# Geometry of Arithmetic Statistics (2022)

Date & Time

Organizers:
Alina Bucur, University of California, San Diego
Jordan Ellenberg, University of Wisconsin, Madison
Ila Varma, University of California, San Diego
David Zureick-Brown, Emory University

The Simons Symposium on Geometry of Arithmetic Statistics brought together experts to explore recent developments in arithmetic statistics, an old subject which in recent years has been recast in a totally new light and represents a sudden influx of geometric ideas to what has traditionally been an analytically-flavored subject. The series gathered the disparate research groups reshaping arithmetic statistics and developed a unified point of view about the geometric future of the subject.

The topics discussed included:

• Advances and recent applications of the geometry of numbers and arithmetic invariant theory, including recent work by Bhargava and Shankar and Bhargava, Skinner and Zhang, especially their development of a more robust theory of sieve methods, which allows a systematic approach to counting in shapes that can be quite unwieldy and “thin”.

• The “Q-method” of Bhargava, Shankar and Wang, which improves mod p sieves into ones that are mod p2 by suitable high dimensional embeddings and has seen applications to squarefree values of polynomial discriminants and asymptotics over thin families, giving additional evidence towards the Cohen-Lenstra-Martinet heuristics.

• New approaches to the Batyrev-Manin conjectures, including Ellenberg, Satriano and Zureick-Brown’s recent development of a theory of heights for rational points on stacks, which unifies the Cohen-Lenstram-Martinet and Batyrev-Manin into a single framework as well as Peyre’s 2018 “Beyond Heights” which introduces “categorification” (or better “latticification”) of the idea of counting by height, and appears to be critical in unraveling the mysterious “accumulating subvarieties” which make the Batyrev-Manin conjectures difficult to state and to computationally assess.

• Meeting Report

The symposium brought together experts in the geometry of numbers and the algebraic geometric points of view in arithmetic statistics. A running theme was the parallel between the Malle-Bhargava and Batyrev-Manin conjectures, including how to reconcile each of them with the known counterexamples, which themselves possess their own geometry, just now coming into focus.

Wood gave an overview of the major problems in arithmetic statistics, including asymptotics of number fields, class groups, elliptic curves, the techniques used to attack them and the most recent developments in the field.

Ellenberg gave a survey of the methods by which problems in arithmetic statistics over global function fields in characteristic p can be rephrased as questions about rational points on moduli spaces and moduli stacks over finite fields, thus bringing arithmetic statistics in contact with questions about the geometry of those moduli spaces.

Varma gave a presentation about the community norms of arithmetic statistics and what we want them to look like in the future.

Browning’s talk focused on a motivic adaptation of the circle method from analytic number theory developed by him and Bilu.

Schindler spoke about computing densities of points on or near manifolds in the Euclidean n-space.

Peyre gave a survey talk about his notion of “freeness” and “slopes” of a rational point on a variety. The original body of work on the Batyrev-Manin conjecture concerned counting rational points in order of height. It was immediately seen that certain special subsets of points were “exceptional”; these special subloci already contained more rational points than (according to the conjecture) the whole variety was supposed to have! Peyre’s notion of “freeness” provides insight into what special geometric properties distinguish these exceptional points from the “ambient” points on the variety in general. This is a story very much in progress (see also recent work of Sawin and Browning-Sawin), but it is clear that freeness will be part of whatever the final story is about exceptional loci.

Gundlach’s talk focused on formulating a new conjecture about counting Galois extensions that specializes to obtain some of the most famous conjectures in arithmetic statistics, including Malle-Bhargava, as well as the latest developments in number field counting, like the result of Shankar-Thorne on cubic fields.

Siad spoke about his work on the arithmetic statistics of monogenized fields: that is, the set of pairs (K,a) where K is a number field and a is an algebraic integer with Z[a] equal to the ring of integers of K. Surprisingly, the class groups of monogenized fields are easier to study than those of general number fields, and in cases where both can be computed, Siad’s results show that the average size is different.

Smith discussed his groundbreaking work on the distribution of 2∞ Selmer groups of quadratic twists of elliptic curves and 2∞ class groups of quadratic fields. This work is geometric in a rather different sense than most of what we saw this week — the theorem involves an intricate process of expressing (or rather approximately expressing) an interval as a disjoint union of multiplicatively defined subsets. There was a lot of discussion here — what would these methods look like over function fields, as in Ellenberg’s talk? Is there a way to see the critical relation among the Selmer groups of the 2d quadratic twists involving d primes and the superficially similar sums over hypercubes that appear in the definition of Gowers norms?

Tanimoto spoke about his work with Brian Lehmann and Akash Sengupta on Geometric Manin’s Conjecture. The main idea is that the regularities predicted by the Batyrev-Manin conjecture, when applied over a global function field K/F_q, strongly suggest analogous geometric facts about the moduli spaces of sections of a fibration X -> P1 of complex varieties. The work of Lehmann-Sengupta-Tanimoto reported here proves these facts, providing strong evidence that the Batyrev-Manin conjectures, which have a somewhat turbulent history, are now formulated correctly.

Yasuda spoke on his very recent work with Darda, which offers a new viewpoint on the heights of rational points on stacks, extending the definition proposed by Ellenberg, Satriano and Zureick-Brown. Such a definition is critical for the interface between algebraic geometry and arithmetic statistics because extensions of number fields and class group elements can each be seen as rational points on certain moduli stacks, whose points you would thus like to count; a definition of height tells you in which order to count these points, without which there is not even a well-defined problem. The Darda-Yasuda approach improves on that of Ellenberg, Satriano and Zureick-Brown in that it allows a geometrically formulated version of the strong Batyrev-Manin conjecture, which specifies not only that the number of points of height at most X grows like X^{a + epsilon} , but predicts that the X^epsilon factor is asymptotic to a specific power of log X.

Alberts spoke about his joint work with O’Dorney and a new project with Lemke-Oliver, Wang and Wood. Both projects are geared towards bootstrapping counts for subgroup extensions of number fields to larger anabelian groups. They can handle a variety of situations: when the group of interest has an abelian normal subgroup, certain wreath products, as well as nilpotent groups whose minimal index elements commute. The technique requires a very delicate analysis of the error terms of the intermediate extensions, geared towards obtaining uniform bounds on these error terms.

Akhtari’s talk was of a somewhat different geometric flavor than most in this workshop, focusing on Diophantine equations that can be addressed using the methods of Diophantine approximation. The focus was the question: how many monogenizations, or approximate monogenizations (in the sense of Siad’s talk) can a number field have? This question is critical for understanding the relationship between the statistics of monogenized fields and the statistics of fields possessing a monogenization.

Newton’s talk concerned the genus group: the part of the class group associated to the maximal unramified extension of K realizable as a compositum with an abelian extension of Q. This is most classically studied in the context of quadratic fields, where the genus group is the whole 2-torsion of the class group and its distribution is very well understood. Newton spoke on her work with Frei and Loughran which computes the distribution of the genus group for a much wider range of abelian extensions. As with classical genus theory, the size of the genus group grows as a power of log; a new wrinkle here is that, as discussed in the talks of Wood and Alberts as well, the order in which you count the fields can make a big difference in the difficulty of the argument.

Vemulapalli talked about her recent work on the shapes of orders in number fields. Her striking work refutes a conjecture of Lenstra’s, then replaces it with one she proves to be correct, showing that, for a large class of n, the successive minima of an order in a degree-n number field lie in a region of R^n delineated by an explicit (albeit complicated) list of equations and inequalities she provides. In the unified story of Batyrev-Manin and Malle conjectures being discussed at this workshop, the successive minima studied by Vemulapalli are quite closely related to the slopes discussed by Peyre; the talk engendered a spirited discussion of the extent to which Vemulapalli’s work could be extended to the case where only maximal orders are considered, which is most relevant to the proposed unified conjectures.

In closing, Thorne talked about using Poisson summation to count objects of interest in number theory.

Additionally, there were two problem sessions whose goal was to formulate the short- and medium-term goals in the field of arithmetic statistics and counting points on stacks, especially taking into account the parallels between the two points of view.

A few themes which were agreed to be important directions for the future:

• Arithmetic statistics over global fields in “bad” characteristics. We still don’t have a clear picture of what Malle’s conjecture should say about G-extensions of K in cases where the characteristic of K divides |G|; similarly, we don’t have a clear picture about how inseparable phenomena affect what the Batyrev-Manin conjecture should say about K-points on varieties when K is a global field in low characteristic. The important geometric advances of Lehmann-Sengupta-Tanimoto involve characteristic 0 algebraic geometry in critical ways and it remains to be understood how and whether things change in positive characteristic.
• A specific question in this direction: how far are we from proving the “q goes to infinity” version of the Batyrev-Manin conjectures? That is: if X is a Fano variety over Q and N(q,X,b) is the number of points on X over F_q(t) of height at most qb, excluding some appropriate locus, can one prove theorems about the behavior of N(q,X,b) as q goes to infinity with b fixed? Theorems in this genre have a long history on the arithmetic statistics side (Achter on q -> infinity Cohen-Lenstra and the recent paper of Wood, Yuan, and Zureick-Brown on q -> infinity for nonabelian Cohen-Lenstra.)
• There was a general flavor of “count by all possible invariants” in the talks of Thorne, Gundlach, Wood, Peyre and Vemulapallli. This might mean counting by different types of ramified primes separately (Thorne, Wood, Gundlach) or by something along the lines of “slope” or “shape” (Peyre, Vemulapalli). We are still not in possession of a general philosophy of counting that incorporates all these invariants and makes hypotheses about their interrelations.
• Agenda

#### MONDAY, AUGUST 29

 10:00 - 11:00 AM Melanie Matchett Wood | What's Happening in Arithmetic Statistics 11:30 - 12:30 PM Jordan Ellenberg | Why is there Geometry in Arithmetic Statistics? 5:00 - 6:00 PM Problem session 6:15 - 7:15 PM Ila Varma | The Future of Arithmetic Statistics: Who has Access?

#### TUESDAY, AUGUST 30

 10:00 - 11:00 AM Tim Browning | A Motivic Circle Method 11:30 - 12:30 PM Damaris Schindler | Density of Rational Points Near Certain Manifolds 5:00 - 6:00 PM Emmanuel Peyre | About Slopes 6:15 - 7:15 PM Fabian Gundlach | Counting Galois Extensions

#### WEDNESDAY, AUGUST 31

 5:00 - 6:00 PM Artane Siad | Even Degree Fields with Prescribed 2Cl[2∞] 6:15 - 7:15 PM Alex Smith | 2^k-Selmer Groups and Goldfeld's Conjecture

#### THURSDAY, SEPTEMBER 1

 10:00 - 11:00 AM Sho Tanimoto | From Exceptional Sets to Non-Free Curves 11:30 - 12:30 PM Takehiko Yasuda | Another Height Function for Deligne-Mumford Stacks via Twisted Sectors 5:00 - 6:00 PM Brandon Alberts | Inductively Counting Number Fields 6:15 - 7:15 PM Shabnam Akhtari | Monogenic Orders in Cubic and Quartic Number Fields and Classical Diophantine Equations

#### FRIDAY, SEPTEMBER 2

 10:00 - 11:00 AM Rachel Newton | Distribution of Genus Numbers of Abelian Number Fields 11:30 - 12:30 PM Problem session 5:00 - 6:00 PM Sameera Vemulapalli | Bounds on Successive Minima of Orders in Number Fields and Scrollar Invariants of Curves 6:15 - 7:15 PM Frank Thorne | The Geometry of Equidistribution
• Abstracts & Slides

Shabnam Akhtari
University of Oregon

Monogenic Orders in Cubic and Quartic Number Fields and Classical Diophantine Equations

An order $$\mathcal O$$ in an algebraic number field is called monogenic if it can be generated by one element over $$\mathbb Z$$. Since $$\mathbb Z[\alpha] = \mathbb Z[±\alpha + c]$$, for any integer $$c$$, we call two algebraic integers $$\alpha$$ and $$\alpha’$$ equivalent if $$\alpha + \alpha’$$ or $$\alpha – \alpha’$$ is a rational integer. By a monogenization of $$\mathcal O$$, we mean an equivalence class of monogenizers of $$\mathcal O$$. Győry has shown that there are finitely many monogenizations for a given order. An interesting (and open) problem is to count the number of monogenizations of a given monogenic order. First we will note, for a given order $$\mathcal O$$, that $$\mathcal O = \mathbb Z[\alpha] \hspace{15pt} \text{in} \enspace \alpha,$$ is indeed a Diophantine equation, namely an index form equation. Then we will modify some algorithmic approaches for finding solutions of index form equations in cubic and quartic number fields to obtain new and improved upper bounds for the number of monogenizations of a cubic or quartic order.

Brandon Alberts
University of California San Diego

Inductively Counting Number Fields

Brandon Alberts will discuss a method for counting G-extensions ordered by discriminant by partitioning the set of extensions with respect to a normal subgroup of G. We will focus on the necessary ingredients for this method, in particular when the normal subgroup is either abelian or generated by the conjugates of an S₃ subgroup. This is joint work with Robert Lemke Oliver, Jiuya Wang, and Melanie Matchett Wood.

Timothy Browning
IST Austria

A Motivic Circle Method

The circle method has been a versatile too in the study of rational points on hypersurfaces. More recently, a version of the method over function fields, combined with spreading out techniques, has led to a range of results about moduli spaces of rational curves on hypersurfaces. I will report on joint work (in progress) with Margaret Bilu on implementing a circle method with an even more geometric flavour, where the computations take place in a suitable Grothendieck ring of varieties. We establish analogues for the key steps of the method, which enable us to approximate the classes of the above moduli spaces directly without relying on point counting.

Jordan Ellenberg

Why is there Geometry in Arithmetic Statistics?

Continuing from Wood’s talk, Jordan Ellenberg will try to give a short introduction on the main themes of the week, with a special emphasis on the analogy between number fields and function fields and how it connects natural questions about arithmetic statistics to geometric questions about moduli spaces.

Fabian Gundlach
Harvard University

Counting Galois Extensions

Fabian Gundlach will present the problem of counting Galois extensions by all Artin conductors at once, instead of just by one invariant. Gundlach will provide an example for Galois extensions of Q whose Galois group is the quaternion group.

Melanie Matchett Wood
Harvard University

What’s Happening in Arithmetic Statistics
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Melanie Matchett Wood will give an overview of the major problems in arithmetic statistics, including asymptotics of number fields, class groups, elliptic curves. Matchett Wood will outline recent developments and new directions, as well as suggest open problems that will expand the frontiers of our current knowledge.

Rachel Newton
King’s College London

Distribution of Genus Numbers of Abelian Number Fields

Let K be a number field and let L/K be an abelian extension. The genus field of L/K is the largest extension of L which is unramified at all places of L and abelian as an extension of K. The genus group is its Galois group over L, which is a quotient of the class group of L, and the genus number is the size of the genus group. We study the distribution of genus numbers as one varies over abelian extensions L/K with fixed Galois group. This is joint work with Christopher Frei and Daniel Loughran.

Emmanuel Peyre
Université Grenoble-Alpes

View Slides (PDF)

Emmanuel Peyre will present a survey on the information provided by slopes for rational points of varieties and the limitations of this information.

Damaris Schindler
Goettingen University

Density of Rational Points Near Certain Manifolds

Abstract: Given a bounded submanifold M in Rⁿ, how many rational points with common bounded denominator are there in a small thickening of M? Under what conditions can we count them asymptotically as the size of the denominator goes to infinity? We discuss some recent work of Huang in this direction as well as a generalization for certain families of manifolds in higher codimension in work of the speaker and Shuntaro Yamagishi. We show that under certain curvature conditions we obtain stronger results for manifolds in higher codimension than for hypersurfaces, and discuss relations to Serre’s dimension growth conjecture.

Princeton University

Even degree fields with prescribed $$2Cl[2^\infty]$$

For an extension of number fields K/F, the relative class group of K over F is the kernel in Cl(K) of the norm map from Cl(K) to Cl(F). For good primes p, the Cohen–Lenstra–Martinet–Malle heuristics offer predictions for the distribution of the p-primary part of the relative class group as K varies in extensions, of a given degree and signature, of a fixed base F. In this talk, Artane Siad will explain how to bound the average number of 2-torsion elements in the relative class group of monogenised extensions, of a given degree and signature, of a number field F and discuss consequences of this result to the existence of fields of high degree with prescribed $$2Cl[2^\infty]$$.

Alexander Smith
Stanford University

2ᵏ-Selmer Groups and Goldfeld’s Conjecture

Take E to be an elliptic curve over a number field whose four torsion obeys certain technical conditions. In this talk, Alexander Smith will outline a proof that 100% of the quadratic twists of E have rank at most one. To do this, he will find the distribution of 2ᵏ-Selmer ranks in this family for every positive integer k.

Sho Tanimoto
Nagoya University

From Exceptional Sets to Non-Free Curves

Manin’s conjecture predicts the asymptotic formula for the counting function of rational points on a smooth Fano variety after removing the contribution of exceptional sets. With Brian Lehmann and Akash Sengupta, Sho Tanimoto studied birational geometry of exceptional sets and proved that these are thin sets in the sense of Serre after providing geometric definitions of exceptional sets conjecturally. In this talk, Tanimoto will spend the first half of his talk discussing the exceptional sets in Manin’s conjecture. The second half he will overview recent joint work with Brian Lehmann and Eric Riedl on the structures of non-free curves on Fano varieties. This is joint work with Lehmann-Sengupta and Lehmann-Riedl.

Frank Thorne
University of South Carolina

The Geometry of Equidistribution
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Many problems in arithmetic statistics involve the following technical question: count lattice points in some vector space, up to equivalence given by some group action, satisfying a finite set of congruence conditions. One obtains the best error terms if these congruence conditions are equidistributed, and this can be measured in terms of Fourier transforms.

Often, these Fourier transforms can be studied in terms of the geometry of the underlying group action. I will give a survey of some past, ongoing, and prospective work investigating and applying these geometric connections.

Sameera Vemulapalli
Princeton University

Bounds on Successive Minima of Orders in Number Fields and Scrollar Invariants of Curves

Let $$n \geq 2$$ be any integer. A number field $$K$$ of degree $$n$$ with embeddings $$\sigma_1,\dots,\sigma_{n}$$ into $$\mathbb C$$ has a norm given by $$| x | = \sqrt{\frac{1}{n}\sum_{i = 1}^{n}| \sigma_i(x)|^2}$$ for $$x \in K$$. With respect to this norm, an order $$\mathcal{O}$$ in a number field of degree $$n$$ has successive minima $$1=\lambda_0 \leq \dots \leq \lambda_{n-1}$$. Motivated by a conjecture of Lenstra, we aim to determine which inequalities of the form $$1 \ll_{\mathcal{S}} \lambda_1^{f_1}\dots\lambda_{n-1}^{f_{n-1}}$$ for $$f_1,\dots,f_{n-1} \in \mathbb R$$ hold for an infinite set $$\mathcal{S}$$ of isomorphism classes of orders in degree $$n$$ fields. In many cases — such as when $$\mathcal{S}$$ is the set of isomorphism classes of orders in degree $$n$$ fields or $$\mathcal{S}$$ is the set of isomorphism classes of orders in degree $$n$$ fields with fixed subfield degrees — we provide a complete classification of which inequalities hold. Moreover, when $$n < 18$$, $$n$$ is a prime power, or $$n$$ is a product of $$2$$ distinct primes, we asymptotically describe all tuples $$(\lambda_1,\dots,\lambda_{n-1})$$ that occur as successive minima of an order in a degree $$n$$ number field. We also prove analogous theorems about successive minima of ideals in number fields and scrollar invariants of curves.

Takehiko Yasuda
Nagoya University

Another Height Function for Deligne-Mumford Stacks via Twisted Sectors
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Takehiko Yasuda will introduce a new height function for Deligne-Mumford stacks. The main ingredient in its definition is twisted sectors, that is, connected components of the stack of twisted 0-jets (also known as the cyclotomic inertia stack). The new height function generalizes the height function associated to a vector bundle introduced by Ellenberg–Satriano–Zureick-Brown and the height function on weighted projective stacks considered by Darda as well as a generalized discriminant by Ellenberg–Venkatesh when specialized to the case of classifying stacks. Yasuda will also discuss how to unify conjectures of Manin, Batyrev–Manin and Malle by using our new height function and the so-called a- and b-invariants which are properly generalized to stacks. This is a joint work with Ratko Darda.

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