## Magma Number Theory Day 2011

This is a day-long meeting on number theory hosted by the Magma computational algebra group at the University of Sydney, Australia on Thursday, May 12, 2011. The theme of the conference will be number theory, in part in gratitude for the time spent by Claus Fieker with the Magma group over the last decade. There will be 7 talks, given by various visitors and Magma personnel, and it will run from approximately 9:30am until 6pm. There will likely be a plan for the pub and dinner afterward -- the current picks are: The Lansdowne Hotel and the Spicy Sichuan Restaurant (in Glebe).

Photos

A small selection of photos is now available.

List of speakers:

    Frank Calegari --- Northwestern University
Brendan Creutz --- Magma
Daniel Delbourgo --- Monash University
Claus Fieker --- Magma
Eric Mortenson ---  The University of Queensland
Frederick Vercauteren --- Katholieke Universiteit Leuven
Jared Weinstein --- Boston University


Schedule.

Thursday, May 12, 2011 (Civil Engineering Building, Drawing Office, Room 361)
9:45am-10:30am   F. Vercauteren: Fully Homomorphic Encryption using
Principal Ideals in Number Rings
10:30am-11:15am   B. Creutz:      Second isogeny descents and the Birch and
Swinnerton-Dyer conjectural formula
11:15am-11:45am   *** Morning Break ***
11:45am-12:30pm   J. Weinstein:   Varieties over finite fields with
many many points
12:30pm-2:30pm    *** Lunch Break ***
2:30pm-3:15pm    C. Fieker:      Working in the multiplicative group
of a number field
3:15pm-3:45pm    E. Mortenson:   Ramanujan, partitions, and mock
theta functions.
3:45pm-4:15pm    *** Afternoon Break ***
4:15pm-5:00pm    D. Delbourgo:   How the shape of K_1 affects the growth
of Mordell-Weil ranks
5:00pm-5:45pm    F. Calegari:    Even Galois representations


Talk titles and abstracts

Frank Calegari: Even Galois representations.

What are the Galois extensions K/Q unramified away from p such that
G = Gal(K/Q) admits an irreducible representation into GL_2(Fbar_p)?
When K is complex, the answer is given by the theory of modular forms,
and Serre's conjecture. In this talk, we discuss the case when K/Q is
real, both from a theoretical and computational perspective.


Brendan Creutz: Second isogeny descents and the Birch and Swinnerton-Dyer conjectural formula.

Abstract: The Birch and Swinnerton-Dyer conjectural formula relates the
leading term of the Taylor expansion of the L-function of an elliptic curve
at s = 1 to a host of arithmetic invariants of the curve. Using various
theoretical and computational tools one can verify this formula for specific
elliptic curves. I will discuss one such computational tool, called second
isogeny descents. This was recently used to complete the verification of the
formula for all elliptic curves of rank 0 or 1 and conductor less than 5000.


Daniel Delbourgo: How the shape of K_1 affects the growth of Mordell-Weil ranks.

We explain how the Ritter-Weiss congruences between arithmetic elements in
certain K-groups, can yield information on the growth of Mordell-Weil ranks
of elliptic curves (as you climb the layers in a p-adic Lie extension).
We also mention some computations using L-series that strongly support
these K_1-congruences.


Claus Fieker: Working in the multiplicative group of a number field.

The multiplicative group of a number field is rather large and difficult
to work in on a computer: we clearly do not have a finitely generated
Z-module structure that can be used. Therefore to use the multiplicative
group in application, one frequently starts by creating a finitely
generated sub-group that is large enough to contain a solution but
small enough to allow effective manipulation. Apart from the
finite generation, a second problem comes from the (necessary) use
of logarithms to linearize the structure, it implies that the
linear structure is only approximated and not exactly represented.

In this context there are a few important problems to solve:
- given a finite number of non-zero number field elements, can
we compute the Z-module structure?
- given a tentative sub-group - can we enlarge it systematically?
- given a particular problem, can we find an effective set of generators
for the part that we are interested in?
- given an element in the finitely generated group, can we find
nicer representatives?

Examples here are the computation of the class group, S-unit group,
solution of norm equations, splitting of co-cycles in cohomology groups and
p-Selmer group computations.

I will indicate algorithmic solutions to some of the problems, classical
solutions as well as new ones based on p-adic techniques.


Eric Mortenson: Ramanujan, partitions, and mock theta functions.

We will give a brief overview of the history of Ramanujan and give
samplings of areas such as partitions, partition congruences, ranks,
modular forms, and mock theta functions.

For example:  A partition of a positive number $n$ is a non-increasing
sequence of positive integers whose sum is $n$.  There are five partitions
of the number four: 4, 3+1, 2+2, 2+1+1, 1,1,1,1.  If we let $p(n)$ be the
number of partitions of $n$, it turns out that $p(5n+4)\equiv 0 \pmod{5}$.
How does one explain this?

We will then discuss new results on mock theta functions and will show how
they relate to old and recent results.


Fre Vercauteren: Fully Homomorphic Encryption using Principal Ideals in Number Rings.

In this talk, I will review the concept and possibilities of fully
homomorphic encryption and describe one possible realization of such system
based on principal ideals in number rings.  The security of the proposed
system is then related to well known problems in number theory such as
computing generators of principal ideals and finding short vectors in
lattices.


Jared Weinstein: Varieties over finite fields with many many points.

Over a finite field with $q^2$ elements, the projective curve with
equation $y^q-y=x^{q+1}$ is maximal" in the sense that it has the
maximum possible number of rational points relative to its genus.
(This curve is called the Hermitian curve", and it is sometimes used
in coding theory. There are also minimal" curves.) For higher-dimensional
varieties, there is a suitable generalization of this notion:  a variety is
maximal when it has the maximum number of rational points allowed by the
Weil conjectures (relative to its topology).  We will discuss a new recipe
for constructing interesting higher-dimensional varieties which we conjecture
to be maximal. If our conjecture is true, these varieties constitute evidence
for Tate's conjecture as well.


Venue: The talks will be held in the Drawing Office of the Civil Engineering Building (Room 361), which is sometimes called the Civil and Mining Engineering Building. This is in the lower right corner of the map, and is almost the first point you reach when coming from Redfern Station.