. We attach appropriate signed Selmer groups to an elliptic curve $E$, where $E$ is assumed to have semistable reduction at all primes above $p$. We then compare the Iwasawa $λ$-invariants of these signed Selmer groups for two congruent elliptic curves over the cyclotomic $\mathbb{Z}_p$-extension in the spirit of Greenberg-Vatsal and B. D. Kim. As an application of our comparsion formula, we show that if the $p$-parity conjecture is true for one of the congruent elliptic curves, then it is also true for the other elliptic curve. In the midst of proving this latter result…
We prove new sharp asymptotic for counting the semistable elliptic curves with two marked Weierstrass points at $\infty$ and $0$ and also the cases where $0$ is a 2-torsion or a 3-torsion marked Weierstrass point over $\mathbb{F}_q(t)$ by the bounded height of discriminant $Δ(X)$. We consider the motivic probabilities over any basefield $K$ with $\text{char}(K) \neq 2,3$ of picking a nonsingular semistable elliptic surface over $\mathbb{P}^{1}$ with two marked Weierstrass sections at $\infty$ and $0$ such that marked Weierstrass section at $0$ is 2-torsion or 3-torsion. In the end, we formulat…
We study generalisations to totally real fields of methods originating with Wiles and Taylor-Wiles. In view of the results of Skinner-Wiles on elliptic curves with ordinary reduction, we focus here on the case of supersingular reduction. Combining these, we then obtain some partial results on the modularity problem for semistable elliptic curves, and end by giving some applications of our results, for example proving the modularity of all semistable elliptic curves over $\mathbb{Q}(\sqrt{2})$.
We prove a new sharp asymptotic with the lower order term of zeroth order on $\mathcal{Z}_{\mathbb{F}_q(t)}(\mathcal{B})$ for counting the semistable elliptic curves over $\mathbb{F}_q(t)$ by the bounded height of discriminant $Δ(X)$. The precise count is acquired by considering the moduli of nonsingular semistable elliptic fibrations over $\mathbb{P}^{1}$, also known as semistable elliptic surfaces, with $12n$ nodal singular fibers and a distinguished section. We establish a bijection of $K$-points between the moduli functor of semistable elliptic surfaces and the stack of morphisms $\mathcal…
Using a combination of several powerful modularity theorems and class field theory we derive a new modularity theorem for semistable elliptic curves over certain real abelian fields. We deduce that if $K$ is a real abelian field of conductor $n<100$, with $5 \nmid n$ and $n \ne 29$, $87$, $89$, then every semistable elliptic curve $E$ over $K$ is modular. Let $\ell$, $m$, $p$ be prime, with $\ell$, $m \ge 5$ and $p \ge 3$.To a putative non-trivial primitive solution of the generalized Fermat $x^{2\ell}+y^{2m}=z^p$ we associate a Frey elliptic curve defined over $\mathbb{Q}(ζ_p)^+$, and study i…
Under suitable, fairly weak hypotheses on an elliptic curve $E/\mathbb{Q}$ and a primitive non-trivial Dirichlet character $χ$, we show that the algebraic $L$-value $\mathscr{L}(E,χ)$ at $s=1$ is an algebraic integer. For instance, for semistable curves $\mathscr{L}(E,χ)$ is integral whenever $E$ admits no isogenies defined over $\mathbb{Q}$. Moreover we give examples illustrating that our hypotheses are necessary for integrality to hold.
Let $\mathcal{M}$ be a pure motive over $\mathbb{Q}$ of odd weight $w\geq 3$, even rank $d\geq 2$, and global conductor $N$ whose $L$-function $L(s,\mathcal{M})$ coincides with the $L$-function of a self-dual algebraic tempered cuspidal symplectic representation of $\mathrm{GL}_{d}(\mathbb{A}_{\mathbb{Q}})$. We show that a certain polynomial which generates special values of $L(s,\mathcal{M})$ (including all of the critical values) has all of its zeros equidistributed on the unit circle, provided that $N$ or $w$ are sufficiently large with respect to $d$. These special values have arithmetic s…
In 1972, Serre showed that the adelic Galois representation associated to a non-CM elliptic curve over a number field has open image in GL_2(\hat{Z}). In Greicius' thesis, he develops necessary and sufficient criteria for determining when this representation is actually surjective and exhibits such an example. However, verifying these criteria turns out to be difficult in practice; Greicius describes tests for them that apply only to semistable elliptic curves over a specific class of cubic number fields. In this paper, we extend Greicius' methods in several directions. First, we consider the …
Fix $p$ an odd prime. Let $E$ be an elliptic curve over $\mathbb{Q}$ with semistable reduction at $p$. We show that the adjoint $p$-adic $L$-function of $E$ evaluated at infinitely many integers prime to $p$ completely determines up to a quadratic twist the isogeny class of $E$. To do this, we prove a result on the determination of isobaric representations of $\text{GL}(3, \mathbb{A}_\mathbb{Q})$ by certain $L$-values of $p$-power twists.
We prove local results on the $p$-adic density of elliptic curves over $\mathbb{Q}_p$ with different reduction types, together with global results on densities of elliptic curves over $\mathbb{Q}$ with specified reduction types at one or more (including infinitely many) primes. These global results include: the density of integral Weierstrass equations which are minimal models of semistable elliptic curves over $\mathbb{Q}$ (that is, elliptic curves with square-free conductor) is $1/ζ(2)\approx60.79\%$, the same as the density of square-free integers; the density of semistable elliptic curves …
Let E be an elliptic curve over a number field K which admits a cyclic p-isogeny with p odd and semistable at primes above p. We determine the root number and the parity of the p-Selmer rank for E/K, in particular confirming the parity conjecture for such curves. We prove the analogous results for p=2 under the additional assumption that E is not supersingular at primes above 2.
We construct an integral model for Lubin-Tate curves as moduli of finite subgroups of formal deformations over complete Noetherian local rings. They are p-adic completions of the modular curves X_0(p) at a mod-p supersingular point. Our model is semistable in the sense that the only singularities of its special fiber are normal crossings. Given this model, we obtain a uniform presentation for the Dyer-Lashof algebra of Morava E-theories at height 2 as local moduli of power operations in elliptic cohomology.
Let $E$ be an elliptic curve defined over $\mathbb Q$. Let $Γ$ be a subgroup of $E(\mathbb Q)$ and $P\in E(\mathbb Q)$. In [1], it was proved that if $E$ has no nontrivial rational torsion points, then $P\inΓ$ if and only if $P\in Γ$ mod $p$ for finitely many primes $p$. In this note, assuming the General Riemann Hypothesis, we provide an explicit upper bound on these primes when $E$ does not have complex multiplication and either $E$ is a semistable curve or $E$ has no exceptional prime.
Given two semistable, non potentially isotrivial elliptic surfaces over a curve $C$ defined over a field of characteristic zero or finitely generated over its prime field, we show that any compatible family of effective isometries of the N{é}ron-Severi lattices of the base changed elliptic surfaces for all finite separable maps $B\to C$ arises from an isomorphism of the elliptic surfaces. Without the effectivity hypothesis, we show that the two elliptic surfaces are isomorphic. We also determine the group of universal automorphisms of a semistable elliptic surface. In particular, this includes…
Let $E/\mathbb{Q}$ be a semistable elliptic curve such that $\mathrm{ord}_{s=1}L(E,s) = 1$. We prove the $p$-part of the Birch and Swinnerton-Dyer formula for $E/\mathbb{Q}$ for each prime $p \geq 5$ of good reduction such that $E[p]$ is irreducible: $$ \mathrm{ord}_p \left (\frac{L'(E,1)}{Ω_E\cdot\mathrm{Reg}(E/\mathbb{Q})} \right ) = \mathrm{ord}_p \left (\#\mathrm{Sha}(E/\mathbb{Q})\prod_{\ell\leq \infty} c_\ell(E/\mathbb{Q}) \right ). $$ This formula also holds for $p=3$ provided $a_p(E)=0$ if $E$ has supersingular reduction at $p$.
In a recent preprint, F. Calegari has shown that for $\ell = 2, 3, 5$ and 7 there exist 2-dimensional surjective representations $ρ$ of $\Gal(\bar{\Q}/\Q)$ with values in $\F_\ell$ coming from the $\ell$-torsion points of an elliptic curve defined over $\Q$, but not minimally, i.e., so that any elliptic curve giving rise to $ρ$ has prime-to-$\ell$ conductor greater than the (prime-to-$\ell$) conductor of $ρ$. In this brief note, we will show that the same is true for any prime $\ell >7$, concretely, we will show that for any such $\ell$ the elliptic curve $$E^\ell: \qquad Y^2 = X (X- 3^\ell ) …
It is known, that for every elliptic curve over Q there exists a quadratic extension in which the rank does not go up. For a large class of elliptic curves, the same is known with the rank replaced by the 2-Selmer group. We show, however, that there exists a large supply of semistable elliptic curves E/Q whose 2-Selmer group goes up in every bi-quadratic extension and for any odd prime p, the p-Selmer group goes up in every D_{2p}-extension and every elementary abelian p-extension of rank at least 2. We provide a simple criterion for an elliptic curve over an arbitrary number field to exhibit …
Let $E$ be a rational elliptic curve and let $p$ be an odd prime of additive reduction. Let $K$ be an imaginary quadratic field and fix a positive integer $c$ prime to the conductor of $E$. The main goal of the present article is to define an anticyclotomic $p$-adic $L$-function $Ł$ attached to $E/K$ when $E/\QQ_p$ attains semistable reduction over an abelian extension. We prove that $Ł$ satisfies the expected interpolation properties; namely, we show that if $χ$ is an anticyclotomic character of conductor $cp^n$ then $χ(Ł)$ is equal (up to explicit constants) to $L(E,χ,1)$ or $L'(E,χ,1)$.
We describe a simple, but effective, method for deriving families of elliptic curves, with high rank, all of whose members have the same torsion subgroup structure.
Let $n>1$ be an integer such that $X_{0}\!\left( n\right) $ has genus $0$, and let $K$ be a field of characteristic $0$ or relatively prime to $6n$. In this article, we explicitly classify the isogeny graphs of all rational elliptic curves that admit a non-trivial isogeny over $\mathbb{Q}$. We achieve this by introducing $56$ parameterized families of elliptic curves $\mathcal{C}_{n,i}(t,d)$ defined over $K(t,d)$, which have the following two properties for a fixed $n$: the elliptic curves $\mathcal{C}_{n,i}(t,d)$ are isogenous over $K(t,d)$, and there are integers $k_{1}$ and $k_{2}$ such tha…
We show that the minimum $h_{\text{min}}$ of the stable Faltings height on elliptic curves found by Deligne is followed by a gap. This means that there is a constant $C >0$ such that for every elliptic curve $E/K$ with everywhere semistable reduction over a number field $K$, we either have $h(E/K)=h_{\text{min}}$ or $h(E/K)\geq h_{\text{min}} +C$. We determine such an absolute constant explicitly. On the contrary, we show that there is no such gap for elliptic curves with unstable reduction.
For an elliptic curve $E$ over a number field $K$, one consequence of the Birch and Swinnerton-Dyer conjecture is the parity conjecture: the global root number matches the parity of the Mordell-Weil rank. Assuming finiteness of $\mathrm{Sha}(E/K)[p^\infty]$ for a prime $p$ this is equivalent to the $p$-parity conjecture: the global root number matches the parity of the $\mathbb{Z}_p$-corank of the $p^\infty$-Selmer group. We complete the proof of the $p$-parity conjecture for elliptic curves that have a $p$-isogeny for $p > 3$ (the cases $p \le 3$ were known). T. and V. Dokchitser have showed …
Let $E/\mathbf{Q}$ be a semistable elliptic curve of analytic rank one, and let $p>3$ be a prime for which $E[p]$ is irreducible. In this note, following a slight modification of the methods of Jetchev-Skinner-Wan, we use Iwasawa theory to establish the $p$-part of the Birch and Swinnerton-Dyer formula for $E$. In particular, we extend the main result of loc.cit. to primes of multiplicative reduction.
In this paper we show that the map % $$\partial:CH^2(E_1 \times E_2,1)\otimes \Q \longrightarrow PCH^1(\XX_v)$$ % is surjective, where $E_1$ and $E_2$ are two non-isogenous semistable elliptic curves over a local field, $CH^2(E_1 \times E_2,1)$ is one of Bloch's higher Chow groups and $PCH^1(\XX_v)$ is a certain subquotient of a Chow group of the special fibre $\XX_{v}$ of a semi-stable model $\XX$ of $E_1 \times E_2$. On one hand, this can be viewed as a non-Archimedean analogue of the Hodge-$\D$-conjecture of Beilinson - which is known to be true in this case by the work of Chen and Lewis \c…
We introduce the twisted $\boldsymbolμ_4$-normal form for elliptic curves, deriving in particular addition algorithms with complexity $9\mathbf{M} + 2\mathbf{S}$ and doubling algorithms with complexity $2\mathbf{M} + 5\mathbf{S} + 2\mathbf{m}$ over a binary field. Every ordinary elliptic curve over a finite field of characteristic 2 is isomorphic to one in this family. This improvement to the addition algorithm, applicable to a larger class of curves, is comparable to the $7\mathbf{M} + 2\mathbf{S}$ achieved for the $\boldsymbolμ_4$-normal form, and replaces the previously best known complexit…
