AUTOMORPHIC FORMS: THEORY AND COMPUTATION
ABSTRACTS
Samuele Anni (University of Warwick)
Constructing hyperelliptic curves with surjective Galois representations
Given a hyperelliptic curve over Q of genus g, it is possible to define a Galois
representation from the absolute Galois group of an algebraic closure of Q to the
general symplectic group in GSp2g (F` ), corresponding to the action of the absolute
Galois group on the `-torsion of the Jacobian variety. If this representation is
surjective, then we realize GSp2g (F` ) as a Galois group over the rationals. In this
talk I will describe how to show that for each g there exists a positive density of
polynomialsf (x) in Z[x] such that simultaneously for all odd primes ` we have that
Gal(Q(Jac(y 2 = f (x))[`])/Q) is isomorphic to GSp2g (F` ). This is joint work with
Vladimir Dokchitser.
Kevin Buzzard (Imperial College London)
Computing exotic automorphic forms
We will talk around and about two relatively recent computations. The first, joint
work with Alan Lauder, is some sort of an attempt at a definitive approach for
computing weight 1 modular forms. The second is a computation of automorphic
forms for a definite quaternion algebra, at the boundary of weight space, a characteristic p point of the adic space for which the theory of overconvergent forms is
still in its infancy.
Frank Calegari (University of Chicago)
Hypergeometric q-series, K-theory, and modularity
The Rogers-Ramanujan identity:
1+
q4
q9
q
+
+
+ ...
2
(1 − q) (1 − q)(1 − q ) (1 − q)(1 − q 2 )(1 − q 3 )
1
=
4
(1 − q)(1 − q )(1 − q 6 )(1 − q 9 ) . . .
says that a certain q-hypergeometric function (the left hand side) is equal to a modular form (the right hand side). To what extent can one classify all q-hypergeometric
functions which are modular? We discuss this question and its relation to conjectures in knot theory and K-theory. This is joint work with Stavros Garoufalidis
and Don Zagier.
John Cremona (University of Warwick)
The LMFDB
I will give a short demonstration of the LMFDB website, including its main features,
some highlights, and recent developments.
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AUTOMORPHIC FORMS ABSTRACTS
Lassina Dembélé (University of Warwick)
Explicit Inertial Langlands correspondence for GL2 and arithmetic applications
In this talk we will give an algorithmic description of the inertial Langlands correspondence for GL2 . Then, we will give several arithmetic applications of this.
(This is joint work with Nuno Freitas and John Voight.)
Tim Dokchitser (University of Bristol)
Reconstructing Weil-Deligne representations
Given a variety V over a global field, say we wish to determine the conductor, Lfunction, epsilon-factor etc. for its etale cohomology groups. This requires understanding of the action of the local Galois groups on etale cohomology at all primes,
notably those of bad reduction. One method to do this is to go to extensions where
V acquires good (or semistable) reduction, and use representation theory to recover
the action downstairs. I will describe this method, and its applications to curves.
This is joint work with Vladimir Dokchitser.
Xavier Guitart (University of Barcelona)
Modular forms over fields of mixed signature and algebraic points in elliptic curves
I will describe a conjectural construction of algebraic points on certain modular
elliptic curves defined over fields of mixed signature. The points are defined by
means of integrals of the modular form attached to the elliptic curve in a way that
resembles, and is inspired by, a construction of Darmon in the totally real field
case. I will also discuss some numerical computations that give evidence for the
conjecture. This is joint work with Marc Masdeu and Haluk Sengun.
Kiran Kedlaya (University of California, San Diego)
Mod 2 linear algebra and tabulation of rational eigenforms
Can Cremona’s tabulation of rational elliptic curves be accelerated by judicious
use of mod 2 linear algebra? After explaining the application of linear algebra we
have in mind, we describe some initial results of an investigation into this question.
We focus on the cases of prime level and of general odd level, combining extensive
numerical experiments with some applications of Serre’s conjecture; in the process,
we bump into some questions about mod 2 modular forms that have been studied
by many authors but not completely resolved. Joint work with Anna Medvedovsky.
Marc Masdeu (University of Warwick)
p-adic periods of abelian varieties attached to GL2 -automorphic forms
Let F be a number field, and let f be a normalized eigenform modular form of
weight 2 and level N for GL(2, F ). It is conjectured that attached to f there is
an abelian variety Af . This abelian variety should have dimension equal to the
degree of the field of Hecke eigenvalues, and should have good reduction outside
N . In those instances where the Eichler–Shimura construction is not available (for
example when F is not totally-real) little is known about how to find Af . In joint
work with Xavier Guitart, we present a p-adic conjectural construction (subject to
several restrictions, in particular p should divide N ) of Af , and illustrate how in
favourable situations it can be used to find equations for abelian surfaces Af as
jacobians of hyperelliptic curves.
AUTOMORPHIC FORMS ABSTRACTS
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Anna Medvedovsky (Max Planck Institute for Mathematics, Bonn)
Lower bounds on dimensions of mod p Hecke algebras
In 2012, Nicolas and Serre revived interest in the study of mod p Hecke algebras
when they proved that the Hecke algebra acting on the space of all modular forms of
level one mod 2 is the power-series ring F2 [[T3 , T5 ]]. Their technical yet elementary
arguments do not appear to generalize directly to p > 2, but their tools the Hecke
recursion, the nilpotence filtration serve as the backbone of a new method, uniform
and entirely in characteristic p, for obtaining lower bounds on dimensions of mod p
Hecke algebras. I will present this new method, currently implemented in the genuszero case only; compare it with others (viz. Bellaiche-Khare); and discuss future
directions and applications. The key technical result is pure algebra, combinatorial
in flavor; and may be of independent interest.
Ariel Pacetti (University of Buenos Aires)
Congruences between 2-dimensional Galois representations and applications
Given a strong compatible system of 2-dimensional Galois representations over a
totally real number field, we will show how to modify the level and weight of the
family via a chain of congruences to land in a family of controlled weight and
level. As an application of our result, we will prove base change for some small real
quadratic fields. This is a joint work with Luis Dieulefait.
Aurel Page (University of Warwick)
Computing good covers of compact arithmetic manifolds
I will present a new algorithm to compute the cohomology of arithmetic groups
using coverings of arithmetic manifolds by geodesic balls centered at points from a
Hecke orbit. This is joint work in progress with Michael Lipnowski.
Robert Pollack (Boston University)
Slopes of modular forms and the ghost conjecture
In this talk, we present a conjecture on slopes of p-adic modular forms. We write
down a relatively simple and explicit power series over weight space and conjecture, in the Buzzard-regular case, that this power series knows the slopes of the
Up operator acting on all spaces of overconvergent modular forms. Precisely, we
conjecture that the Newton polygon of our series evaluated at a weight k (classical
or not) matches the Newton polygon of the characteristic power series of Up acting
on weight k overconvergent modular forms. We call this power series the "ghost
series" as its spectral curve hovers around the true spectral curve.
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AUTOMORPHIC FORMS ABSTRACTS
David Roberts (University of Minnesota, Morris)
PGL2 (`) number fields with rational companion forms
The polynomial
x12 − 4x11 − 4x10 + 16x9 + 24x8 − 30x7
−78x6 − 18x5 + 72x4 + 86x3 + 52x2 + 16x + 2
has Galois group PGL2 (11) and field discriminant −214 310 119 . It captures the
projective modulo 11 Galois representation associated to newforms of weight four
and eight on Γ0 (24). It is an attractive example of the theory of companion forms
because both these newforms have rational coefficients.
We will give a list of PGL2 (`) number fields for ` ≥ 11 which likewise have rational
companion forms. Our list contains 52 fields and seems likely to be complete.
Some of these fields have the smallest known discriminant for their Galois group,
and finding the true minimum via modular methods seems within reach.
Sandra Rozensztajn (École Normale Supérieure, Lyon)
Computing the reduction modulo p of 2-dimensional crystalline representations
In this lecture, I will talk about an algorithm that allows to compute the reduction modulo p of 2-dimensional crystalline representations of GQp with distinct
Hodge-Tate weights (at least for small values of p), using the p-adic Langlands correspondence. I will also explain how additional information can be obtained from
this algorithm, such as local constancy results.
Mehmet Haluk Şengün (University of Sheffield)
K-theory of arithmetic groups
The ordinary cohomology of arithmetic groups, endowed with the action of Hecke
operators, plays an important role in the theory of automorphic forms and in the
Langlands programme. Could replacing ordinary cohomology with K-theory offer
new insight or reveal new phenomena? In this talk, I will present some highlights
of joint work with Bram Mesland (Hannover) in which we take the first steps to
attack the above question in the case of Bianchi groups.
AUTOMORPHIC FORMS ABSTRACTS
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Hironori Shiga (Chiba University)
To the Hilbert 12th problem via the hypergeometric modular function
We discuss the Hilbert 12th Problem: Construct the Hilbert class field of a given
CM field by the adjunction of singular values of modular functions. We can use the
Shimura complex multiplication theorem (1967), but it is only an existence theorem.
There has been no nontrivial explicit example supporting the above problem, except
an experimental approach of J. Voight (2006). Conclusions:
(1) We have determined the generating modular functions (i.e. the Shimura
canonical model) in all Takeuchi arithmetic triangle cases with the triangle
unit group = Our Main theorem,
(2) We made an explicit theta representation for it in the case with index (3,3,5)
(basically due to Kenji Koike (2003)).
(3) By an approximate calculation we find several explicit defining equations
of the Hilbert class fields of CM fields of higher degree.
For the full argument, see A. Nagano and H. Shiga, To the Hilbert class field from
the hypergeometric modular function, J. Number Theory, 165 (2016), 408 – 430.
Olivier Taïbi (Imperial College London)
On the formulation of Arthur’s multiplicity formula for automorphic representations
in the case of inner forms
Computing with automorphic forms for a reductive group over a number field is
easier when the group is definite, that is compact at all Archimedean places. Such a
group is not quasisplit, and so it is a non-trivial inner form of a quasisplit reductive
group, having the same Langlands dual group. Formulating the (conjectural in
general) local Langlands correspondence and Arthur’s multiplicity formula for such
groups was only recently achieved by Tasho Kaletha, using certain local and global
Galois gerbes. After recalling Kaletha’s work, I will explain a proof of the fact that
cocycles for these global Galois gerbes are almost everywhere unramified, and how
this can be used for explicit computations.
John Voight (Dartmouth College)
Explicit modularity for genus 2 curves
We discuss what it means for a genus 2 curve to be modular. In joint work with
Andrew Booker, Jeroen Sijsling, Drew Sutherland, and Dan Yasaki, to every genus
2 curve X we discuss conjectures (and some theorems) that attach to X a modular
form with a matching L-function. The precise description depends on the structure
of the endomorphism algebra of the Jacobian of X — it turns out there are many
variations on the theme ‘GL2 -type’! To explore this conjecture, we built a database
of genus 2 curves with associated data including geometric and arithmetic invariants
of the curve and its Jacobian, available on the LMFDB (and with some further
data to be added). Finally, when X is typical, we arrive at the Brumer-Kramer
paramodularity conjecture; in joint work with Armand Brumer, Cris Poor, and
David Yuen, we prove paramodularity for an abelian surface of conductor 277.
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AUTOMORPHIC FORMS ABSTRACTS
Chris Williams (University of Warwick)
p-adic L-functions for GL2
The first constructions of p-adic L-functions for classical modular forms, or automorphic forms for GL2 /Q, were given almost forty years ago by Mazur and
Swinnerton-Dyer, and have been followed by a number of other constructions. The
picture for GL2 over more general number fields, however, has been very fragmented, and constructions have previously been given only in isolated cases. In
this talk, I will discuss joint work with Daniel Barrera in which we give a construction of p-adic L-functions for ‘small slope’ automorphic forms for GL2 over
general number fields. Our methods, which use overconvergent modular symbols,
generalise work of Pollack and Stevens over the rationals.