Number Theory

We show that the reduction of a projective endomorphism modulo a discrete valuation naturally takes the form of a set-theoretic correspondence. This raises the possibility of classifying "reduction types" of such dynamical systems, reminiscent of the additive/multiplicative dichotomy for elliptic curves. These correspondences facilitate the exact evaluation of certain integrals of dynamical Green's functions, which arise as local factors in the context of counting rational points ordered by the Call-Silverman canonical height. No prior knowledge of arithmetic dynamics will be assumed.

The Polya group of a number field K is a specific subgroup of the ideal class group Cl(K) of K, generated by all classes of Ostrowski ideals of K. In this talk, I will discuss the equality Po(K)=Cl(K) in two directions. First, we will see this equality happens for infinitely many "non-Galois fields'' K. Accordingly, I prove two conjectures presented by Chabert and Halberstadt concerning the Polya groups of some families of non-Galois fields. Then, I present some "finiteness theorems" for the equality Po(K)=Cl(K) for some families of "Galois" fields K obtained in joint work with Amir Akbary (University of Lethbridge).

The Polya group P o ( K ) of a Galois number field K coincides with the subgroup of the ideal class group C l ( K ) of K consisting of all strongly ambiguous ideal classes. We prove that there are only finitely many imaginary abelian number fields K whose "Polya index" [ C l ( K ) : P o ( K ) ] is a fixed integer. Accordingly, under GRH, we completely classify all imaginary quadratic fields with the Polya indices 1 and 2. Also, we unconditionally classify all imaginary biquadratic and imaginary tri-quadratic fields with the Polya index 1. In another direction, we classify all real quadratic fields K of extended R-D type (with possibly only one more field K ) for which P o ( K ) = C l ( K ) . Our result generalizes Kazuhiro's classification of all real quadratic fields of narrow R-D type whose narrow genus numbers are equal to their narrow class numbers.

This is a joint work with Amir Akbary (University of Lethbridge).

I will discuss recent work with Chantal David, Alexander Dunn, and Joshua Stucky, in which we prove that a positive proportion of Hecke L-functions associated to the cubic residue symbol modulo square-free Eisenstein integers do not vanish at the central point. Our principal new contribution is the asymptotic evaluation of the mollified second moment. No such asymptotic formula was previously known for a cubic family (even over function fields).

Our new approach makes crucial use of Patterson's evaluation of the Fourier coefficients of the cubic metaplectic theta function, Heath-Brown's cubic large sieve, and a Lindelöf-on-average upper bound for the second moment of cubic Dirichlet series that we establish. The significance of our result is that the family considered does not satisfy a perfectly orthogonal large sieve bound. This is quite unlike other families of Dirichlet L-functions for which unconditional results are known (namely the family of quadratic characters and the family of all Dirichlet characters modulo q). Consequently, our proof has fundamentally different features from the corresponding works of Soundararajan and of Iwaniec and Sarnak.

Let ord𝑝(𝑎)be the order of 𝑎in (ℤ/𝑝ℤ)∗. In 1927, Artin conjectured that the set of primes 𝑝for which an integer 𝑎≠−1,◻is a primitive root (i.e. ord𝑝(𝑎)=𝑝−1) has a positive asymptotic density among all primes. In 1967 Hooley proved this conjecture assuming the Generalized Riemann Hypothesis (GRH). In this talk, we will study the behaviour of ord𝑝(𝑎)as 𝑝varies over primes. In particular, we will show, under GRH, that the set of primes 𝑝for which ord𝑝(𝑎)is “𝑘prime factors away” from 𝑝−1− 1 has a positive asymptotic density among all primes, except for particular values of 𝑎and 𝑘. We will interpret being “𝑘prime factors away” in three different ways:
𝑘=𝜔(𝑝−1ord𝑝(𝑎)),𝑘=Ω(𝑝−1ord𝑝(𝑎)),𝑘=𝜔(𝑝−1)−𝜔(ord𝑝(𝑎)).

We will present conditional results analogous to Hooley’s in all three cases and for all integer 𝑘. From this, we will derive conditionally the expectation for these quantities.

Furthermore, we will provide partial unconditional answers to some of these questions.

This is joint work with Leo Goldmakher and Greg Martin.