* uschian groups of the first kind * Automorphic forms and functions * Hecke operators and the zeta-functions associated with modular forms * Elliptic curves * Abelian extensions of imaginary quadratic fields and complex multiplication of elliptic curves * Modular functions of higher level * Zeta-functions of algebraic curves and abelian varieties * The cohomology group assoicated with cusp forms * Arithmetic Fuschian groups

Introduction to the arithmetic theory of automorphic functions

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Introduction To Cyclotomic Fields

One of the first concepts one meets in elementary number theory is that of the multiplicative order. We give a survey of the literature on this topic emphasizing the Artin primitive root conjecture (1927). The first part of the survey is intended for a rather general audience and rather colloquial, whereas the second part is intended for number theorists and ends with several open problems. The contributions in the survey on ‘elliptic Artin’ are due to Alina Cojocaru. Wojciec Gajda wrote a section on ‘Artin for K-theory of number fields’, and Hester Graves (together with me) on ‘Artin’s conjecture and Euclidean

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Artin's primitive root conjecture -a survey -

Using a multidimensional large sieve inequality, we obtain a bound for the mean-square error in the Chebotarev theorem for division fields of elliptic curves that is as strong as what is implied by the Generalized Riemann Hypothesis. As an application we prove that, according to height, almost all elliptic curves are Serre curves, where a Serre curve is an elliptic curve whose torsion subgroup, roughly speaking, has as much Galois symmetry as possible.

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Almost all elliptic curves are serre curves

Under suitable hypotheses, we construct a probability measure on the set of closed maximal isotropic subspaces of a locally compact quadratic space over F_p. A random subspace chosen with respect to this measure is discrete with probability 1, and the dimension of its intersection with a fixed compact open maximal isotropic subspace is a certain nonnegative-integer-valued random variable.
We then prove that the p-Selmer group of an elliptic curve is naturally the intersection of a discrete maximal isotropic subspace with a compact open maximal isotropic subspace in a locally compact quadratic space over F_p. By modeling the first subspace as being random, we can explain the known phenomena regarding distribution of Selmer ranks, such as the theorems of Heath-Brown, Swinnerton-Dyer, and Kane for 2-Selmer groups in certain families of quadratic twists, and the average size of 2- and 3-Selmer groups as computed by Bhargava and Shankar. Our model is compatible with Delaunay's heuristics for p-torsion in Shafarevich-Tate groups, and predicts that the average rank of elliptic curves over a fixed number field is at most 1/2. Many of our results generalize to abelian varieties over global fields.

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Random maximal isotropic subspaces and Selmer groups

We compute the averages over elliptic curves of the constants occurring in the Lang–Trotter conjecture, the Koblitz conjecture, and the cyclicity conjecture. The results obtained confirm the consistency of these conjectures with the corresponding “theorems on average” obtained recently by various authors.

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Averages of elliptic curve constants

We compute the averages over elliptic curves of the constants occurring in the Lang-Trotter conjecture, the Koblitz conjecture, and the cyclicity conjecture. The results obtained confirm the consistency of these conjectures with the corresponding ``theorems on average'' obtained recently by various authors.

Let E be an elliptic curve over Q and let Q(E[n]) be its nth division field. In 1972, Serre showed that if E is without complex multiplication, then the Galois group of Q(E[n])/Q is as large as possible, that is, GL2(Z/nZ), for all integers n coprime to a constant integer m(E,Q) depending (at most) on E/Q. Serre also showed that the best one can hope for is to have |GL2(Z/nZ) : Gal(Q(E[n])/Q)| ! 2 for all positive integers n. We study the frequency of this optimal situation in a one-parameter family of elliptic curves over Q, and show that in essence, for almost all one-parameter families, almost all elliptic curves have this optimal behavior.

One-parameter families of elliptic curves over ℚ with maximal Galois representations

There is a modular curve X'(6) of level 6 defined over Q whose Q-rational points correspond to j-invariants of elliptic curves E over Q for which Q(E[2]) is a subfield of Q(E[3]). In this note we characterize the j-invariants of elliptic curves with this property by exhibiting an explicit model of X'(6). Our motivation is two-fold: on the one hand, X'(6) belongs to the list of modular curves which parametrize non-Serre curves (and is not well-known), and on the other hand, X'(6)(Q) gives an infinite family of examples of elliptic curves with non-abelian "entanglement fields," which is relevant to the systematic study of correction factors of various conjectural constants for elliptic curves over Q.

Nathan Jones

Papers

Almost all elliptic curves are serre curves

Averages of elliptic curve constants

Averages of elliptic curve constants

One-parameter families of elliptic curves over ℚ with maximal Galois representations

Elliptic curves with 2-torsion contained in the 3-torsion field