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Ila Varma

Publications and source records attributed to Ila Varma.

14 recordsLinked to original sources
arXiv2026

Counting number fields of fixed degree by their smallest defining polynomial

When do two irreducible polynomials with integer coefficients define the same number field? One can define an action of $\mathrm{GL}_2 \times \mathrm{GL}_1$ on the space of polynomials of degree $n$ so that for any two polynomials $f$ and $g$ in the same orbit, the roots of $f$ may be expressed as rational linear transformations of the roots of $g$; thus, they generate the same field. In this article, we show that almost all polynomials of degree $n$ with size at most $X$ can only define the same number field as another polynomial of degree $n$ with size at most $X$ if they lie in the same orbit for this group action. (Here we measure the size of polynomials by the greatest absolute value of their coefficients.) This improves on work of Bhargava, Shankar, and Wang, who proved a similar statement for a positive proportion of polynomials. Using this result, we prove that the number of degree $n$ fields such that the smallest polynomial defining the field has size at most $X$ is asymptotic to a constant times $X^{n+1}$ as long as $n\geq 3$. For $n = 2$, we obtain a precise asymptotic of the form $\frac{27}{\pi^2} X^2$.

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arXiv2025

Malle's Conjecture for Galois octic fields over $\mathbb Q$

We compute the asymptotic number of octic number fields whose Galois groups over $\mathbb Q$ are isomorphic to $D_4$, the symmetries of a square, when ordering such fields by their absolute discriminants. In particular, we verify the strong form of Malle's conjecture for such octic $D_4$-fields and obtain the constant of proportionality. Our result answers the question of whether a positive proportion of Galois octic extensions of $\mathbb Q$ have non-abelian Galois group in the negative. We further demonstrate that the constant of proportionality satisfies the Malle--Bhargava principle of being a product of local masses, despite the fact that this principle does {\em not} hold for discriminants of quartic $D_4$-fields. This is the first instance of asymptotics being recovered for a non-concentrated family of number fields of Galois group neither abelian nor symmetric. Previously, this was only known for abelian fields, degree-$n$ $S_n$-fields for $n=3,4,5$, and degree-$6$ $S_3$-fields.

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arXiv2022

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

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 Kl\"uners 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.

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arXiv2021

Geometry-of-numbers methods in the cusp

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.

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arXiv2021

The mean number of 3-torsion elements in ray class groups of quadratic fields

We determine the average number of $3$-torsion elements in the ray class groups of fixed (integral) conductor $c$ of quadratic fields ordered by absolute discriminant, generalizing Davenport and Heilbronn's theorem on class groups. A consequence of this result is that a positive proportion of such ray class groups of quadratic fields have trivial 3-torsion subgroup whenever the conductor $c$ is taken to be a squarefree integer having very few prime factors none of which are congruent to $1 \bmod 3$. Additionally, we compute the second main term for the number of $3$-torsion elements in ray class groups with fixed conductor of quadratic fields with bounded discriminant.

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arXiv2018

Differential operators and families of automorphic forms on unitary groups of arbitrary signature

In the 1970's, Serre exploited congruences between $q$-expansion coefficients of Eisenstein series to produce $p$-adic families of Eisenstein series and, in turn, $p$-adic zeta functions. Partly through integration with more recent machinery, including Katz's approach to $p$-adic differential operators, his strategy has influenced four decades of developments. Prior papers employing Katz's and Serre's ideas exploiting differential operators and congruences to produce families of automorphic forms rely crucially on $q$-expansions of automorphic forms. The overarching goal of the present paper is to adapt the strategy to automorphic forms on unitary groups, which lack $q$-expansions when the signature is of the form $(a, b)$, $a\neq b$. In particular, this paper completely removes the restrictions on the signature present in prior work. As intermediate steps, we achieve two key objectives. First, partly by carefully analyzing the action of the Young symmetrizer on Serre-Tate expansions, we explicitly describe the action of differential operators on the Serre-Tate expansions of automorphic forms on unitary groups of arbitrary signature. As a direct consequence, for each unitary group, we obtain congruences and families analogous to those studied by Katz and Serre. Second, via a novel lifting argument, we construct a $p$-adic measure taking values in the space of $p$-adic automorphic forms on unitary groups of any prescribed signature. We relate the values of this measure to an explicit $p$-adic family of Eisenstein series. One application of our results is to the recently completed construction of $p$-adic $L$-functions for unitary groups by the first named author, Harris, Li, and Skinner.

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arXiv2018

Odd degree number fields with odd class number

For every odd integer $n \geq 3$, we prove that there exist infinitely many number fields of degree $n$ and associated Galois group $S_n$ whose class number is odd. To do so, we study the class groups of families of number fields of degree $n$ whose rings of integers arise as the coordinate rings of the subschemes of $\mathbb{P}^1$ cut out by integral binary $n$-ic forms. By obtaining upper bounds on the mean number of $2$-torsion elements in the class groups of fields in these families, we prove that a positive proportion (tending to $1$ as $n$ tends to $\infty$) of such fields have trivial $2$-torsion subgroup in their class groups and narrow class groups. Conditional on a tail estimate, we also prove the corresponding lower bounds and obtain the exact values of these averages, which are consistent with the heuristics of Cohen-Lenstra-Martinet-Malle and Dummit-Voight. Additionally, for any order $\mathcal{O}_f$ of degree $n$ arising from an integral binary $n$-ic form $f$, we compare the sizes of $\mathrm{Cl}_2(\mathcal{O}_f)$, the $2$-torsion subgroup of ideal classes in $\mathcal{O}_f$, and $\mathcal{I}_2(\mathcal{O}_f)$, the $2$-torsion subgroup of ideals in $\mathcal{O}_f$. For the family of orders arising from integral binary $n$-ic forms and contained in fields with fixed signature $(r_1,r_2)$, we prove that the mean value of the difference $|\mathrm{Cl}_2(\mathcal{O}_f)| - {2^{1-r_1-r_2}}|\mathcal{I}_2(\mathcal{O}_f)|$ is equal to $1$, generalizing a result of Bhargava and the third-named author for cubic fields. Conditional on certain tail estimates, we also prove that the mean value of $|\mathrm{Cl}_2(\mathcal{O}_f)| - {2^{1-r_1-r_2}}|\mathcal{I}_2(\mathcal{O}_f)|$ remains $1$ for certain families obtained by imposing local splitting and maximality conditions.

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arXiv2017

The number of quartic $D_4$-fields ordered by conductor

We consider families of number fields of degree 4 whose normal closures over $\mathbb{Q}$ have Galois group isomorphic to $D_4$, the symmetries of a square. To any such field $L$, one can associate the Artin conductor of the corresponding 2-dimensional irreducible Galois representation with image $D_4$. We determine the asymptotic number of such quartic $D_4$-fields ordered by conductor, and compute the leading term explicitly as a mass formula, verifying heuristics of Kedlaya and Wood. Additionally, we are able to impose any local splitting conditions at any finite number of primes (sometimes, at an infinite number of primes), and as a consequence, we also compute the asymptotic number of order 4 elements in class groups and narrow class groups of quadratic fields ordered by discriminant. Traditionally, there have been two approaches to counting quartic fields, using arithmetic invariant theory in combination with geometry-of-number techniques, and applying Kummer theory together with L-function methods. Both of these strategies fall short in the case of $D_4$-fields ordered by conductor since counting quartic fields containing a quadratic subfield with large discriminant is difficult. However, when ordering by conductor, we utilize additional algebraic structure arising from the outer automorphism of $D_4$ combined with both approaches mentioned above to obtain exact asymptotics.

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arXiv2015

p-adic q-expansion principles on unitary Shimura varieties

We formulate and prove certain vanishing theorems for p-adic automorphic forms on unitary groups of arbitrary signature. The p-adic q-expansion principle for p-adic modular forms on the Igusa tower says that if the coefficients of (sufficiently many of) the q-expansions of a p-adic modular form f are zero, then f vanishes everywhere on the Igusa tower. There is no p-adic q-expansion principle for unitary groups of arbitrary signature in the literature. By replacing q-expansions with Serre-Tate expansions (expansions in terms of Serre-Tate deformation coordinates) and replacing modular forms with automorphic forms on unitary groups of arbitrary signature, we prove an analogue of the p-adic q-expansion principle. More precisely, we show that if the coefficients of (sufficiently many of) the Serre-Tate expansions of a p-adic automorphic form f on the Igusa tower (over a unitary Shimura variety) are zero, then f vanishes identically on the Igusa tower. This paper also contains a substantial expository component. In particular, the expository component serves as a complement to Hida's extensive work on p-adic automorphic forms.

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arXiv2014

Local-global compatibility for regular algebraic cuspidal automorphic representation when $\ell \neq p$

We prove the compatibility of local and global Langlands correspondences for $GL_n$ up to semisimplification for the Galois representations constructed by Harris-Lan-Taylor-Thorne and Scholze. More precisely, let $r_p(π)$ denote an $n$-dimensional $p$-adic representation of the Galois group of a CM field $F$ attached to a regular algebraic cuspidal automorphic representation $π$ of $GL_n(\mathbb{A}_F)$. We show that the restriction of $r_p(π)$ to the decomposition group of a place $v\nmid p$ of $F$ corresponds up to semisimplification to $rec(π_v)$, the image of $π_v$ under the local Langlands correspondence. Furthermore, we can show that the monodromy of the associated Weil-Deligne representation of $.r_p(π)|_{G_{F_v}}$ is `more nilpotent' than the monodromy of $rec(π_v)$.

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arXiv2014

On the mean number of 2-torsion elements in the class groups, narrow class groups, and ideal groups of cubic orders and fields

Given any family of cubic fields defined by local conditions at finitely many primes, we determine the mean number of 2-torsion elements in the class groups and narrow class groups of these cubic fields when ordered by their absolute discriminants. For an order $\cal O$ in a cubic field, we study the three groups: $\rm Cl_2(\cal O)$, the group of ideal classes of $\cal O$ of order 2; $\rm Cl^+_2(\cal O)$, the group of narrow ideal classes of $\cal O$ of order 2; and ${\cal I}_2(\cal O)$, the group of ideals of $\cal O$ of order 2. We prove that the mean value of the difference $|\rm Cl_2({\cal O})|-\frac14|{\cal I}_2(\cal O)|$ is always equal to $1$, whether one averages over the maximal orders in real cubic fields, over all orders in real cubic fields, or indeed over any family of real cubic orders defined by local conditions. For the narrow class group, we prove that the mean value of the difference $|\rm Cl^+_2({\cal O})|-|{\cal I}_2(\cal O)|$ is equal to $1$ for any such family. For any family of complex cubic orders defined by local conditions, we prove similarly that the mean value of the difference $|\rm Cl_2(\mathcal O)|-\frac12|{\cal I}_2(\cal O)|$ is always equal to $1$, independent of the family. The determination of these mean numbers allows us to prove a number of further results as by-products. Most notably, we prove---in stark contrast to the case of quadratic fields---that: 1) a positive proportion of cubic fields have odd class number; 2) a positive proportion of real cubic fields have isomorphic 2-torsion in the class group and the narrow class group; and 3) a positive proportion of real cubic fields contain units of mixed real signature. We also show that a positive proportion of real cubic fields have narrow class group strictly larger than the class group, and thus a positive proportion of real cubic fields do not possess units of every possible real signature.

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arXiv2014

The mean number of 3-torsion elements in the class groups and ideal groups of quadratic orders

We determine the mean number of 3-torsion elements in the class groups of quadratic orders, where the quadratic orders are ordered by their absolute discriminants. Moreover, for a quadratic order $\mathcal{O}$ we distinguish between the two groups: $\mathrm{Cl}_3(\mathcal{O})$, the group of ideal classes of order $3$; and $\mathcal{I}_3(\mathcal{O})$, the group of ideals of order $3$. We determine the mean values of both $|\mathrm{Cl}_3(\mathcal{O})|$ and $|\mathcal{I}_3(\mathcal{O})|$, as $\mathcal{O}$ ranges over any family of orders defined by finitely many (or in suitable cases, even infinitely many) local conditions. As a consequence, we prove the surprising fact that the mean value of the difference $|\mathrm{Cl}_3(\mathcal{O})|-|\mathcal{I}_3(\mathcal{O})|$ is equal to $1$, regardless of whether one averages over the maximal orders in complex quadratic fields or over all orders in such fields or, indeed, over any family of complex quadratic orders defined by local conditions. For any family of real quadratic orders defined by local conditions, we prove similarly that the mean value of the difference $|\mathrm{Cl}_3(\mathcal{O})|-\frac13|\mathcal{I}_3(\mathcal{O})|$ is always equal to $1$, independent of the family.

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arXiv2011

Finding elementary formulas for theta functions associated to even sums of squares

This article discusses the classical problem of how to calculate $r_n(m)$, the number of ways to represent an integer $m$ by a sum of $n$ squares from a computational efficiency viewpoint. Although this problem has been studied in great detail, there are very few formulas given for the purpose of computing $r_n(m)$ quickly. More precisely, for fixed $n$, we want a formula for $r_n(m)$ that computes in log-polynomial time (with respect to $m$) when the prime factorization of $m$ is given. Restricting to even $n$, we can view $θ_n(q)$, the theta function associated to sums of $n$ squares, as a modular form of weight $n/2$ on $Γ_1(4)$. In particular, we show that for only a small finite list of $n$ can $θ_n$ be written as a linear combination consisting entirely of Eisenstein series and cusp forms with complex multiplication. These are the only $n$ that give rise to "elementary" formulas for $r_n(m)$, i.e. formulas such that for a prime $p$, $r_n(p)$ can be calculated in $\cO(\log(p))$-time. Viewing $θ_n(q)$ as one of the simpler examples of modular forms that are not strictly Eisenstein, this result motivates the necessity of a log-polynomial time algorithm that directly calculates the Fourier coefficients of modular forms in the generic situation when there is no such formula, as described in Couveignes and Edixhoven's forthcoming book (for level 1 cases) and Peter Bruin's Ph.D. thesis (for higher level, including 4).

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