Preprints
- Entropy of Cohen—Lenstra measures: the u-aspect
Preprint (2024)
+ We carry out several calculations of entropies associated to Cohen—Lenstra measures on finite abelian p-groups. We show that unit rank u=0 is an entropy maximizer amongst all Cohen-Lenstra measures of varying unit rank. - Spin structures, biextensions and class groups (with A. Venkatesh)
+ Motivated by anomalous class group statistics, we propose an arithmetic analogue of the topological story of quadratic enhancements associated with spin structures on closed oriented 2- and 3-manifolds: a choice of spin structure provides, respectively, a quadratic refinement of the mod 2 intersection form and of the linking pairing on the first torsion homology. - Global fields with odd ordinary, narrow, and oriented relative class numbers
+ I extend the techniques of my thesis to study the Z/2-moment of the relative class group of relatively monogenised extensions of global fields and closely related groups. The results of this paper prove the existence of number fields with prescribed 2-class groups in composite degrees in great generality. - Even degree number fields with odd class number
arXiv:2011.08842 (2020)
+ The second part of my thesis in which I obtain information about the Z/2-moment of the class group of monogenized fields in even degree. A corollary of the main result is the the existence, in any even degree, of infinitely many fields with odd class number. This statement concludes a historical sequence of results on the parity of class numbers, a topic first explored by Gauss in his Disquisitiones Arithmeticae through his genus theory. - Effect of monogenicity on 2-torsion in the class group of number fields of odd degree
arXiv:2011.08834 (2020)
+ This paper and the one above are essentially my PhD thesis. Inspired by work of Bhargava-Hanke-Shankar, I show that the Z/2-moment of the class group of monogenized fields in any odd degree and independent of signature, conditional on a tail estimate. In the second part of the paper, I carry out the strategy for N-monogenized fields of odd degree.
Papers
- Counting integral points on symmetric varieties with applications to arithmetic statistics (with A. Shankar, A. Swaminathan)
Proceedings of the London Mathematical Society (2023), accepted pending revisions
+ In this article, we combine Bhargava's geometry-of-numbers methods with the dynamical point-counting methods of Eskin--McMullen and Benoist--Oh to develop a new technique for counting integral points on symmetric varieties lying within fundamental domains for coregular representations. As applications, we study the distribution of the 2-torsion subgroup of the class group in thin families of cubic number fields and elliptic curves over ℚ. - Geometry-of-numbers methods in the cusp (with A. Shankar, A. Swaminathan, I. Varma)
Algebra and Number Theory (2021), to appear
+ In this article, we develop new methods for counting integral orbits having bounded invariants that lie inside the cusps of fundamental domains for coregular representations. We illustrate these methods for a representation of cardinal interest in number theory, namely that of the split orthogonal group acting on the space of quadratic forms.
The years above indicate when the articles were submitted to the arXiv.