ALG6 - ALGEBRAIC NUMBER THEORY AND RELATED
TOPICS
Organizers
Elisa Lorenzo, Leiden University, Holland
Guillermo Mantilla, Universidad de los Andes, Colombia
ALG6-1 Francesca Bergamaschi, Concordia University, Canada
Bad reduction of Hilbert modular varieties
Hilbert modular varieties can be seen as a generalization of the modular curve; they can be roughly described
as spaces parametrizing abelian varieties under the action of a given totally real number field. Their relation
to number theory is strong, as Hilbert modular forms can be seen as sections of line bundles over them. In
particular, the many interesting phenomena arising in positive characteristic provide us with powerful tools
to study their geometry. In this talk we give a description of the Hilbert modular variety Mp with Γ0 (p)-level
in ramified characteristic. By classifying the p-torsion of points we will give an understanding of the local geometry of the Hilbert modular variety and define stratifications of the space, which are a natural environment
for constructing modular forms, such as generalized Hasse invariants.
ALG6-2 Victoria Cantoral Farfán, Institut de Mathématiques de Jussieu, France
Torsion for abelian varieties of type III
Mordell-Weil’s theorem states that for an abelian variety A defined over a number field K the group of rational
points over K is finitely generated, i.e. A(K) = A(K)tors × Zr where A(K)tors denotes the finite subgroup of
torsion points defined over K. One can wonder if we can get an uniform bound for |A(L)tors |, depending on
[L : K], when the abelian variety A varies. This question is more commonly known as the “Strong Uniform
Boundedness Conjecture”. For elliptic curves defined over a number field K, Merel proved in 1994 that we
can indeed get a uniform bound using methods developed by Mazur and Kamienny.
A complementary question would be to ask if we can get a bound for |A(L)tors | depending only on the degree
[L : K] when the field extension L/K varies and the abelian variety A is fixed? This question had been already
answered by Hindry and Ratazzi for certain classes of abelian variety.
In this talk we focus our attention on this last question and extend the previous results. We are going to present
some new results on this direction concerning the class of abelian varieties A which are isogenous to a product
of simple abelian vari- eties of type I, II or III in Albert classification and is “fully of Lefschetz type” (i.e. whose
Mumford-Tate group is the group of symplectic or orthogonal similitudes commuting with endomorphisms
and which satisfy the Mumford-Tate conjecture).
After defining all the necessary tools, we show that there is a constant cA such, as
∀L/K, |A(L)tors | ≤ cA · [L : K]γ(A)
The exponent γ(A) is the optimal exponent for this bound. Our results give an explicit expression for γ(A) in
terms of the dimensions of the abelian subvarieties of A and their rings of endomorphisms.
ALG6-3 Chantal David, Concordial University, Canada
One-parameter families of elliptic curves with non-zero average root number
We investigate in this talk the average root number (i.e. sign of the functional equation) of one-parameter families of elliptic curves (i.e elliptic curves over Q(t), or elliptic surfaces over Q). For most one-parameter families
of elliptic curves, the average root number is predicted to be 0. Helfgott showed that under Chowla’s conjecture
and the square-free conjecture, the average root number is 0 unless the curve has no place of multiplicative
reduction over Q(t). We then build families of elliptic curves with no place of multiplicative reduction, and
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compute the average root number of the families. Some families have periodic root number, giving a rational
average, and some other families have an average root number which is expressed as an infinite Euler product.
We also show several density results for the average root number of families of elliptic curves, and exhibit some
surprising examples, for example, non-isotrivial families of elliptic curves with rank r over Q(t) and average
root number −(−1)r , which were not found in previous literature.
Joint work with S. Bettin and C. Delaunay.
ALG6-4 Piper Harron, The liberated mathematician, USA
The Equidistribution of Lattice Shapes of Rings of Integers in Cubic, Quartic, and Quintic Number Fields
Piper Harron presents the delightfully mathematical one woman show that answers questions her audience
may have never asked itself before now! Such as: What is the shape of a number field? And: How do we show
shapes are equidistributed? She will sketch the proof, providing references to old stuff and details to new stuff.
Come one, come all (people, including graduate students, interested in number theory)!
ALG6-5 Robert Harron, University of Hawai’i, USA
Equidistribution of shapes of cubic fields of fixed quadratic resolvent
Building upon work of Bhargava, P. Harron, and Shnidman, I will discuss results on the distribution of shapes
of cubic fields K of fixed quadratic resolvent. The shapes depend on the trace zero form (that is the projection
of the trace form to the trace zero space). For instance, I’ll show that the shapes of complex cubic fields lie on
the geodesic on the modular surface SL(2, Z)\H determined by their trace zero form and that, in a fixed such
geodesic, the shapes are equidistributed with respect to the natural hyperbolic measure. In the case of pure
cubic fields (whose quadratic resolvent field is the third cyclotomic field), the corresponding geodesics have
infinite length and the equidistribution must be considered in a regularized sense. That these geodesics are
of infinite length provides a reason behind the different asymptotic growth rates of pure cubic fields versus
other fields of fixed quadratic resolvent seen in the work of Bhargava–Shnidman and Cohen– Morra. I’ll also
discuss related results such as the fact that the shape is a complete invariant of complex cubic fields.
ALG6-6 Elisa Lorenzo, Leiden University, Holland
On twists of smooth plane curves
Given a smooth curve defined over a field k that admits a non-singular plane model over k, it does not necessarily have a non-singular plane model defined over the field k. We will determine under which conditions
this happens and we will show an example of such phenomenon. Even assuming that such a smooth plane
model exists, we will discuss the existence of non-singular plane models over k for its twists. We characterize
twists possessing such models. We also show an example of a twist not admitting such non-singular plane
model via a non-trivial Brauer-Severi surface. (This is a joint work with E. Badr and F. Bars)
ALG6-7 Piermarco Milione, Universitat de Barcelona, Spain
p-adic uniformization of Shimura curves through Mumford curves
Shimura curves, and in particular their Jacobians, have been key ingredients in the proof of important results
in number theory, such as many relevant cases of Serre’s Conjecture and Fermat’s Last Theorem. Finally, in
more recent years, the study of the p-adic uniformization of Shimura curves has also assumed a fundamental
role in the p-adic Birch and Swinnerton-Dayer Conjecture and in geometric realizations of the p-adic Langlands
correspondence. In this talk I will give a brief introduction to the theory of p-adic uniformization of Shimura
curves and I will present general and explicit results that I have obtained, in a joint work with Laia Amoróos,
of such uniformization.
Let X(pD, N ) be the Shimura curve associated to an Eichler order of level N in an indefinite quaternion Qalgebra of discriminant pD, where p is a fixed odd prime integer. Thanks to the fundamental works of Cerednik
and Drinfel’d (cf. [Cer76] and [Dri76]), we know that the curve X(pD, N ) admits a p-adic uniformization which
can be expressed, inter alia, as a rigid analytic Qp -isomorphism
Γp \(Hp ⊗Qp Qp2 ) ' (X(pD, N ) ⊗Q Qp )rig
where Hp denotes the p-adic upper half-plane over Qp and Γp is a discrete cocompact subgroup of P GL2(Qp ).
As a consequence, the p-adic Shimura curve X(D, N ) ⊗Q Qp is the twist over Qp2 of a finite quotient of some
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Mumford curve associated to a cocompact Schottky group ΓSch
⊂ Γp . Moreover the group Γp arises as the
p
units group of an Eichler order of level N over Z[1/p], inside the definite quaternion algebra of discriminant
D.
In this talk, I will first present the method we developed in order to find the Schottky group ΓSch
⊂ Γp together
p
with a free system of generators for this, (at least) in the cases of those definite Eichler orders having ideal class
number h(D, N ) = 1. This generalizes some nice results of Gerritzen and van der Put [GvDP80] on Mumford
curves arising from the definite quaternion algebra of discriminant 2. Thanks to this method, we are able to
give an explicit description of the rigid analytic structure of the Mumford curve associate to the group ΓSch
p
(such as a good fundamental domain in Hp and its stable reduction graph), as well as of the the rigidification
of the p-adic Shimura curve X(pD, N ) ⊗Q Qp . As an application we can easily obtain formulas describing
the reduction-graphs with lengths of the Shimura curves considered (generalizing some formulas of Kurihara
[Kur79]), and also formulas for the genera of the special fibres of these curves.
Eventually, I will also explain how these results can be applied in order to compute the period matrices of the
Jacobians of the Shimura curves considered using p-adic multiplicative integrals, which is a work in progress
with Iago Giné.
References
[Cer76] I. V. Cerednik, Uniformization of algebraic curves by discrete aithmetic subgroups of PGL2(kw) with compact
quotients, Math. USSR Sbornik 29 (1976), no. 1, 55–78.
[Dri76] V. G. Drinfeld, Coverings of p-adic symmetric regions, Functional Analysis and Its Applications 10 (1976),
no. 2, 107–115.
[GvDP80] L. Gerritzen and M. van Der Put, Schottky groups and Mumford curves, Lecture Notes in Mathematics,
vol. 817, Springer, 1980.
[Kur79] A. Kurihara, On some examples of equations defining Shimura curves and the Mumford uniformization, J. Fac.
Sci. Univ. Tokyo, Sect. IA Math 25 (1979), no. 3, 277–300.
ALG6-8 Marta Narváez-Clauss, Universitat Barcelona, Spain
Quantitative equidistribution of Galois orbits of points of small heght on the algebraic torus
Bilu’s equidistribution theorem establishes that, given a strict sequence of points on the N -dimensional algebraic torus whose Weil height tends to zero, the Galois orbits of the points are equidis- tributed with respect
to the Haar probability measure of the unit poly- circle. For the case of dimension one, quantitative versions
of this re- sult were independently obtained by Petsche, and by Favre and Rivera- Letelier.
We present a quantitative version of Bilu’s result for the case of any dimension. Given a point on the algebraic
torus of dimension N and Weil height less than 1, we give a bound for the integral of a suitable test function
on P1 (C)N with respect to the signed measure defined as the difference of the discrete probability measure
associated to the Galois orbit of the point and the probability measure supported on the unit polycircle, where
it coincides with the normalized Haar measure. This bound is given in terms of a constant depending only on
the test function, the Weil height of the point, and a notion that generalizes to higher dimension the degree of
an algebraic number.
For the proof of this result we use Fourier analysis techniques to decompose the problem and we reduce it, via
projections, to the one- dimensional case where we apply the quantitative version by Favre and Rivera-Letelier.
Joint work with Carlos D’Andrea and Martín Sombra, both at the Universitat de Barcelona.
ALG6-9 Frank Thorne, University of South Carolina, USA
Levels of distribution in arithmetic statistics
A "level of distribution" is, roughly speaking, a bound on the cumulative error terms made when estimating
the distribution of arithmetic sequences. These are an important input in many standard techniques in analytic
number theory.
In this talk I will discuss levels of distribution in the context of prehomogeneous vector spaces. After giving an
overview of what prehomogeneous vector spaces are and why people care, I will describe what sorts of results
one hopes to obtain and how they may be proved. I will also outline a variety of applications in arithmetic
statistics.
This is joint work with Takashi Taniguchi.
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