author={A. I. Vijaya Shankar and Anders S{\"o}dergren and Nicolas Templier},
  journal={Forum of Mathematics, Sigma},
We study various families of Artin $L$ -functions attached to geometric parametrizations of number fields. In each case we find the Sato–Tate measure of the family and determine the symmetry type of the distribution of the low-lying zeros. 

On the asymptotics of cubic fields ordered by general invariants

In this article, we introduce a class of invariants of cubic fields termed “generalized discriminants”. We then obtain asymptotics for the families of cubic fields ordered by these invariants. In

Central values of zeta functions of non-Galois cubic fields

The Dedekind zeta functions of infinitely many non-Galois cubic fields have negative central values.


In this paper, we study lower-order terms of the one-level density of low-lying zeros of quadratic Hecke L-functions in the Gaussian field. Assuming the generalized Riemann hypothesis, our result

An improvement on Schmidt’s bound on the number of number fields of bounded discriminant and small degree

Abstract We prove an improvement on Schmidt’s upper bound on the number of number fields of degree n and absolute discriminant less than X for $6\leq n\leq 94$ . We carry this out by improving and

Omega results for cubic field counts via lower-order terms in the one-level density

In this paper, we obtain a precise formula for the one-level density of L-functions attached to non-Galois cubic Dedekind zeta functions. We find a secondary term which is unique to this context,

Hilbert spaces and low-lying zeros of L-functions

Lower Order Terms for Expected Value of Traces of Frobenius of a Family of Cyclic Covers of $\mathbb{P}^1_{\mathbb{F}_q}$ and One-Level Densities

We consider the expected value of $\mbox{Tr}(\Theta_C^n)$ where $C$ runs over a thin family of $r$-cyclic covers of $\mathbb{P}^1_{\mathbb{F}_q}$ for any $r$. We obtain many lower order terms

Improved error estimates for the Davenport-Heilbronn theorems

We improve the error terms in the Davenport–Heilbronn theorems on counting cubic fields to O(X). This improves on separate and independent results of the authors and Shankar and Tsimerman [BST13,

Low-lying zeros of quadratic Dirichlet $L$ -functions: lower order terms for extended support

We study the $1$ -level density of low-lying zeros of Dirichlet $L$ -functions attached to real primitive characters of conductor at most $X$ . Under the generalized Riemann hypothesis, we give an

Non-vanishing of class group L-functions for number fields with a small regulator

Let $E/\mathbb {Q}$ be a number field of degree $n$. We show that if $\operatorname {Reg}(E)\ll _n |\!\operatorname{Disc}(E)|^{1/4}$ then the fraction of class group characters for which the Hecke



Families of L -Functions and Their Symmetry

A few years ago the first-named author proposed a working definition of a family of automorphic L-functions. Then the work by the second and third-named authors on the Sato–Tate equidistribution for

Low-lying zeros of elliptic curve L-functions: Beyond the Ratios Conjecture

Abstract We study the low-lying zeros of L-functions attached to quadratic twists of a given elliptic curve E defined over $\mathbb{Q}$. We are primarily interested in the family of all twists

Prehomogeneous vector spaces and field extensions

SummaryLetk be an infinite field of characteristic not equal to 2, 3, 5. In this paper, we construct a natural map from the set of orbits of certain prehomogeneous vector spaces to the set of

Enumerating Quartic Dihedral Extensions of ℚ

We give an explicit Dirichlet series for the generating function of the discriminants of quartic dihedral extensions of ℚ. From this series we deduce an asymptotic formula for the number of

Secondary terms in counting functions for cubic fields

We prove the existence of secondary terms of order X^{5/6} in the Davenport-Heilbronn theorems on cubic fields and 3-torsion in class groups of quadratic fields. For cubic fields this confirms a

Geometry-of-numbers methods over global fields I: Prehomogeneous vector spaces

We develop geometry-of-numbers methods to count orbits in prehomogeneous vector spaces having bounded invariants over any global field. As our primary example, we apply these techniques to determine,

Counting dihedral and quaternionic extensions

We give asymptotic formulas for the number of biquadratic extensions of ℚ that admit a quadratic extension which is a Galois extension of ℚ with a prescribed Galois group, for example, with a Galois

Rational Points on Weighted projective Spaces

In this paper,we count the rational points on the weighted projective spaces defined over number fields w.r.t. ``size''. An asymptotic formula which generalizes the result of Schanuel's ``Heights in

The number of ramified primes in number fields of small degree

In this paper we investigate the distribution of the number of primes which ramify in number fields of degree d <= 5. In analogy with the classical Erdos-Kac theorem, we prove for S_d-extensions that

Equality of Polynomial and Field Discriminants

An appendix by the second author gives a conjecture concerning when the discriminant of an irreducible monic integral polynomial is square-free and some computational evidence for it.