arXiv subjects

David Zywina

Publications and source records attributed to David Zywina.

35 records · Page 2Linked to original sources

arXiv2015

On the possible images of the mod ell representations associated to elliptic curves over Q

Consider a non-CM elliptic curve $E$ defined over $\mathbb{Q}$. For each prime $\ell$, there is a representation $ρ_{E,\ell}: G \to GL_2(\mathbb{F}_\ell)$ that describes the Galois action on the $\ell$-torsion points of $E$, where $G$ is the absolute Galois group of $\mathbb{Q}$. A famous theorem of Serre says that $ρ_{E,\ell}$ is surjective for all large enough $\ell$. We will describe all known, and conjecturally all, pairs $(E,\ell)$ such that $ρ_{E,\ell}$ is not surjective. Together with another paper, this produces an algorithm that given an elliptic curve $E/\mathbb{Q}$, outputs the set of such exceptional primes $\ell$ and describes all the groups $ρ_{E,\ell}(G)$ up to conjugacy. Much of the paper is dedicated to computing various modular curves of genus $0$ with their morphisms to the $j$-line.

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arXiv2015

Bounds for the Lang-Trotter conjectures

For a non-CM elliptic curve $E$ defined over the rationals, Lang and Trotter made very deep conjectures concerning the number of primes $p\leq x$ for which $a_p(E)$ is a fixed integer (and for which the Frobenius field at $p$ is a fixed imaginary quadratic field). Under GRH, we use a smoothed version of the Chebotarev density theorem to improve the best known Lang-Trotter upper bounds of Murty, Murty and Saradha, and Cojocaru and David.

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arXiv2015

Modular forms and some cases of the inverse Galois problem

We prove new cases of the inverse Galois problem by considering the residual Galois representations arising from a fixed newform. Specific choices of weight $3$ newforms will show that there are Galois extensions of $\mathbb{Q}$ with Galois group $PSL_2(\mathbb{F}_p)$ for all primes $p$ and $PSL_2(\mathbb{F}_{p^3})$ for all odd primes $p \equiv \pm 2, \pm 3, \pm 4, \pm 6 \pmod{13}$.

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arXiv2014

The inverse Galois problem for orthogonal groups

We prove many new cases of the Inverse Galois Problem for those simple groups arising from orthogonal groups over finite fields. For example, we show that the finite simple groups Omega_{2n+1}(p) and POmega_{4n}^+(p) both occur as the Galois group of a Galois extension of the rationals for all integers n>1 and all primes p>3. We obtain our representations by studying families of twists of elliptic curves and using some known cases of the Birch and Swinnerton-Dyer conjecture along with a big monodromy result of Hall.

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arXiv2013

The inverse Galois problem for PSL_2(F_p)

We show that the simple group PSL_2(F_p) occurs as the Galois group of an extension of the rationals for all primes p>3. We obtain our Galois extensions by studying the Galois action on the second etale cohomology groups of a specific elliptic surface.

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arXiv2011

The splitting of reductions of an abelian variety

Consider an absolutely simple abelian variety A defined over a number field K. For most places v of K, we study how the reduction A_v of A modulo v splits up to isogeny. Assuming the Mumford-Tate conjecture for A and possibly increasing K, we will show that A_v is isogenous to the m-th power of an absolutely simple abelian variety for all places v of K away from a set of density 0, where m is an integer depending only on the endomorphism ring End(A_Kbar). This proves many cases, and supplies justification, for a conjecture of Murty and Patankar. Under the same assumptions, we will also describe the Galois extension of Q generated by the Weil numbers of A_v for most v.

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arXiv2011

Drinfeld modules with maximal Galois action on their torsion points

To each Drinfeld module over a finitely generated field with generic characteristic, one can associate a Galois representation arising from the Galois action on its torsion points. Recent work of Pink and Rütsche has described the image of this representation up to commensurability. Their theorem is qualitative, and the objective of this paper is to complement this theory with a worked out example. In particular, we give examples of Drinfeld modules of rank 2 for which the Galois action on their torsion points is as large as possible. We will follow the approach that Serre used to give explicit examples of his openness theorem for elliptic curves. Using our specific examples, we will numerically test analogues of some well-known elliptic curve conjectures.

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arXiv2011

The Sato-Tate law for Drinfeld modules

We prove an analogue of the Sato-Tate conjecture for Drinfeld modules. Using ideas of Drinfeld, J.-K. Yu showed that Drinfeld modules satisfy some Sato-Tate law, but did not describe the actual law. More precisely, for a Drinfeld module ϕdefined over a field L, he constructs a continuous representation ρ_\infty : W_L \to D^* of the Weil group of L into a certain division algebra, which encodes the Sato-Tate law. When the Drinfeld module has generic characteristic and L is finitely generated, we shall describe the image of this representation up to commensurability. As an application, we give improved upper bounds for the Drinfeld module analogue of the Lang-Trotter conjecture.

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arXiv2011

Explicit Class Field Theory for global function fields

Let F be a global function field and let F^ab be its maximal abelian extension. Following an approach of D.Hayes, we shall construct a continuous homomorphism ρ: Gal(F^ab/F) \to C_F, where C_F is the idele class group of F. Using class field theory, we shall show that our ρis an isomorphism of topological groups whose inverse is the Artin map of F. As a consequence of the construction of ρ, we obtain an explicit description of F^ab. Fix a place \infty of F, and let A be the subring of F consisting of those elements which are regular away from \infty. We construct ρby combining the Galois action on the torsion points of a suitable Drinfeld A-module with an associated \infty-adic representation studied by J.-K. Yu.

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arXiv2011

Bounds for Serre's open image theorem

Let E be an elliptic curve over the rationals without complex multiplication. The absolute Galois group of Q acts on the group of torsion points of E, and this action can be expressed in terms of a Galois representation rho_E:Gal(Qbar/Q) \to GL_2(Zhat). A renowned theorem of Serre says that the image of rho_E is open, and hence has finite index, in GL_2(Zhat). We give the first general bounds of this index in terms of basic invariants of E. For example, the index can be bounded by a polynomial function of the logarithmic height of the j-invariant of E. As an application of our bounds, we settle an open question on the average of constants arising from the Lang-Trotter conjecture.

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arXiv2010

Abelian varieties over large algebraic fields with infinite torsion

Let A be an abelian variety of positive dimension defined over a number field K and let Kbar be a fixed algebraic closure of K. For each element sigma of the absolute Galois group Gal(Kbar/K), let Kbar(sigma) be the fixed field of sigma in Kbar. We shall prove that the torsion subgroup of A(Kbar(sigma)) is infinite for all sigma in Gal(Kbar/K) outside of some set of Haar measure zero. This proves the number field case of a conjecture of Geyer and Jarden from 1978.

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arXiv2010

Hilbert's irreducibility theorem and the larger sieve

We describe an explicit version of Hilbert's irreducibility theorem using a generalization of Gallagher's larger sieve. We give applications to the Galois theory of random polynomials, and to the images of the adelic representation associated to elliptic curves varying in rational families.

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arXiv2010

The Chebotarev invariant of a finite group

We consider invariants of a finite group related to the number of random (independent, uniformly distributed) conjugacy classes which are required to generate it. These invariants are intuitively related to problems of Galois theory. We find group-theoretic expressions for them and investigate their values both theoretically and numerically.

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arXiv2009

A refinement of Koblitz's conjecture

Let E be an elliptic curve over the number field Q. In 1988, Koblitz conjectured an asymptotic for the number of primes p for which the cardinality of the group of F_p-points of E is prime. However, the constant occurring in his asymptotic does not take into account that the distributions of the |E(F_p)| need not be independent modulo distinct primes. We shall describe a corrected constant. We also take the opportunity to extend the scope of the original conjecture to ask how often |E(F_p)|/t is prime for a fixed positive integer t, and to consider elliptic curves over arbitrary number fields. Several worked out examples are provided to supply numerical evidence for the new conjecture.

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arXiv2008

The Large Sieve and Galois Representations

We describe a generalization of the large sieve to situations where the underlying groups are nonabelian, and give several applications to the arithmetic of abelian varieties. In our applications, we sieve the set of primes via the system of representations arising from the Galois action on the torsion points of an abelian variety. The resulting upper bounds require explicit character sum calculations, with stronger results holding if one assumes the Generalized Riemann Hypothesis.

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arXiv2008

Elliptic curves with maximal Galois action on their torsion points

Given an elliptic curve E over a number field k, the Galois action on the torsion points of E induces a Galois representation, ρ_E : Gal(\bar{k}/k) \to GL_2(\hat{Z}). For a fixed number field k, we describe the image of ρ_E for a "random" elliptic curve E over k. In particular, if k\neq Q is linearly disjoint from the cyclotomic extension of Q, then ρ_E will be surjective for "most" elliptic curves over k.

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arXiv2008

Arithmetic E_8 lattices with maximal Galois action

We construct explicit examples of E_8 lattices occurring in arithmetic for which the natural Galois action is equal to the full group of automorphisms of the lattice, i.e., the Weyl group of E_8. In particular, we give explicit elliptic curves over Q(t) whose Mordell-Weil lattices are isomorphic to E_8 and have maximal Galois action. Our main objects of study are del Pezzo surfaces of degree 1 over number fields. The geometric Picard group, considered as a lattice via the negative of the intersection pairing, contains a sublattice isomorphic to E_8. We construct examples of such surfaces for which the action of Galois on the geometric Picard group is maximal.

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