About

Alina Bucur received her Ph.D. in Mathematics from Brown University in 2006. She has held a postdoctoral fellowship at the Institute for Advanced Study and served as a Moore Instructorship at MIT. Her research is in analytic number theory, specifically focusing on multiple Dirichlet series and moments of L-functions.

Research topics

• Mathematics
• Pure mathematics
• Mathematical analysis
• Combinatorics
• Discrete mathematics
• Statistics
• Geometry

Selected publications

• Counting number fields using multiple Dirichlet series

  Open MIND · 2026-02-27

  preprintSenior author

  We provide a method for counting number fields of fixed Galois group ordered by arbitrary inertial invariants using analytic techniques from the study of multiple Dirichlet series. We prove unconditional results for infinitely many new (concentrated and semiconcentrated) groups that were not approachable by previous methods. Conditional on subconvexity bounds bounds for certain Dirichlet series (e.g. the generalized Lindelöf hypothesis), we use these techniques to prove the existence of an asymptotic growth rate for $G$-extensions for infinitely many new groups $G$ for which the minimum index elements of $G$ are contained in a union of proper abelian normal subgroups. In particular, our conditional results include all groups with nilpotency class $2$. Additionally, when $G$ is nilpotent our results give a power saving error term.

  DOI

• Counting number fields using multiple Dirichlet series

  arXiv (Cornell University) · 2026-02-27

  articleOpen accessSenior author

  We provide a method for counting number fields of fixed Galois group ordered by arbitrary inertial invariants using analytic techniques from the study of multiple Dirichlet series. We prove unconditional results for infinitely many new (concentrated and semiconcentrated) groups that were not approachable by previous methods. Conditional on subconvexity bounds bounds for certain Dirichlet series (e.g. the generalized Lindelöf hypothesis), we use these techniques to prove the existence of an asymptotic growth rate for $G$-extensions for infinitely many new groups $G$ for which the minimum index elements of $G$ are contained in a union of proper abelian normal subgroups. In particular, our conditional results include all groups with nilpotency class $2$. Additionally, when $G$ is nilpotent our results give a power saving error term.

  PublisherOA PDF

• Power-Saving Error Terms for the Number of $$D_4$$-Quartic Extensions over a Number Field Ordered by Discriminant

  Association for Women in Mathematics series · 2024-01-01

  book-chapter1st authorCorresponding
  PublisherDOI

• Effective Sato–Tate conjecture for abelian varieties and applications

  Journal of the European Mathematical Society · 2024 · 2 citations

  1st authorCorresponding
  • Mathematics
  • Pure mathematics

  From the generalized Riemann hypothesis for motivic L -functions, we derive an effective version of the Sato–Tate conjecture for an abelian variety A defined over a number field k with connected Sato–Tate group. By effective we mean that we give an upper bound on the error term in the count predicted by the Sato–Tate measure that only depends on certain invariants of A . We discuss three applications of this conditional result. First, for an abelian variety defined over k , we consider a variant of Linnik’s problem for abelian varieties that asks for an upper bound on the least norm of a prime whose normalized Frobenius trace lies in a given interval. Second, for an elliptic curve defined over k with complex multiplication, we determine (up to multiplication by a nonzero constant) the asymptotic number of primes whose Frobenius traces attain the integral part of the Hasse–Weil bound. Third, for a pair of abelian varieties A and A' defined over k with no common factors up to k -isogeny, we find an upper bound on the least norm of a prime at which the respective Frobenius traces of A and A' have opposite sign.

  PublisherOA PDFDOI

• Frobenius sign separation for abelian varieties

  arXiv (Cornell University) · 2023-10-16

  preprintOpen access1st authorCorresponding

  Let A and A' be nonzero abelian varieties defined over a number field k such that Hom(A,A')=0. Under the Generalized Riemann hypothesis for motivic L-functions attached to A and A', we show that there exists a prime p of k of good reduction for A and A' at which the Frobenius traces of A and A' are nonzero and differ by sign, and such that the norm of p is O_{k,g,g'}(log(2NN')^2), where N and N' respectively denote the absolute conductors of A and A'. We also make the dependence of the big-O constant on k and the dimensions g,g' of A,A' explicit up to an effectively computable absolute constant. Our method extends that of Chen, Park, and Swaminathan who considered the case in which A and A' are elliptic curves.

  PublisherOA PDFDOI

• Power-saving error terms for the number of $D_4$-quartic extensions over a number field ordered by discriminant

  arXiv (Cornell University) · 2022-09-27

  preprintOpen access1st authorCorresponding

  We study the asymptotic count of dihedral quartic extensions over a fixed number field with bounded norm of the relative discriminant. The main term of this count (including a summation formula for the constant) can be found in the literature (see Cohen--Diaz y Diaz--Olivier for the statement without proof and see Klüners for a proof), but a power-saving for the error term has not been explicitly determined except in the case that the base field is $\mathbb{Q}$. In this article, we describe the argument for obtaining both the explicit main term and a power-saving error term for the number of $D_4$-quartic extensions over a general base number field ordered by the norms of their relative discriminants. We also give an extensive overview of the history and development of number field asymptotics.

  PublisherOA PDFDOI

• Geometric generalizations of the square sieve, with an application to cyclic covers

  Mathematika · 2022 · 2 citations

  1st authorCorresponding
  • Mathematics
  • Combinatorics
  • Pure mathematics

  Abstract We formulate a general problem: Given projective schemes and over a global field K and a K ‐morphism η from to of finite degree, how many points in of height at most B have a pre‐image under η in ? This problem is inspired by a well‐known conjecture of Serre on quantitative upper bounds for the number of points of bounded height on an irreducible projective variety defined over a number field. We give a nontrivial answer to the general problem when and is a prime degree cyclic cover of . Our tool is a new geometric sieve, which generalizes the polynomial sieve to a geometric setting over global function fields.

  PublisherDOI

• Geometric generalizations of the square sieve, with an application to cyclic covers

  arXiv (Cornell University) · 2021-09-23

  preprintOpen access1st authorCorresponding

  We formulate a general problem: given projective schemes $\mathbb{Y}$ and $\mathbb{X}$ over a global field $K$ and a $K$-morphism $η$ from $\mathbb{Y}$ to $\mathbb{X}$ of finite degree, how many points in $\mathbb{X}(K)$ of height at most $B$ have a pre-image under $η$ in $\mathbb{Y}(K)$? This problem is inspired by a well-known conjecture of Serre on quantitative upper bounds for the number of points of bounded height on an irreducible projective variety defined over a number field. We give a non-trivial answer to the general problem when $K=\mathbb{F}_q(T)$ and $\mathbb{Y}$ is a prime degree cyclic cover of $\mathbb{X}=\mathbb{P}_{K}^n$. Our tool is a new geometric sieve, which generalizes the polynomial sieve to a geometric setting over global function fields.

  PublisherOA PDFDOI

• Effective Sato-Tate conjecture for abelian varieties and applications

  arXiv (Cornell University) · 2020

  1st authorCorresponding
  • Mathematics
  • Combinatorics
  • Pure mathematics

  From the generalized Riemann hypothesis for motivic L-functions, we derive an effective version of the Sato-Tate conjecture for an abelian variety A defined over a number field k with connected Sato-Tate group. By effective we mean that we give an upper bound on the error term in the count predicted by the Sato-Tate measure that only depends on certain invariants of A. We discuss three applications of this conditional result. First, for an abelian variety defined over k, we consider a variant of Linnik's problem for abelian varieties that asks for an upper bound on the least norm of a prime whose normalized Frobenius trace lies in a given interval. Second, for an elliptic curve defined over k with complex multiplication, we determine (up to multiplication by a nonzero constant) the asymptotic number of primes whose Frobenius trace attain the integral part of the Hasse-Weil bound. Third, for a pair of abelian varieties defined over k with no common factors up to k-isogeny, we find an upper bound on the least norm of a prime at which the respective Frobenius traces have opposite sign.

  PublisherOA PDFDOI

• Analytic Methods in Arithmetic Geometry

  Contemporary mathematics - American Mathematical Society · 2019-11-22 · 5 citations

  book
  PublisherDOI

Recent grants

• Arithmetic Statistics: Geometry and Analysis

  NSF · $165k · 2020–2025

Frequent coauthors

• Chantal David

  17 shared

• Brooke Feigon

  13 shared

• Matilde Laĺın

  12 shared

• Kiran S. Kedlaya

  7 shared

• Melanie Matchett Wood

  Harvard University

  4 shared

• Nathan O. Kaplan

  3 shared

• Edgar Costa

  3 shared

• Alina Carmen Cojocaru

  3 shared

Awards & honors

• FRG Research Grant by NSF
• Hellman Fellowship