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The Joint Athens-Atlanta Number Theory Seminar meets once a semester,
usually on a Tuesday, with two talks back to back, at 4:00 and at 5:15.
Participants then go to dinner together.
Fall 2018
Tuesday October 23, 2018, at Emory, in TBA
First talk at
4:00 by Bianca Viray (University of Washington)
TBA
TBA
Second talk at 5:15 by Larry Rolen
(Vanderbilt)
TBA
TBA
Spring 2018
Tuesday February 20, 2018, at UGA, in Boyd Room 304
First talk at 4:00
by David Harbater (University
of Pennsylvania)
Local-global principles for zero-cycles
over semi-global fields
Classical local-global principles are given over global fields. This talk will discuss such principles
over semi-global fields, which are function fields of curves defined over a
complete discretely valued field.
Paralleling a result that Y. Liang proved over number fields, we prove a
local-global principle for zero-cycles on varieties over semi-global
fields. This builds on earlier work
about local-global principles for rational points. (Joint work with J.-L. Colliot-Thélène, J. Hartmann, D. Krashen, R. Parimala, J. Suresh.)
Second talk at 5:15 by Jacob Tsimerman (U. Toronto)
Cohen-Lenstra heuristics in the Presence of Roots of Unity
The class
group is a natural abelian group one can associated to a number field, and it
is natural to ask how it varies in families. Cohen and Lenstra
famously proposed a model for families of quadratic fields based on random
matrices of large rank, and this was later generalized by Cohen-Martinet to
general number fields. However, their model was observed by Malle
to have issues when the base field contains roots of unity. We explain that in
this setting there are naturally defined additional invariants on the class
group, and based on this we propose a refined model in the number field setting
rooted in random matrix theory. Our conjecture is based on keeping track not
only of the underlying group structure, but also certain natural pairings one
can define in the presence of roots of unity. Specifically, if the base field
contains roots of unity, we keep track of the class group G together with a
naturally defined homomorphism G*[n] --> G from the n-torsion of the Pontryagin dual of G to G. Using methods of Ellenberg-Venkatesh-Westerland, we can prove some of our
conjecture in the function field setting.
Fall 2017
Monday October 30, 2017, at Georgia Tech.
First talk at
4:00 by Bjorn Poonen (MIT)
Gonality and the strong uniform boundedness
conjecture for periodic points
The function field case of the strong uniform boundedness conjecture for
torsion points on elliptic curves reduces to showing that classical modular
curves have gonality tending to infinity. We prove an
analogue for periodic points of polynomials under iteration by studying the
geometry of analogous curves called dynatomic curves.
This is joint work with John R. Doyle.
Second talk at 5:15 by Spencer Bloch (U. Chicago)
Periods, motivic Gamma functions, and
Hodge structures
Golyshev and Zagier found an interesting new source of periods
associated to (eventually inhomogeneous) solutions generated by the Frobenius method for Picard Fuchs equations in the
neighborhood of singular points with maximum unipotent
monodromy. I will explain how this works, and how one
can associate "motivic Gamma functions" and generalized Beilinson style variations of mixed Hodge structure to
these solutions. This is joint work with M. Vlasenko.
Spring 2017
Tuesday April 18, 2017, at Emory.
First talk at
4:00 by Rachel Pries (Colorado State)
Galois
action on homology of Fermat curves
We prove a result about the Galois module structure of the Fermat curve
using commutative algebra, number theory, and algebraic topology.
Specifically, we extend work of Anderson about the action of the absolute
Galois group of a cyclotomic field on a relative
homology group of the Fermat curve. By finding explicit formulae for this
action, we determine the maps between several Galois cohomology
groups which arise in connection with obstructions for rational points on the
generalized Jacobian. Heisenberg extensions play a key role in the
result. This is joint work with R. Davis, V. Stojanoska,
and K. Wickelgren.
Second talk at 5:15 by Gopal Prasad (U. Michigan)
Weakly
commensurable Zariski-dense subgroups of semi-simple groups and isospectral locally symmetric spaces
I will discuss the notion of weak
commensurability of Zariski-dense subgroups of
semi-simple groups. This notion was introduced in my joint work with Andrei Rapinchuk (Publ. Math. IHES 109(2009), 113-184), where we determined
when two Zariski-dense S-arithmetic subgroups of absolutely almost simple
algebraic groups over a field of characteristic zero can be weakly
commensurable. These results enabled us to prove that in many situations isospectral locally symmetric spaces of simple real
algebraic groups are necessarily commensurable. This settled the famous
question "Can one hear the shape of a drum?" of Mark Kac for these spaces. The arguments use algebraic and
transcendental number theory.
Fall 2016
Tuesday October 18, 2016, at UGA.
First talk at 4:00 by Florian Pop (University of Pennsylvania)
Local
section conjectures and Artin-Schreier theorems
After a short introduction to (Grothendieck's) section conjecture (SC), I will explain how
the classical Artin Schreier
Thm and its p-adic analog
imply the birational local SC; further, I will mention
briefly local-global aspects of the birational SC.
Finally, I will give an effective "minimalistic" p-adic Artin-Schreier Thm, which is similar in flavor to the classical Artin-Schreier Thm.
Second talk at 5:15 by Ben Bakker (UGA)
Recovering
elliptic curves from their p-torsion
Given an elliptic curve E over the rationals Q, its p-torsion E[p] gives a 2-dimensional
representation of the Galois group G_Q over F_p. The Frey-Mazur conjecture asserts that
for p>17, this representation is essentially a complete invariant: E is determined up to isogeny by
E[p]. In joint work with J. Tsimerman, we prove the analog of the Frey-Mazur conjecture
over characteristic 0 function fields.
The proof uses the hyperbolic repulsion of special subvarieties
in a modular surface to show that families of elliptic curves with many isogenous fibers have large volume. We will also explain how these ideas
relate to other uniformity conjectures about the size of monodromy
representations
Spring 2016
Thursday April 14, 2016, at Georgia Tech.
First talk at 4:00
by Melanie Matchett-Wood (University of Wisconsin)
Nonabelian Cohen-Lenstra
Heuristics and Function Field Theorems
The Cohen-Lenstra
Heuristics conjecturally give the distribution of class groups of imaginary
quadratic fields. Since, by class field theory, the class group is the Galois
group of the maximal unramified abelian extension, we
can consider the Galois group of the maximal unramified
extension as a non-abelian generalization of the class group. We will explain
non-abelian analogs of the Cohen-Lenstra heuristics
due to Boston, Bush, and Hajir and joint work with
Boston proving cases of the non-abelian conjectures in the function field
analog.
Second talk at 5:15 by Zhiwei Yun (Stanford University)
Intersection
numbers and higher derivatives of L-functions for function fields
In joint work with Wei Zhang, we
prove a higher derivative analogue of the Waldspurger
formula and the Gross-Zagier formula in the function
field setting under the assumption that the relevant objects are everywhere unramified. Our formula relates the self-intersection
number of certain cycles on the moduli of Shtukas for
GL(2) to higher derivatives of automorphic L-functions
for GL(2).
Fall 2015
Tuesday November 17, 2015, at Emory.
First talk at 4:00 by John Duncan (Emory)
K3
Surfaces, Mock Modular Forms and the Conway Group
In their famous “Monstrous
Moonshine” paper of 1979, Conway—Norton also described an
association of modular functions to the automorphism
group of the Leech lattice (a.k.a. Conway’s group). In analogy with the
monstrous case, there is a distinguished vertex operator superalgebra
that realizes these functions explicitly. More recently, it has come to light
that this Conway moonshine module may be used to compute equivariant
enumerative invariants of K3 surfaces. Conjecturally, all such invariants can
be computed in this way. The construction attaches explicitly computable mock
modular forms to automorphisms of K3 surfaces.
One expects the Brauer-Manin
obstruction to control rational points on 1-parameter families of conics and
quadrics over a number field when the base curve has genus 0. Results in this
direction have recently been obtained as a consequence of progress inanalytic number theory. On the other hand, it is easy to
construct a family of 2-dimensional quadrics over a curve with just one
rational point over Q, which is a counterexample to the Hasse
principle not detected by the \'etale Brauer-Manin obstruction. Conic bundles with similar
properties exist over real quadratic fields, though most certainly not over Q.
Second talk at 5:15 by Alexei Skorobogatov (Imperial
College)
Local-to-global
principle for rational points on conic and quadric bundles over curves
One expects the Brauer-Manin
obstruction to control rational points on 1-parameter families of conics and
quadrics over a number field when the base curve has genus 0. Results in this
direction have recently been obtained as a consequence of progress inanalytic number theory. On the other hand, it is easy to
construct a family of 2-dimensional quadrics over a curve with just one
rational point over Q, which is a counterexample to the Hasse
principle not detected by the \'etale Brauer-Manin obstruction. Conic bundles with similar
properties exist over real quadratic fields, though most certainly not over Q.
Spring 2015
Thursday April 9, 2015, at UGA, Room 304 in Boyd Graduate Studies Building.
First talk at 4:00 by Dick Gross (Harvard)
Pencils of
quadrics
Quadric hypersurfaces, defined by
homogeneous equations of degree 2, are the simplest projective varieties other
than linear subspaces. In this talk I will review the theory of quadratic forms
over a general field, and discuss the smooth intersection of two quadric
hypersurfaces in projective space. The Fano scheme of
maximal linear subspaces contained in this intersection is either finite or is
a principal homogeneous space for the Jacobian of a hyperelliptic
curve. This gives an important tool
for the arithmetic study of these curves.
Second talk at 5:15 by Ted Chinburg
(University of Pennsylvania)
When is an
error term not really an error term?
The classical case of Iwasawa
theory has to do with how quickly the p-parts of the ideal class groups of
number fields grow in certain towers of number fields. I will discuss the
connection of these growth rates to Chern classes,
and how the "error terms" in various formulas have to do with higher Chern classes. I will then describe a result linking second
Chern classes in Iwasawa
theory over imaginary quadratic fields to pairs of p-adic
L-functions. This is joint work
with F. Bleher, R. Greenberg, M. Kakde,
G. Pappas, R Sharifi and M. Taylor.
Fall 2014
Tuesday, November 4, 2014, at Georgia Tech.
First talk at 4:00 by Arul Shankar (Harvard University)
Geometry-of-numbers
methods over number fields
We discuss the necessary modifications
required to apply Bhargava's geometry-of-numbers methods to representations
over number fields. As an application, we derive upper bounds on the average
rank of elliptic curves over any number field. This is joint work with Manjul Bhargava and Xiaoheng
Jerry Wang.
Second talk at 5:15 by Wei Zhang (Columbia University)
Kolyvagin's conjecture on Heegner
points
We recall a conjecture of Kolyvagin on Heegner points for
elliptic curves of arbitrary analytic rank, and present some recent results on
this conjecture for elliptic curves satisfying some technical conditions.
Spring 2014
Tuesday, February 25, 2014, at Emory.
First talk at 4:00 by Paul Pollack (UGA)
Solved and
unsolved problems in elementary number theory
This will be a survey of certain
easy-to-understand problems in elementary number theory about which "not
enough" is known. We will start with a discussion of the infinitude of
primes, then discuss the ancient concept of perfect numbers (and related notions),
and then branch off into other realms as the spirit of Paul Erdös
leads us.
Second talk at 5:15 by James Maynard (Université
de Montréal)
Bounded gaps between primes
It
is believed that there should be infinitely many pairs of primes which differ
by 2; this is the famous twin prime conjecture. More generally, it is believed
that for every positive integer $m$ there should be infinitely many sets of $m$
primes, with each set contained in an interval of size roughly $m\log{m}$. We
will introduce a refinement of the `GPY sieve method' for studying these
problems. This refinement will allow us to show (amongst other things) that $\liminf_n(p_{n+m}-p_n)<\infty$ for any integer $m$, and so there are infinitely
many bounded length intervals containing $m$ primes.
Fall 2013
Tuesday, November 5, 2013, at UGA, Room 303 in Boyd Graduate Studies
Building.
First talk at 4:00 by Joe Rabinoff (Georgia
Tech)
Lifting
covers of metrized complexes to covers of curves
Let K be a complete and algebraically
closed non-Archimedean field and let X be a smooth K-curve. Its Berkovich analytification X^an
deformation retracts onto a metric graph Gamma, called a skeleton of X^an. The
collection of different skeleta of X are in natural
bijective correspondence with the collection of semistable
models of X over the valuation ring R of K. We prove that, given a finite morphism
of curves f: Y -> X, there exists a skeleton Gamma_X
of X^an whose inverse image
is a skeleton Gamma_Y of Y^an. This can be seen as a
"skeletal" simultaneous semistable
reduction theorem, which can in fact be used to give new, more general proofs
of foundational results of Liu, Liu-Lorenzini, and
Coleman on simultaneous semistable reductions. We then consider the following problem:
given X, a skeleton Gamma_X, and a finite harmonic
morphism of metric graphs Gamma' -> Gamma_X, can
we find a curve Y and a finite morphism f: Y -> X such that f^{-1}(Gamma_X) is a skeleton of Y^an and is isomorphic to Gamma'? In general the answer is no: one must
enrich the skeleton with the structure of a metrized
complex of curves. In this context
the answer is yes, and moreover the map Gamma' -> Gamma_X can be used to calculate the finitely many
isomorphism classes of Y -> X, as well as their automorphisms. We give an application to component
groups of Jacobians, answering a question of Ribet.
Second talk at 5:15 by Kirsten Wickelgren
(Georgia Tech)
Splitting
varieties for triple Massey products in Galois cohomology
The Brauer-Severi
variety a x^2 + b y^2 = z^2 has a rational point if and only if the cup product
of cohomology classes associated to a and b vanish. The cup product is the order-2 Massey
product. Higher Massey products give further structure to Galois cohomology, and more generally, they measure information
carried in a differential graded algebra which can be lost on passing to the associated
cohomology ring. For example, the cohomology
of the Borromean rings is isomorphic to that of three unlinked circles, but
non-trivial Massey products of elements of H^1 detect the more complicated
structure of the Borromean rings. Analogues of this example exist in Galois cohomology due to work of Morishita,
Vogel, and others. This talk will first introduce Massey products and some
relationships with non-abelian cohomology. We will
then show that b x^2 = (y_1^2 - ay_2^2 + c y_3^2 - ac y_4^2)^2 - c(2 y_1 y_3 -
2 a y_2 y_4)^2 is a splitting variety for the triple Massey product <a,b,c>, and that this variety satisfies the Hasse principle. The method could produce splitting
varieties for higher order Massey products. It follows that all triple Massey products
over global fields vanish when they are defined. More generally, one can show
this vanishing over any field of characteristic different from 2; Jan Minac and Nguyen Duy Tan, and
independently Suresh Venapally, found an explicit
rational point on X(a,b,c). Minac
and Tan have other nice results in this direction. This is joint work with Michael Hopkins.
Spring 2013
Tuesday, April 16, 2013, at Georgia Tech
First talk at 4:00 by Dick Gross (Harvard)
The
arithmetic of hyperelliptic curves
Hyperelliptic curves over Q have equations of the form y^2 = F(x),
where F(x) is a polynomial with rational coefficients which has simple roots
over the complex numbers. When the degree of F(x) is at least 5, the genus of
the hyperelliptic curve is at least 2 and Faltings has proved that there are only finitely many
rational solutions. In this talk, I will describe methods which Manjul Bhargava and I have developed to quantify this
result, on average.
Second talk at 5:15 by Jordan Ellenberg
(Wisconsin)
Arithmetic
statistics over function fields
What is the probability that
a random integer is squarefree? Prime? How many
number fields of degree d are there with discriminant at most X? What does the
class group of a random quadratic field look like? These questions, and
many more like them, are part of the very active subject of arithmetic
statistics. Many aspects of the subject are well-understood, but many more
remain the subject of conjectures, by Cohen-Lenstra, Malle, Bhargava, Batyrev-Manin,
and others. In this talk, I explain what arithmetic statistics looks like when
we start from the field Fq(x)
of rational functions over a finite field instead of the field Q of rational
numbers. The analogy between function fields and
number fields has been a rich source of insights throughout the modern
history of number theory. In this setting, the analogy reveals a surprising
relationship between conjectures in number theory and conjectures in topology
about stable cohomology of moduli spaces, especially
spaces related to Artin's braid group. I will discuss
some recent work in this area, in which new theorems about the topology of
moduli spaces lead to proofs of arithmetic conjectures over function fields,
and to new, topologically motivated questions about counting arithmetic
objects.
Fall 2012
Thursday, October 25, 2012, at Emory.
First talk at 4:00 by Karl Rubin (UCI)
Ranks of
elliptic curves
I will discuss some recent
conjectures and results on the distribution of Mordell-Weil
ranks and Selmer ranks of elliptic curves. After some general background, I
will specialize to families of quadratic twists, and describe some recent
results in detail.
Second talk at 5:15 by Jayce Getz (Duke)
An
approach to nonsolvable base change for GL(2)
Motivated by Langlands' beyond endoscopy idea, the speaker will present
a conjectural trace identity that is essentially equivalent to base change and
descent of automorphic representations of GL(2) along
a nonsolvable extension of fields.
Spring 2012
Tuesday, April 10, 2012, at UGA, Room 323 in Boyd Graduate Studies Building
First talk at 4:00 by Max Lieblich (Univ. of
Washington)
Finiteness
of K3 surfaces and the Tate conjecture
Fix a finite field k. It is
well known that there are only finitely many smooth projective curves of a
given genus over k. It turns out that there are also only a finite number of
abelian varieties of a given dimension over k. What about other classes of
varieties? I will review the history of these results and describe joint work
with Maulik and Snowden that links the finiteness of
K3 surfaces over k to the Tate conjecture for K3 surfaces over k. The key is a
link between certain lattices in the l-adic cohomology of K3 surfaces and derived categories of sheaves
on certain algebraic stacks. I will not assume you know anything about any of
this.
Second talk at 5:15 by Frank Calegari
(Northwestern Univ.)
Even
Galois Representations
What Galois representations
"come" from algebraic geometry? The Fontaine-Mazur conjecture gives a
very precise conjectural answer to this question. A simplified version of this conjecture
in the case of two dimensional representations says that "all nice
representations come from modular forms". Yet, by construction, all
representations coming from modular forms are "odd", that is, complex
conjugation acts by a 2x2 matrix of determinant -1. What happened to all the
even Galois representations?
Fall 2011
Wednesday, November 2, 2011, at Georgia Tech in Skiles room 005 (ground
floor).
First talk at 4:00 by Jared Weinstein (IAS)
Maximal
varieties over finite fields
This is joint work with Mitya Boyarchenko. We
construct a special hypersurface X over a finite field, which has the property
of "maximality", meaning that it has the
maximum number of rational points relative to its topology. Our variety
is derived from a certain unipotent algebraic group,
in an analogous manner as Deligne-Lusztig varieties
are derived from reductive algebraic groups. As a consequence, the cohomology of X can be shown to realize a piece of the
local Langlands correspondence for certain wild Weil
parameters of low conductor.
Second talk at 5:15 by David Brown (Emory)
Random Dieudonne modules and the Cohen-Lenstra
conjectures.
Knowledge of the distribution of class groups is
elusive -- it is not even known if there are infinitely many number fields with
trivial class group. Cohen and Lenstra noticed a
strange pattern --experimentally, the group $\mathbb{Z}/(9)$
appears more often than $\mathbb{Z{/(3) \times \mathbb{Z}/(3)$ as the 3-part of the classgroup
of a real quadratic field $\Q(\sqrt{d})$ - and refined
this observation into concise conjectures on the manner in which class groups
behave randomly. Their heuristic says roughly that $p$-parts of class groups
behave like random finite abelian $p$-groups, rather than like random numbers;
in particular, when counting one should weight by the size of the automorphism group, which explains why $\mathbb{Z}/(3) \times \mathbb{Z}/(3)$
appears much less often than $\mathbb{Z}/(9)$ (in
addition to many other experimental observations).
While proof of the Cohen-Lenstra conjectures remains
inaccessible, the function field analogue -- e.g., distribution of class groups
of quadratic extensions of \mathbb{F}_p(t) -- is more
tractable. Friedman and Washington modeled the $\ell$-power part
(with $\ell \neq p) of such class groups as
random matrices and derived heuristics which agree with experiment. Later, Achter refined these heuristics, and many cases have been
proved (Achter, Ellenberg
and Venkatesh).
When $\ell = p$, the $\ell$-power torsion of abelian varieties, and thus the
random matrix model, goes haywire. I will explain the correct linear algebraic
model -- Dieudone\'e modules. Our main result is an
analogue of the Cohen-Lenstra/Friedman-Washington
heuristics – a theorem about the distributions of class numbers of Dieudone\'e modules (and other invariants particular to
$\ell = p$). Finally, I'll present experimental evidence which mostly agrees
with our heuristics and explain the connection with rational points on
varieties.
Spring 2011
Tuesday, February 1, 2011, at Emory
First talk at 4:00 by K. Soundararajan
(Stanford)
Moments of
zeta and L-functions
An important theme in number
theory is to understand the values taken by the Riemann zeta-function and
related L-functions. While much progress has been made, many of the basic
questions remain unanswered. I will discuss what is known about this question,
explaining in particular the work of Selberg, random
matrix theory and the moment conjectures of Keating and Snaith,
and recent progress towards estimating the moments of zeta and L-functions.
Second talk at 5:15 by Matthew Baker (Georgia Institute of Technology)
Complex
dynamics and adelic potential theory
I will discuss the following theorem: for any fixed
complex numbers a and b, the set of complex numbers c for which both a and
b both have finite orbit under iteration of the map z -->z^2 + c
is infinite if and only if a^2 = b^2. I will explain the motivation
for this result and give an outline of the proof. The main
arithmetic ingredient in the proof is an adelic equidistribution theorem for preperiodic
points over product formula fields, with non-archimedean Berkovich spaces playing an essential role. This is joint
work with Laura DeMarco, relying on earlier joint work with Robert Rumely.
Fall 2010
Tuesday, September 21, 2010, at UGA
First talk at 4:00 by Ken Ono (Emory)
Mock
modular periods and L-values
Recent works have shed light on the enigmatic mock
theta functions of Ramanujan. These strange power
series are now known to be pieces of special "harmonic" Maass forms. The speaker will discuss recent joint work in
the subject with regard to special values of L-functions. This will include the
study of values and derivatives of elliptic curve L-functions, as well as
general critical values of modular L-functions. In addition, the speaker will
derive new Eichler-Shimura isomorphisms,
and will derive new relations among the "even" periods of modular
L-functions. This is joint work with Jan Bruinier,
Kathrin Bringmann, Zach Kent, and Pavel Guerzhoy.
Second talk at 5:15 by Armand Brumer
(Fordham)
Abelian
Surfaces and Siegel Paramodular Forms
This expository talk will survey recent progress on
modularity of abelian surfaces. After a brief review of the history, I'll
describe work of Cris Poor and David Yuen on the
modular side and Ken Kramer and me on the arithmetic side.
Spring
2010
Tuesday, April 13, 2010
First talk at
4:00 by Venapally Suresh (Emory)
Degree three cohomology of function fields of
surfaces
Let
k be a global field or a local field. Class field theory says that every
central division algebra over k is cyclic. Let l be a prime not equal to the
characteristic of k. If k contains a primitive l-th
root of unity, then this leads to the fact that every element in H^2(k,
µ_l ) is a symbol. A natural question is a
higher dimensional analogue of this result: Let F be a function field in one
variable over k which contains a primitive l-th root
of unity. Is every element in H^3(F, µ_l ) a
symbol? In this talk we answer this question in affirmative for k a p-adic field or a global field of positive characteristic.
The main tool is a certain local global principle for elements of H^3(F,
µ_l ) in terms of symbols in H^2(F µ_l ).
We also show that this local-global principle is equivalent to the vanishing of
certain unramified cohomology
groups of 3-folds over finite fields.
Second talk at
5:15 by Antoine Chambert-Loir (IAS and University of Rennes)
Some applications of potential theory to number theoretical problems on
analytic curves
Slides available at http://perso.univ-rennes1.fr/antoine.chambert-loir/publications/pdf/atlanta2010.pdf
Fall
2009
Tuesday, October 20, 2009
First talk at 4:00 by Doug Ulmer (GA Tech).
Constructing elliptic curves of
high rank over function fields
There
are now several constructions of elliptic curves of high rank over function
fields, most involving high-tech things like L- functions, cohomology,
and the Tate or BSD conjectures. I'll review some of this and then give a very
down-to-earth, low-tech construction of elliptic curves of high ranks over the
rational function field Fp(t).
Second talk at 5:15 by Jonathan Hanke (UGA).
Using Mass formulas to Enumerate Definite
Quadratic Forms of Class Number One
This talk will describe some recent results using exact
mass formulas to determine all definite quadratic forms of small class number
in n>=3 variables, particularly those of class number one. The mass of a
quadratic form connects the class number (i.e. number of classes in the genus)
of a quadratic form with the volume of its adelic
stabilizer, and is explicitly computable in terms of special values of zeta
functions. Comparing this with known results about the sizes of automorphism groups, one can make precise statements about
the growth of the class number, and in principle determine those quadratic
forms of small class number. We will describe some known results about masses
and class numbers (over number fields), then present some new computational
work over the rational numbers, and perhaps over some totally real number
fields.