arXiv subjects
Publications and source records attributed to Tom Weston.
We give an algorithm to determine factorization types of primes in the number fields generated by a single point of odd order on an elliptic curve. We apply this to compute coefficients of the Dedekind zeta function of the field.
Let $K$ be a number field and $\ell \geq 5$ a prime number. Mazur and Rubin introduced the notion of diophantine stability for a variety $X_{/K}$ at a prime $\ell$. We show that there is a positive density set of elliptic curves $E_{/\mathbb{Q}}$ of rank $1$ such that $E_{/K}$ is diophantine stable at $\ell$. This has implications for Hilbert's Tenth Problem over $\mathscr{O}_K$. This problem asks whether there exists an algorithm that decides in finite time whether a finite system of Diophantine equations over $\mathscr{O}_K$ has a solution.
Let $K$ be an imaginary quadratic field and $p$ be an odd prime which splits in $K$. Let $E_1$ and $E_2$ be elliptic curves over $K$ such that the $Gal(\bar{K}/K)$-modules $E_1[p]$ and $E_2[p]$ are isomorphic. We show that under certain explicit additional conditions on $E_1$ and $E_2$, the anticyclotomic $\mathbb{Z}_p$-extension $K_{anti}$ of $K$ is integrally diophantine over $K$. When such conditions are satisfied, we deduce new cases of Hilbert's tenth problem. In greater detail, the conditions imply that Hilbert's tenth problem is unsolvable for all number fields that are contained in $K_{anti}$. We illustrate our results by constructing an explicit example for $p=3$ and $K=\mathbb{Q}(\sqrt{-5})$.
We prove that the Mazur-Tate elements of an eigenform $f$ sit inside the Fitting ideals of the corresponding dual Selmer groups along the cyclotomic $\mathbb Z_p$-extension (up to scaling by a single constant). Our method begins with the construction of local cohomology classes built via the $p$-adic local Langlands correspondence. From these classes, we build algebraic analogues of the Mazur-Tate elements which we directly verify sit in the appropriate Fitting ideals. Using Kato's Euler system and explicit reciprocity laws, we prove that these algebraic elements divide the corresponding Mazur-Tate elements, implying our theorem.
For a prime $p$ and a rational elliptic curve $E_{/\mathbb{Q}}$, set $K=\mathbb{Q}(E[p])$ to denote the torsion field generated by $E[p]:=\operatorname{ker}\{E\xrightarrow{p} E\}$. The class group $\operatorname{Cl}_K$ is a module over $\operatorname{Gal}(K/\mathbb{Q})$. Given a fixed odd prime number $p$, we study the average non-vanishing of certain Galois stable quotients of the mod-$p$ class group $\operatorname{Cl}_K/p\operatorname{Cl}_K$. Here, $E$ varies over rational elliptic curves, ordered according to \emph{height}. Our results are conditional and rely on predictions made by Delaunay and Poonen-Rains for the statistical variation of the $p$-primary parts of Tate-Shafarevich groups of elliptic curves. We also prove results in the case when the elliptic curve $E_{/\mathbb{Q}}$ is fixed and the prime $p$ is allowed to vary.
Given an elliptic curve $E$ defined over the rational numbers and a prime $p$ at which $E$ has good reduction, we consider the Galois deformation ring parametrizing lifts of the residual representation on the $p$-torsion group $E[p]$. For a fixed elliptic curve without complex multiplication, it is shown that these deformation rings are unobstructed for all but finitely many primes. For a fixed prime $p$ and varying elliptic curve $E$, we relate the problem to the question of how often $p$ does not divide the modular degree. Heuristics due to M.Watkins based on those of Cohen and Lenstra indicate that this proportion should be $\prod_{i\geq 1} \left(1-\frac{1}{p^i}\right)\approx 1-\frac{1}{p}-\frac{1}{p^2}$. This heuristic is supported by computations which indicate that most elliptic curves (satisfying further conditions) have smooth deformation rings at a given prime $p\geq 5$, and this proportion comes close to $100\%$ as $p$ gets larger.
We prove, for many cuspidal automorphic representations for GSp(4), that the local obstructions to the deformation theory of the associated residual Galois representations generically vanish.
We establish formulae for the Iwasawa invariants of Mazur--Tate elements of cuspidal eigenforms, generalizing known results in weight 2. Our first theorem deals with forms of "medium" weight, and our second deals with forms of small slope . We give examples illustrating the strange behavior which can occur in the high weight, high slope case.
We prove that, on average, elliptic curves over Q have finitely many primes p for which they possess a p-adic point of order p. We include a discussion of applications to companion forms and the deformation theory of Galois representations.
Let f be a modular form of weight 2 and trivial character. Fix also an imaginary quadratic field K. We use work of Bertolini-Darmon and Vatsal to study the mu-invariant of the p-adic Selmer group of f over the anticyclotomic Zp-extension of K. In particular, we verify the mu-part of the main conjecture in this context. The proof of this result is based on an analysis of congruences of modular forms, leading to a conjectural quantitative version of level-lowering (which we verify in the case that Mazur's principle applies).
Fix m >= 1 and let E be an elliptic curve over Q with complex multiplication. We formulate conjectures on the density of primes p (congruent to one modulo m) for which the pth Fourier coefficient of E is an mth power modulo p; often these densities differ from the naive expectation of 1/m. We also prove our conjectures for m dividing the number of roots of unity lying in the CM field of E; the most involved case is m=4 and complex multiplication by Q(i).
We prove a formula (analogous to that of Kida in classical Iwasawa theory and generalizing that of Hachimori-Matsuno for elliptic curves) giving the analytic and algebraic p-adic Iwasawa invariants of a modular eigenform over an abelian p-extension of Q to its p-adic Iwasawa invariants over Q. On the algebraic side our methods, which make use of congruences between modular forms, yield a Kida-type formula for a very general class of ordinary Galois representations. We are further able to deduce a Kida-type formula for elliptic curves at supersingular primes.
We study the behavior of Iwasawa invariants among ordinary deformations of a fixed residual Galois representation taking values in a reductive algebraic group G. In particular, under the assumption that these Selmer groups are cotorsion modules over the Iwasawa algebra, we show that the vanishing of the mu invariant is independent of the deformation, while the lambda invariant depends only on the ramification. This generalizes work of Greenberg-Vatsal and Emerton-Pollack and the author in the case G = GL(2).
Let r : G_Q -> GL_2(Fpbar) be a p-ordinary and p-distinguished irreducible residual modular Galois representation. We show that the vanishing of the algebraic or analytic Iwasawa mu-invariant of a single modular form lifting r implies the vanishing of the corresponding mu-invariant for all such forms. Assuming that the mu-invariant vanishes, we also give explicit formulas for the difference in the algebraic or analytic lambda-invariants of modular forms lifting r. In particular, our formula shows that the lambda-invariant is constant on branches of the Hida family of r. We further show that our formulas are identical for the algebraic and analytic invariants, so that the truth of the main conjecture of Iwasawa theory for one form in the Hida family of r implies it for the entire Hida family.
Let r : G_Q -> GL_n Q_l be a motivic l-adic Galois representation. For fixed m > 1 we initiate an investigation of the density of the set of primes p such that the trace of the image of an arithmetic Frobenius at p under r is an m^th power residue modulo p. Based on numerical investigations with modular forms we conjecture (with Ramakrishna) that this density equals 1/m whenever the image of r is open. We further conjecture that for such r the set of these primes p is independent of any set defined by Cebatorev-style Galois theoretic conditions (in an appropriate sense). We then compute these densities for certain m in the complementary case of modular forms of CM-type with rational Fourier coefficients; our proofs are a combination of the Cebatorev density theorem (which does apply in the CM case) and reciprocity laws applied to Hecke characters. We also discuss a potential application (suggested by Ramakrishna) to computing inertial degrees at p in abelian extensions of imaginary quadratic fields unramified away from p.
Let f be a newform of weight at least 3 with Fourier coefficients in a number field K. We show that the universal deformation ring of the mod lambda Galois representation associated to f is unobstructed, and thus isomorphic to a power series ring in three variables over the Witt vectors, for all but finitely many primes lambda of K. We give an explicit bound on such lambda for the six known cusp forms of level 1, trivial character, and rational Fourier coefficients. We also prove a slightly weaker result for weight 2.
Let f be a newform of weight at least 2 and squarefree level with Fourier coefficients in a number field K. We give explicit bounds, depending on congruences of f with other newforms, on the set of primes lambda of K for which the deformation problem associated to the mod lambda Galois representation of f is obstructed. We include some explicit examples.
Let A be an abelian variety over a number field F with End(A/F) commutative. Let S be a subgroup of A(F) and let x be a point of A(F). Suppose that for almost all places v of F the reduction of x modulo v lies in the reduction of S modulo v. In this paper we prove that x must then lie in S + A(F)_tors. This provides a partial answer to a generalization (due to W. Gajda) of the support problem of Erdos.