Manjul Bhargava

Manjul Bhargava is a renowned Canadian-American mathematician known for his profound contributions to number theory and algebra. Born on August 8, 1974, he grew up in Canada and the United States and is of Indian descent. Bhargava is the R. Brandon Fradd Professor of Mathematics at Princeton University.

Academic Contributions:

Higher Composition Laws: Bhargava is celebrated for his work on higher composition laws, which extend the classical composition of binary quadratic forms studied by Carl Friedrich Gauss. His innovative approach has provided new insights into the composition of more complex algebraic structures.

Geometry of Numbers: He has developed novel techniques in the geometry of numbers, leading to significant advancements in counting number fields and understanding their properties.

Elliptic Curves and Rational Points: His research has impacted the understanding of rational points on elliptic curves, which has implications for solving Diophantine equations—equations seeking integer solutions.

Awards and Honors:

Fields Medal (2014): Bhargava was awarded the Fields Medal, often regarded as the highest honor in mathematics, for "developing powerful new methods in the geometry of numbers" and applying them to count rings of small rank and to bound the average rank of elliptic curves.

Cole Prize in Number Theory (2008): Presented by the American Mathematical Society for outstanding contributions to number theory.

Memberships: He is a fellow of prestigious societies including the Royal Society (FRS), the National Academy of Sciences (NAS), and the American Academy of Arts and Sciences.

Educational Background:

Harvard University: Bhargava completed his A.B. degree in mathematics in 1996.

Princeton University: He earned his Ph.D. in 2001 under the supervision of Andrew Wiles, famous for proving Fermat's Last Theorem.

Teaching and Outreach:

Bhargava is known for his dedication to teaching and mentorship, inspiring a new generation of mathematicians.

He actively participates in mathematical outreach programs and gives lectures aimed at making complex mathematical ideas accessible to a broader audience.

Cultural Interests:

In addition to his mathematical pursuits, Bhargava is an accomplished musician skilled in playing the tabla, an Indian percussion instrument. He often explores connections between mathematics and music.

Manjul Bhargava's work has significantly advanced the field of mathematics, particularly in number theory, and continues to influence ongoing research and applications within mathematics and related disciplines.

All published works

Action	Title	Year	Authors
+ PDF Chat	Rank stability in quadratic extensions and Hilbert's tenth problem for the ring of integers of a number field	2025	Levent Alpöge Manjul Bhargava Wei Ho Ari Shnidman
+	A positive proportion of cubic fields are not monogenic yet have no local obstruction to being so	2024	Levent Alpöge Manjul Bhargava Ari Shnidman
+ PDF Chat	A proof of van der Waerden's Conjecture on random Galois groups of polynomials	2024	Manjul Bhargava
+	Improved error estimates for the Davenport–Heilbronn theorems	2023	Manjul Bhargava Takashi Taniguchi Frank Thorne
+ PDF Chat	A positive proportion of quartic fields are not monogenic yet have no local obstruction to being so	2023	Levent Alpöge Manjul Bhargava Ari Shnidman
+	Hermite equivalence of polynomials	2023	Manjul Bhargava Jan‐Hendrik Evertse Kálmán Győry László Remete Ashvin Swaminathan
+ PDF Chat	A proof of van der Waerden’s Conjecture on random Galois groups of polynomials	2023	Manjul Bhargava
+	The local-global principle for integral points on stacky curves	2022	Manjul Bhargava Bjorn Poonen
+	On the number of integral binary n$n$‐ic forms having bounded Julia invariant	2022	Manjul Bhargava Andrew Yang
+	On the number of monogenizations of a quartic order (with an appendix by Shabnam Akhtari)	2022	Manjul Bhargava
+ PDF Chat	The density of polynomials of degree n$n$ over Zp${\mathbb {Z}}_p$ having exactly r$r$ roots in Qp${\mathbb {Q}}_p$	2022	Manjul Bhargava J. E. Cremona Tom Fisher Stevan Gajović
+ PDF Chat	Squarefree values of polynomial discriminants I	2022	Manjul Bhargava Arul Shankar Xiaoheng Wang
+ PDF Chat	An improvement on Schmidt’s bound on the number of number fields of bounded discriminant and small degree	2022	Manjul Bhargava Arul Shankar Xiaoheng Wang
+	On average sizes of Selmer groups and ranks in families of elliptic curves having marked points	2022	Manjul Bhargava Wei Ho
+	Squarefree values of polynomial discriminants II	2022	Manjul Bhargava Arul Shankar Xiaoheng Wang
+	Integers expressible as the sum of two rational cubes	2022	Levent Alpöge Manjul Bhargava Ari Shnidman
+	An improvement on Schmidt's bound on the number of number fields of bounded discriminant and small degree	2022	Manjul Bhargava Arul Shankar Xiaoheng Wang
+	Galois groups of random integer polynomials and van der Waerden's Conjecture.	2021	Manjul Bhargava
+	On the number of monogenizations of a quartic order.	2021	Manjul Bhargava
+	The second moment of the size of the $2$-Selmer group of elliptic curves	2021	Manjul Bhargava Arul Shankar Ashvin Swaminathan
+ PDF Chat	Elements of given order in Tate–Shafarevich groups of abelian varieties in quadratic twist families	2021	Manjul Bhargava Zev Klagsbrun Robert J. Lemke Oliver Ari Shnidman
+	The density of polynomials of degree n over Zp having exactly r roots in Qp	2021	Stevan Gajović Manjul Bhargava J. E. Cremona Tom Fisher
+	The density of polynomials of degree $n$ over $\mathbb{Z}_p$ having exactly $r$ roots in $\mathbb{Q}_p$	2021	Manjul Bhargava J. E. Cremona Tom Fisher Stevan Gajović
+	Galois groups of random integer polynomials and van der Waerden's Conjecture	2021	Manjul Bhargava
+	On the number of monogenizations of a quartic order	2021	Manjul Bhargava
+	The second moment of the size of the $2$-Selmer group of elliptic curves	2021	Manjul Bhargava Arul Shankar Ashvin Swaminathan
+	Hermite equivalence of polynomials	2021	Manjul Bhargava Jan‐Hendrik Evertse Kálmán Győry László Remete Ashvin Swaminathan
+	The density of polynomials of degree $n$ over $\mathbb{Z}_p$ having exactly $r$ roots in $\mathbb{Q}_p$	2021	Manjul Bhargava J. E. Cremona Tom Fisher Stevan Gajović
+	A positive proportion of quartic fields are not monogenic yet have no local obstruction to being so	2021	Levent Alpöge Manjul Bhargava Ari Shnidman
+	Improved error estimates for the Davenport-Heilbronn theorems	2021	Manjul Bhargava Takashi Taniguchi Frank Thorne
+	Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves	2020	Manjul Bhargava Arul Shankar Takashi Taniguchi Frank Thorne Jacob Tsimerman Zhao Yang
+	The proportion of genus one curves over ℚ defined by a binary quartic that everywhere locally have a point	2020	Manjul Bhargava J. E. Cremona Tom Fisher
+	The proportion of genus one curves over $\mathbb{Q}$ defined by a binary quartic that everywhere locally have a point	2020	Manjul Bhargava J. E. Cremona Tom Fisher
+	The mean number of 2-torsion elements in the class groups of $n$-monogenized cubic fields	2020	Manjul Bhargava Jonathan Hanke Arul Shankar
+	A positive proportion of cubic fields are not monogenic yet have no local obstruction to being so	2020	Levent Alpöge Manjul Bhargava Ari Shnidman
+	The proportion of genus one curves over $\mathbb{Q}$ defined by a binary quartic that everywhere locally have a point	2020	Manjul Bhargava J. E. Cremona Tom Fisher
+	The local-global principle for integral points on stacky curves	2020	Manjul Bhargava Bjorn Poonen
+ PDF Chat	3-Isogeny Selmer groups and ranks of Abelian varieties in quadratic twist families over a number field	2019	Manjul Bhargava Zev Klagsbrun Robert J. Lemke Oliver Ari Shnidman
+ PDF Chat	The average size of the 3‐isogeny Selmer groups of elliptic curves y2=x3+k	2019	Manjul Bhargava Noam D. Elkies Ari Shnidman
+ PDF Chat	A positive proportion of Thue equations fail the integral Hasse principle	2019	Shabnam Akhtari Manjul Bhargava
+	Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves	2017	Manjul Bhargava Arul Shankar Takashi Taniguchi Frank Thorne Jacob Tsimerman Yongqiang Zhao
+	A positive proportion of locally soluble hyperelliptic curves over ℚ have no point over any odd degree extension	2016	Manjul Bhargava Benedict H. Gross Xiaoheng Wang
+ PDF Chat	The equidistribution of lattice shapes of rings of integers in cubic, quartic, and quintic number fields	2016	Manjul Bhargava Piper Harron
+	A positive proportion of Thue equations fail the integral Hasse principle.	2016	Shabnam Akhtari Manjul Bhargava
+ PDF Chat	The mean number of 3-torsion elements in the class groups and ideal groups of quadratic orders	2016	Manjul Bhargava Ila Varma
+	Squarefree values of polynomial discriminants I	2016	Manjul Bhargava Arul Shankar Xiaoheng Wang
+ PDF Chat	Coregular spaces and genus one curves	2016	Manjul Bhargava Wei Ho
+ PDF Chat	ORBIT PARAMETRIZATIONS FOR K3 SURFACES	2016	Manjul Bhargava Wei Ho Abhinav Kumar
+ PDF Chat	What is the Probability that a Random Integral Quadratic Form in<i>n</i>Variables has an Integral Zero?	2015	Manjul Bhargava J. E. Cremona Tom Fisher Nick G. Jones Jonathan P. Keating
+ PDF Chat	The proportion of plane cubic curves over ℚ that everywhere locally have a point	2015	Manjul Bhargava J. E. Cremona Tom Fisher
+ PDF Chat	On the mean number of 2 -torsion elements in the class groups, narrow class groups, and ideal groups of cubic orders and fields	2015	Manjul Bhargava Ila Varma
+	Geometry-of-numbers methods over global fields I: Prehomogeneous vector spaces	2015	Manjul Bhargava Arul Shankar Xiaoheng Wang
+	Arithmetic invariant theory II: Pure inner forms and obstructions to the existence of orbits	2015	Manjul Bhargava Benedict H. Gross Xiaoheng Wang
+	What is the probability that a random integral quadratic form in $n$ variables has an integral zero?	2015	Manjul Bhargava J. E. Cremona Tom Fisher Nick G. Jones Jonathan P. Keating
+ PDF Chat	Modeling the distribution of ranks, Selmer groups, and Shafarevich–Tate groups of elliptic curves	2015	Manjul Bhargava Daniel M. Kane H. W. Lenstra Bjorn Poonen Eric M. Rains
+ PDF Chat	Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0	2014	Manjul Bhargava Arul Shankar
+ PDF Chat	On a notion of “Galois closure” for extensions of rings	2014	Manjul Bhargava Matthew Satriano
+ PDF Chat	Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves	2014	Manjul Bhargava Arul Shankar
+ PDF Chat	On the number of cubic orders of bounded discriminant having automorphism group <i>C</i><sub>3</sub>, and related problems	2014	Manjul Bhargava Ariel Shnidman
+	A positive proportion of plane cubics fail the Hasse principle	2014	Manjul Bhargava
+	The mean number of 3-torsion elements in the class groups and ideal groups of quadratic orders	2014	Manjul Bhargava Ila Varma
+	The geometric sieve and the density of squarefree values of invariant polynomials	2014	Manjul Bhargava
+ PDF Chat	Arithmetic invariant theory	2014	Manjul Bhargava Benedict H. Gross
+	A positive proportion of elliptic curves over $\mathbb{Q}$ have rank one	2014	Manjul Bhargava Christopher Skinner
+	A majority of elliptic curves over $\mathbb Q$ satisfy the Birch and Swinnerton-Dyer conjecture	2014	Manjul Bhargava Christopher Skinner Wei Zhang
+	A positive proportion of elliptic curves over Q have rank one	2014	Manjul Bhargava Christopher Skinner
+	Rational points on elliptic and hyperelliptic curves	2014	Manjul Bhargava
+	A positive proportion of plane cubics fail the Hasse principle	2014	Manjul Bhargava
+	The mean number of 3-torsion elements in the class groups and ideal groups of quadratic orders	2014	Manjul Bhargava Ila Varma
+	Pencils of quadrics and the arithmetic of hyperelliptic curves	2013	Manjul Bhargava Benedict H. Gross Xiaoheng Wang
+	Coregular spaces and genus one curves	2013	Manjul Bhargava Wei Ho
+	The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1	2013	Manjul Bhargava Arul Shankar
+	The average number of elements in the 4-Selmer groups of elliptic curves is 7	2013	Manjul Bhargava Arul Shankar
+	What is the probability that a random integral quadratic form in $n$ variables is isotropic?	2013	Manjul Bhargava J. E. Cremona Tom Fisher
+	Most hyperelliptic curves over Q have no rational points	2013	Manjul Bhargava
+	The proportion of plane cubic curves over ${\mathbb Q}$ that everywhere locally have a point	2013	Manjul Bhargava J. E. Cremona Tom Fisher
+	A positive proportion of locally soluble hyperelliptic curves over $\mathbb Q$ have no point over any odd degree extension	2013	Manjul Bhargava Benedict H. Gross Xiaoheng Wang
+	Coregular spaces and genus one curves	2013	Manjul Bhargava Wei Ho
+	Arithmetic invariant theory II	2013	Manjul Bhargava Benedict H. Gross Xiaoheng Wang
+	Quadratic and Higher Degree Forms	2013	Krishnaswami Alladi Manjul Bhargava David Savitt Pham Huu Tiep
+	On the number of integral binary $n$-ic forms having bounded Julia invariant	2013	Manjul Bhargava Andrew Yang
+ PDF Chat	On the Davenport–Heilbronn theorems and second order terms	2012	Manjul Bhargava Arul Shankar Jacob Tsimerman
+	The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point	2012	Manjul Bhargava Benedict H. Gross
+	Arithmetic invariant theory	2012	Manjul Bhargava Benedict H. Gross
+ PDF Chat	The density of discriminants of quintic rings and fields	2010	Manjul Bhargava
+	Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0	2010	Manjul Bhargava Arul Shankar
+	On a notion of "Galois closure" for extensions of rings	2010	Manjul Bhargava Matthew Satriano
+	Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves	2010	Manjul Bhargava Arul Shankar
+	On the Davenport-Heilbronn theorems and second order terms	2010	Manjul Bhargava Arul Shankar Jacob Tsimerman
+ PDF Chat	Error estimates for the Davenport-Heilbronn theorems	2010	Karim Belabas Manjul Bhargava Carl Pomerance
+	Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0	2010	Manjul Bhargava Arul Shankar
+	On the Davenport-Heilbronn theorems and second order terms	2010	Manjul Bhargava Arul Shankar Jacob Tsimerman
+	Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves	2010	Manjul Bhargava Arul Shankar
+	On a notion of "Galois closure" for extensions of rings	2010	Manjul Bhargava Matthew Satriano
+	The density of discriminants of quintic rings and fields	2010	Manjul Bhargava
+	On 𝑃-orderings, rings of integer-valued polynomials, and ultrametric analysis	2009	Manjul Bhargava
+	Finite generation properties for various rings of integer-valued polynomials	2009	Manjul Bhargava Paul-Jean Cahen Julie Yeramian
+	On the average number of octahedral newforms of prime level	2009	Manjul Bhargava Eknath Ghate
+ PDF Chat	Higher composition laws IV: The parametrization of quintic rings	2008	Manjul Bhargava
+ PDF Chat	The density of discriminants of $S_3$-sextic number fields	2007	Manjul Bhargava Melanie Matchett Wood
+	Higher composition laws and applications	2007	Manjul Bhargava
+ PDF Chat	Mass Formulae for Extensions of Local Fields, and Conjectures on the Density of Number Field Discriminants	2007	Manjul Bhargava
+	Higher composition laws and applications	2006	Manjul Bhargava
+ PDF Chat	The density of discriminants of quartic rings and fields	2005	Manjul Bhargava
+ PDF Chat	Higher composition laws III: The parametrization of quartic rings	2004	Manjul Bhargava
+ PDF Chat	Higher composition laws II: On cubic analogues of Gauss composition	2004	Manjul Bhargava
+ PDF Chat	Higher composition laws I: A new view on Gauss composition, and quadratic generalizations	2004	Manjul Bhargava
+	Gauss Composition and Generalizations	2002	Manjul Bhargava
+	The Factorial Function and Generalizations	2000	Manjul Bhargava
+ PDF Chat	On the Conway-Schneeberger fifteen theorem	2000	Manjul Bhargava
+	Factoring Dickson Polynomials over Finite Fields	1999	Manjul Bhargava Michael E. Zieve
+ PDF Chat	Continuous functions on compact subsets of local fields	1999	Manjul Bhargava Kiran Kedlaya
+	Generalized Factorials and Fixed Divisors over Subsets of a Dedekind Domain	1998	Manjul Bhargava
+	P-orderings and polynomial functions on arbitrary subsets of Dedekind rings.	1997	Manjul Bhargava
+	Congruence preservation and polynomial functions from Zn to Zm	1997	Manjul Bhargava

Common Coauthors

Coauthor	Papers Together
Arul Shankar	22
J. E. Cremona	12
Tom Fisher	12
Xiaoheng Wang	11
Benedict H. Gross	8
Ari Shnidman	7
Levent Alpöge	6
Jacob Tsimerman	5
Frank Thorne	4
Stevan Gajović	4
Takashi Taniguchi	4
Ila Varma	4
Ashvin Swaminathan	4
Wei Ho	4
Bjorn Poonen	3
Christopher Skinner	3
Matthew Satriano	3
Shabnam Akhtari	2
Jonathan P. Keating	2
Robert J. Lemke Oliver	2
Wei Ho	2
Ari Shnidman	2
László Remete	2
Zev Klagsbrun	2
Andrew Yang	2
Kálmán Győry	2
Jan‐Hendrik Evertse	2
Nick G. Jones	2
Paul-Jean Cahen	1
Julie Yeramian	1
Eknath Ghate	1
Jonathan Hanke	1
Daniel M. Kane	1
Piper Harron	1
Pham Huu Tiep	1
Noam D. Elkies	1
Eric M. Rains	1
Michael E. Zieve	1
Carl Pomerance	1
Krishnaswami Alladi	1
Karim Belabas	1
Ariel Shnidman	1
Wei Zhang	1
Zhao Yang	1
Yongqiang Zhao	1
Abhinav Kumar	1
H. W. Lenstra	1
David Savitt	1
Kiran Kedlaya	1
Melanie Matchett Wood	1

Commonly Cited References

Action	Title	Year	Authors	# of times referenced
+ PDF Chat	The density of discriminants of quartic rings and fields	2005	Manjul Bhargava	30
+	On the density of discriminants of cubic fields. II	1971	H. Davenport H. Heilbronn	28
+	The Theory of Irrationalities of the Third Degree	2009	B. Delone D. K. Faddeev	26
+	On the Class-Number of Binary Cubic Forms (II)	1951	H. Davenport	21
+ PDF Chat	Higher composition laws I: A new view on Gauss composition, and quadratic generalizations	2004	Manjul Bhargava	21
+ PDF Chat	Higher composition laws III: The parametrization of quartic rings	2004	Manjul Bhargava	21
+	Fourier coefficients of modular forms on G2	2002	Wee Teck Gan Benedict H. Gross Gordan Savin	16
+ PDF Chat	On the Davenport–Heilbronn theorems and second order terms	2012	Manjul Bhargava Arul Shankar Jacob Tsimerman	15
+ PDF Chat	A classification of irreducible prehomogeneous vector spaces and their relative invariants	1977	Masahide Sato T. Kimura	15
+ PDF Chat	The density of discriminants of quintic rings and fields	2010	Manjul Bhargava	14
+ PDF Chat	Finiteness Theorems for Binary Forms with Given Discriminant	1972	B. J. Birch J. R. Merriman	13
+ PDF Chat	Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves	2014	Manjul Bhargava Arul Shankar	13
+ PDF Chat	On Dirichlet series whose coefficients are class numbers of integral binary cubic forms	1972	Takuro Shintani	12
+ PDF Chat	Higher composition laws II: On cubic analogues of Gauss composition	2004	Manjul Bhargava	12
+	Prehomogeneous vector spaces and field extensions	1992	D. J. Wright Akihiko Yukie	12
+	Notes on elliptic curves. II.	1965	B. J. Birch H. P. F. Swinnerton-Dyer	11
+	Most hyperelliptic curves over Q have no rational points	2013	Manjul Bhargava	10
+ PDF Chat	Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves	2010	J. E. Cremona Tom Fisher Michael Stoll	10
+ PDF Chat	Higher composition laws IV: The parametrization of quintic rings	2008	Manjul Bhargava	10
+	The average number of elements in the 4-Selmer groups of elliptic curves is 7	2013	Manjul Bhargava Arul Shankar	10
+	On the Class-Number of Binary Cubic Forms (I)	1951	H. Davenport	10
+	Random Matrices, Frobenius Eigenvalues, and Monodromy	1998	Nicholas M. Katz Peter Sarnak	10
+ PDF Chat	The Arithmetic of Elliptic Curves	1986	Joseph H. Silverman	10
+	Density of discriminants of cubic extensions.	1988	B. Datskovsky D. J. Wright	10
+ PDF Chat	Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0	2014	Manjul Bhargava Arul Shankar	9
+	The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point	2012	Manjul Bhargava Benedict H. Gross	9
+ PDF Chat	Arithmetic Subgroups of Algebraic Groups	1962	Armand Borel Harish-chandra Harish-Chandra	9
+ PDF Chat	Rings and ideals parameterized by binary <i>n</i> -ic forms	2011	Melanie Matchett Wood	9
+ PDF Chat	The invariants of a genus one curve	2008	Tom Fisher	9
+	The rank of elliptic curves	1977	Armand Brumer Kenneth Kramer	8
+	Jacobians of Genus One Curves	2001	Sang Yook An Seog Young Kim David C. Marshall Susan Marshall William G. McCallum Alexander R. Perlis	8
+ PDF Chat	Reduction of Binary Cubic and Quartic Forms	1999	J. E. Cremona	8
+	An Infinite Version of the Chinese Remainder Theorem	1991	Torsten Ekedahl	8
+	Disquisitiones arithmeticae	1801	Carl Friedrich Gauß	8
+	The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1	2013	Manjul Bhargava Arul Shankar	8
+ PDF Chat	On the Birch-Swinnerton-Dyer quotients modulo squares	2010	Tim Dokchitser Vladimir Dokchitser	8
+	Random maximal isotropic subspaces and Selmer groups	2011	Bjorn Poonen Eric M. Rains	8
+ PDF Chat	Average ranks of elliptic curves: Tension between data and conjecture	2007	Baur Bektemirov Barry Mazur William Stein Mark E. Watkins	7
+	The average rank of elliptic curves I	1992	Armand Brumer	7
+	Étude heuristique des groupes de classes des corps de nombres.	1990	Henri Cohen Jacques Martinet	7
+ PDF Chat	Mass Formulae for Extensions of Local Fields, and Conjectures on the Density of Number Field Discriminants	2007	Manjul Bhargava	7
+	Moduli Spaces for Rings and Ideals	2009	Melanie Eggers Matchett Wood	7
+ PDF Chat	Most odd degree hyperelliptic curves have only one rational point	2014	Bjorn Poonen Michael Stoll	7
+	The geometric sieve and the density of squarefree values of invariant polynomials	2014	Manjul Bhargava	7
+	The Average Measure of Quadratic Forms With Given Determinant and Signature	1944	Carl Ludwig Siegel	7
+ PDF Chat	Arithmetic invariant theory	2014	Manjul Bhargava Benedict H. Gross	7
+ PDF Chat	The Average Analytic Rank of Elliptic Curves	2004	D. R. Heath‐Brown	7
+	Maximal linear spaces contained in the base loci of pencils of quadrics	2013	Xiaoheng Wang	6
+	On the density of discriminants of quartic fields.	1980	Andrew Marc Baily	6
+	Coregular spaces and genus one curves	2013	Manjul Bhargava Wei Ho	6