About

Manjul Bhargava is a professor at Princeton University in the Department of Mathematics. The provided page text does not include specific details about his research focus, background, or key contributions.

Research topics

• Mathematics
• Combinatorics
• Artificial Intelligence
• Computer Science
• Mathematical analysis
• Algorithm
• Pure mathematics
• Discrete mathematics
• Geometry

Selected publications

• The existence of infinitely many cubic fields with class group of exact 2-rank 1

  Open MIND · 2026-02-06

  preprint1st authorCorresponding

  We show that infinitely many cubic fields have class group of 2-rank 1.

  DOI

• The existence of infinitely many cubic fields with class group of exact 2-rank 1

  arXiv (Cornell University) · 2026-02-06

  articleOpen access1st authorCorresponding

  We show that infinitely many cubic fields have class group of 2-rank 1.

  PublisherOA PDF

• Geometry-of-numbers methods over global fields II: Coregular representations

  arXiv (Cornell University) · 2026-04-18

  preprintOpen access1st authorCorresponding

  We develop geometry-of-numbers methods to count orbits in coregular vector spaces having bounded invariants over any global field. We apply these techniques to bound the average ranks and determine average Selmer group sizes of elliptic curves and Jacobians of hyperelliptic curves over any base global field $F$ of characteristic not $2$, $3$ or $5$.

  PublisherDOI

• Geometry-of-numbers methods over global fields II: Coregular representations

  arXiv (Cornell University) · 2026-04-18

  articleOpen access1st authorCorresponding

  We develop geometry-of-numbers methods to count orbits in coregular vector spaces having bounded invariants over any global field. We apply these techniques to bound the average ranks and determine average Selmer group sizes of elliptic curves and Jacobians of hyperelliptic curves over any base global field $F$ of characteristic not $2$, $3$ or $5$.

  PublisherOA PDF

• Galois groups of random integer polynomials and van der Waerden's Conjecture

  Annals of Mathematics · 2025-03-01 · 6 citations

  articleOpen access1st authorCorresponding

  Of the $(2H+1)^n$ monic integer polynomials $f(x)=x^n+a_1 x^{n-1}+\cdots+a_n$ with $\mathrm{max}\{|a_1|,\ldots,|a_n|\}\leq H$, how many have associated Galois group that is not the full symmetric group $S_n$? There are clearly $\gg H^{n-1}$ such polynomials, as may be obtained by setting $a_n=0$. In 1936, van der Waerden conjectured that $O(H^{n-1})$ should in fact also be the correct upper bound for the count of such polynomials. The conjecture has been known previously for degrees $n\leq 4$, due to work of van der Waerden and Chow and Dietmann. The purpose of this paper is to prove van der Waerden's Conjecture for all degrees $n$.

  PublisherOA PDFDOI

• Rank stability in quadratic extensions and Hilbert’s tenth problem for the ring of integers of a number field

  Inventiones mathematicae · 2025-12-01

  articleOpen access

  Abstract We show that for any quadratic extension of number fields $K/F$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> <mml:mo>/</mml:mo> <mml:mi>F</mml:mi> </mml:math> , there exists an abelian variety $A/F$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> <mml:mo>/</mml:mo> <mml:mi>F</mml:mi> </mml:math> of positive rank whose rank does not grow upon base change to $K$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> . This result implies that Hilbert’s tenth problem over the ring of integers of any number field has a negative solution. That is, for the ring ${\mathcal{O}}_{K}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>K</mml:mi> </mml:msub> </mml:math> of integers of any number field $K$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> , there does not exist an algorithm that answers the question of whether a polynomial equation in several variables over ${\mathcal{O}}_{K}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>K</mml:mi> </mml:msub> </mml:math> has solutions in ${\mathcal{O}}_{K}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>K</mml:mi> </mml:msub> </mml:math> .

  PublisherOA PDFDOI

• Squarefree values of polynomial discriminants II

  Forum of Mathematics Pi · 2025-01-01 · 2 citations

  articleOpen access1st authorCorresponding

  Abstract We determine the density of integral binary forms of given degree that have squarefree discriminant, proving for the first time that the lower density is positive. Furthermore, we determine the density of integral binary forms that cut out maximal orders in number fields. The latter proves, in particular, an ‘arithmetic Bertini theorem’ conjectured by Poonen for ${\mathbb {P}}^1_{\mathbb {Z}}$ . Our methods also allow us to prove that there are $\gg X^{1/2+1/(n-1)}$ number fields of degree n having associated Galois group $S_n$ and absolute discriminant less than X , improving the best previously known lower bound of $\gg X^{1/2+1/n}$ . Finally, our methods correct an error in and thus resurrect earlier (retracted) results of Nakagawa on lower bounds for the number of totally unramified $A_n$ -extensions of quadratic number fields of bounded discriminant.

  PublisherDOI

• Rank stability in quadratic extensions and Hilbert's tenth problem for the ring of integers of a number field

  ArXiv.org · 2025-01-30 · 1 citations

  preprintOpen access

  We show that for any quadratic extension of number fields $K/F$, there exists an abelian variety $A/F$ of positive rank whose rank does not grow upon base change to $K$. This result implies that Hilbert's tenth problem over the ring of integers of any number field has a negative solution. That is, for the ring $\mathcal{O}_K$ of integers of any number field $K$, there does not exist an algorithm that answers the question of whether a polynomial equation in several variables over $\mathcal{O}_K$ has solutions in $\mathcal{O}_K$.

  PublisherOA PDFDOI

• The second moment of the size of the $2$-class group of monogenized cubic fields

  ArXiv.org · 2025-06-05

  preprintOpen access1st authorCorresponding

  We prove that when totally real (resp., complex) monogenized cubic number fields are ordered by height, the second moment of the size of the $2$-class group is at most $3$ (resp., at most $6$). In the totally real case, we further prove that the second moment of the size of the narrow $2$-class group is at most $9$. This result gives further evidence in support of the general observation, first made in work of Bhargava--Hanke--Shankar and recently formalized into a set of heuristics in work of Siad--Venkatesh, that monogenicity has an altering effect on class group distributions. All of the upper bounds we obtain are tight, conditional on tail estimates.

  PublisherOA PDFDOI

• A positive proportion of cubic fields are not monogenic yet have no local obstruction to being so

  Mathematische Annalen · 2024-12-12 · 1 citations

  articleOpen access

  Abstract We show that a positive proportion of cubic fields are not monogenic, despite having no local obstruction to being monogenic. Our proof involves the comparison of 2-descent and 3-descent in a certain family of Mordell curves $$E_k :y^2 = x^3 + k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:msup> <mml:mi>y</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:math> . As a by-product of our methods, we show that, for every $$r \ge 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , a positive proportion of curves $$E_k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> have Tate–Shafarevich group with 3-rank at least r .

  PublisherOA PDFDOI

Recent grants

• The parameterization of algebraic structures, and applications

  NSF · $381k · 2010–2014

• Number theory, representation theory, and arithmetic geometry

  NSF · $690k · 2013–2020

Frequent coauthors

• Arul Shankar

  University of Toronto

  22 shared

• Robert J. Lemke Oliver

  17 shared

• Ari Shnidman

  17 shared

• Zev Klagsbrun

  CCI Reprographics (United States)

  17 shared

• J. E. Cremona

  University of Warwick

  12 shared

• Tom Fisher

  Nottingham Trent University

  12 shared

• Xiaoheng Wang

  Ministry of Industry and Information Technology

  11 shared

• Benedict H. Gross

  8 shared

Education

• B.A., Mathematics

  Harvard University

  1996

• M.A., Mathematics

  Harvard University

  1998

• Ph.D., Mathematics

  Harvard University

  2001