
Lillian
Pierce
Publications
On a conjecture for \(\ell\)torsion in class groups of number fields: from the perspective of moments
with C. TurnageButterbaugh and M. M. Wood
(submitted)
(abstract)
(arXiv)
It is conjectured that within the class group of any number
field, for every integer \(\ell \geq 1\), the
\(\ell\)torsion subgroup is very small
(in an appropriate sense, relative to the discriminant
of the field).
In nearly all settings, the full strength of this
conjecture remains open,
and even partial progress is limited. Significant recent
progress toward average
versions of the \(\ell\)torsion conjecture has crucially
relied on counts for number fields,
raising interest in how these two types of question
relate. In this paper we make
explicit the quantitative relationships between the
\(\ell\)torsion conjecture and
other wellknown conjectures: the CohenLenstra
heuristics, counts for number
fields of fixed discriminant, counts for number fields
of bounded discriminant
(or related invariants), and counts for elliptic curves
with fixed conductor.
All of these considerations reinforce that we expect the
\(\ell\)torsion conjecture
is true, despite limited progress toward it. Our
perspective focuses
on the relation between pointwise bounds, averages, and
higher moments,
and demonstrates the broad utility of the method of moments.
(hide abstract)
An effective Chebotarev density theorem for families of
number fields,
with an application to \(\ell\)torsion in class groups
with C. TurnageButterbaugh and M. M. Wood
(submitted)
(abstract)
(arXiv)
In this work we prove a new effective Chebotarev density
theorem, independent of GRH, that improves the previously
known unconditional error term and enables the counting of
small primes; this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions.
The new effective Chebotarev theorem is expected to have many
applications, of which we demonstrate two. In particular,
we provide the first nontrivial upper bounds for
\(\ell\)torsion, for all integers \( \ell \geq 1 \), applicable to infinite families of fields of arbitrarily large degree.
(hide abstract)
The Vinogradov Mean Value Theorem [after Wooley, and
Bourgain, Demeter, Guth]
(Bourbaki Seminar, June 2017)
Asterisque, to appear.
(abstract)
(arXiv)
(seminar)
This is the expository essay to accompany my June 2017 Bourbaki seminar
on the resolution of the Vinogradov Mean Value Theorem. 80 pages.
(hide abstract)
Endpoint Sobolev and BV Continuity for Maximal Operators
with coauthors E. Carneiro and J. Madrid
J. Functional Analysis, 273 (2017) 32623294.
(submitted)
On \(\ell\)torsion in class groups of number fields
with coauthors J. Ellenberg and M. Matchett Wood
Algebra and Number Theory, to appear.
(abstract)
(arXiv)
For each integer n, we prove an unconditional upper bound
on the size of the ntorsion subgroup of the class group,
which holds for all but a zerodensity set of field
extensions of the rationals of degree d, for any fixed d=2,3,4,5
(with the additional restriction in the case d=4 that the
field be nonD4). For sufficiently large n (specified
explicitly), these results are as strong as a previously
known bound that is conditional on GRH. As part of our
argument, we develop a probabilistic "Chebyshev sieve,"
and give uniform, powersaving error terms for the
asymptotics of quartic (nonD4) and quintic fields with
chosen splitting types at a finite set of primes. 25 pages.
(hide abstract)
A polynomial Carleson operator along the paraboloid
with coauthor PoLam Yung
Revista Mat. Ibero., to appear.
(abstract)
(arXiv)
We consider a polynomial phase Carleson operator that is
of Radon type, namely it involves integration over a
paraboloid in Euclidean space of dimension at least 3. We
show that this operator is bounded on L^p for p strictly
between 1 and infinity, for certain classes of polynomial phase (in particular with
restrictions on the first and second order terms). 76 pages.
(hide abstract)
Polynomial Carleson operators along monomial curves in the plane
with coauthors Shaoming Guo, Joris Roos, PoLam Yung
Journal of Geometric Analysis, to appear.
(abstract)
(arXiv)
We prove L^p bounds for partial polynomial Carleson operators
along monomial curves in the plane with a phase
polynomial consisting of a single monomial. These
operators are "partial" in the sense that we consider
linearizing stoppingtime functions that depend on only
one of the two ambient variables. A motivation for
studying these partial operators is the curious feature
that, despite their apparent limitations, for certain
combinations of curve and phase, L^2 bounds for partial
operators along curves imply the full strength of the L^2
bound for a onedimensional Carleson operator, and for a
quadratic Carleson operator. 27 pages.
(hide abstract)
Averages and moments associated to class numbers of
imaginary quadratic fields
with coauthor D.R. HeathBrown
Compositio. Math., to appear.
(abstract)
(arXiv)
For any odd prime q, consider the qpart of
the class number of the imaginary quadratic field
with squarefree discriminant d. Nontrivial pointwise upper
bounds are known only for q=3; nontrivial upper bounds
for averages of the qpart have previously been known
only for q=3,5 In this paper we prove nontrivial upper
bounds for the average of the qpart for all primes at
least 7, as well as nontrivial upper bounds for certain
higher moments for all primes at least 3. 25 pages.
(hide abstract)
Representations of integers by systems of three quadratic forms
with coauthors D. Schindler and M. Matchett Wood
Proceedings of the London Mathematical Society (2016) doi: 10.1112/plms/pdw027
(abstract)
(arXiv)
(journal)
We apply the circle method to count simultaneous
representations of "almost any"
tuple of three integers by a system of three quadratic
forms. In particular, we develop a threedimensional
application of Kloostermanstyle cancellation on the minor
arcs; inexplicit geometric criteria for smoothness play a
particularly important role. 66 pages.
(hide abstract)
Burgess bounds for multidimensional short mixed character sums
Journal of Number Theory, 163 (2016) 172210.
(abstract)
(arXiv)
(journal)
This paper applies multidimensional Vinogradov mean value
theorems in order to prove Burgess bounds for
multidimensional short mixed character sums. 33 pages.
(hide
abstract)
Lower bounds for the truncated Hilbert transform
with coauthors R. Alaifari and S. Steinerberger
Revista Mat. Ibero., 32, no. 1 (2016) 23  56.
(abstract)
(arXiv)
(journal)
Given two intervals I and J on the real line,
we ask whether it is possible to reconstruct a
realvalued function supported on I from knowing its Hilbert transform
on the interval J.
When neither interval is fully contained in the other, this problem
has a unique answer (the nullspace is trivial) but is
severely illposed. We isolate the difficulty and show
that by restricting the problem to functions with controlled total variation, reconstruction becomes stable.
(hide abstract)
Burgess bounds for short mixed character sums
with coauthor D.R. HeathBrown
Journal of the London Math. Soc., 91, no. 3 (2015) 693708.
(abstract)
(arXiv)
(journal)
In this paper we introduce Vinogradov mean value theorems in
order to prove Burgess bounds for short mixed character sums.
(hide abstract)
Simultaneous integer values of pairs of quadratic forms
with coauthor D.R. HeathBrown
J. Reine Angew. Math., in press
(abstract)
(arXiv)
(journal)
We prove that a pair of integral quadratic forms in 5 or more
variables will simultaneously represent almost all pairs of integers
that satisfy the necessary local conditions, provided that the forms
satisfy a suitable nonsingularity condition. In particular such
forms simultaneously attain prime values if the obvious local
conditions hold. The proof uses the circle method, and
in particular pioneers a twodimensional version of a Kloosterman
refinement.
(hide abstract)
Counting rational points on smooth cyclic covers
with coauthor D.R. HeathBrown
Journal of Number Theory, 132 (2012), pp. 17411757.
(abstract)
(pdf)
(arXiv)
(journal)
A conjecture of Serre concerns the number of rational
points of bounded height on a finite cover of (n1)dimensional projective
space. In this paper, we achieve Serre's conjecture in the
special case of smooth cyclic covers of any degree when n
is at least 10 and surpass it for covers of degree at
least 3 when n is at least 11. This is achieved by a new bound for the number of perfect rth power values of a polynomial with nonsingular leading form, obtained via a combination of an rth power sieve and the qanalogue of van der Corput's method.
(hide abstract)
On a discrete version of Tanaka's theorem for maximal functions
with coauthors Jonathan Bober, Emanuel Carneiro, and Kevin Hughes
Proceedings of the Amer. Math. Soc. 140 (2012) 16691680.
(abstract)
(journal)
revised version on (arXiv)
In this paper we prove a discrete version of Tanaka's Theorem for the HardyLittlewood maximal operator in one dimension, both in the noncentered and centered cases. Up to date
version available on arXiv.
(hide abstract)
A note on discrete fractional integral operators on the Heisenberg group
Internat. Math. Res. Not. Vol. 2012, No. 1, (2012) 1733.
(abstract)
(pdf)
(journal link)
We consider the discrete analogue of a fractional integral operator on the Heisenberg group, for which we are able to prove nearly sharp results by means of a simple argument of a combinatorial nature.
(hide abstract)
Discrete fractional Radon transforms and quadratic forms
Duke Math. Journal 161, No. 1 (2012)
69106. Correction 162 (2013) 12031204.
(abstract)
(pdf)
We consider discrete analogues of fractional Radon
transforms involving integration over paraboloids defined by
positive definite quadratic forms. We prove certain
conditions are sufficient for them to extend to bounded
operators from l^p to l^q for certain p,q. The method involves an intricate spectral decomposition according to major and minor arcs, motivated by ideas from the circle method of Hardy and Littlewood. Techniques from harmonic analysis, in particular Fourier transform methods and oscillatory integrals, as well as the number theoretic structure of quadratic forms, exponential sums, and theta functions, play key roles in the proof.
(hide abstract)
On discrete fractional integral operators and mean values of Weyl sums
Bull. London Math. Soc., 43 (2011) 597612.
(abstract)
(pdf)
(journal)
In this paper we prove new operator norms for a discrete fractional integral operator by applying techniques motivated by the circle method of Hardy and Littlewood to the Fourier multiplier of the operator.
From a different perspective, we describe explicit interactions between the Fourier multiplier and mean values of Weyl sums. These mean values express the average behaviour of the number of representations of a positive integer as a sum of s positive kth powers. Recent deep results within the context of Waring's problem and Weyl sums enable us to prove a further range of complementary results for the discrete operator under consideration.
(hide abstract)
A note on discrete twisted singular Radon transforms
Mathematical Research Letters 17, no. 4 (2010) 701720.
(abstract)
(pdf)
(journal)
In this paper we consider three types of discrete
operators stemming from singular Radon transforms. We
first extend an l^p result for translation
invariant discrete singular Radon transforms to a class of
twisted operators including an additional oscillatory
component, via a simple method of descent
argument. Second, we note an l^2 bound for
quasitranslation invariant discrete twisted Radon
transforms. Finally, we extend an existing l^2 bound for a
closely related nontranslation invariant discrete
oscillatory integral operator with singular kernel to an
l^p bound. This requires an intricate induction argument involving layers of decompositions of the operator according to the Diophantine properties of the coefficients of its polynomial phase function.
(hide abstract)
A bound for the 3part of class numbers of quadratic fields
by means of the square sieve
Forum Math. 18 (2006) no. 4, 677698.
(abstract)
(pdf)
(journal)
This paper proves a nontrivial bound for the 3part of the class number of a
real or imaginary quadratic field, by developing a variant of the square sieve and
the qanalogue of van der Corput's method.
(hide abstract)
Companion paper by N. M. Katz
On a question of Lillian Pierce
by Nicholas M. Katz
Forum Math. 18 (2006) no. 4, 699710.
(pdf)
The 3part of class numbers of quadratic fields
J. London Math. Soc. (2) 71 (2005) 579598.
(abstract)
(pdf)
(journal)
This paper proves nontrivial upper bounds for the 3part
of the class number of real and imaginary
quadratic fields, as well as consequent upper bounds for the number of
elliptic curves with given conductor.
(hide abstract)
Note: preprints may differ slightly from final published versions.
Oberwolfach report 2016 (link)
Oberwolfach report 2014 (link)
Oberwolfach report 2013 (link)
On torsion subgroups in class groups of number fields, JMM
2019 Sampler, Notices of the AMS, vol. 66 no. 1, January
(2019) 9798
Princeton Lectures in Analysis, by Elias M. Stein and Rami Shakarchi — A Book Review
Book Review in the
Notices of the AMS, May 2012 issue
Reviewed by Charles Fefferman and Robert Fefferman, with contributions from Paul Hagelstein, Nataśa Pavlović, and Lillian Pierce
Discrete Analogues in Harmonic Analysis
PhD Thesis, Princeton University, 2009
Advisor: E. M. Stein
The
3Part of Class Numbers of Quadratic Fields
MSc Thesis, Oxford University, 2004
Advisor: D. R. HeathBrown
The
Pair Correlation of the Zeroes of the Riemann Zeta Function
Undergraduate Senior Thesis, Princeton University, 2002
Advisor: E. M. Stein
