# The VI^{th} Israeli Algebra and Number Theory Day

### Location:

Hebrew University of Jerusalem, Campus Edmond J. Safra, Manchester building, lecture hall 2
(campus map);

how to get here

### Date:

December 17, 2017

## Schedule:

#### 10:45-11:35: Philipp Habegger (University of Basel)

11:35-12:00: coffee

#### 12:00-12:50: Özlem Imamoglu (ETH Zürich)

13:00-14:30: lunch

#### 14:30-15:20: Farrell Brumley (Université Paris 13)

15:20- : coffee

## Talk titles and abstracts:

#### Philipp Habegger: On small sums of roots of unity

Let k be a fixed positive integer. Myerson (and others) asked
how small the modulus of a non-zero sum of k roots of unity can be. If
the roots of unity have order dividing N, then an elementary argument
shows that the modulus decreases at most exponentially in N. Moreover it
is known that the decay is at worst polynomial if k at most 4. But no
general sub-exponential bound is known if k=5.
In this talk I will present evidence that the modulus decreases at most
polynomially for prime values of N by showing that counterexamples must
be very sparse. We do this by counting rational points that approximate
a set that is definable in an o-minimal structure. This is motivated by
the counting results of Bombieri-Pila and Pila-Wilkie.
I will also discuss progress on Myerson's related conjecture on Gaussian
periods, as well as strong equidistribution properties of tuples of
roots of unity, and connections to an ergodic result of
Lind-Schmidt-Verbitskiy.
#### Özlem Imamoglu: Modular integrals, Dirichlet series and linking numbers

#### Farrell Brumley: Concentration properties of theta lifts

The classical conjectures of Ramanujan-Petersson and Sato-Tate on the Fourier coefficients of modular forms, or more generally on the Satake parameters of automorphic representations, are highly sensitive to questions of functoriality. For example, the coefficients of CM modular forms are equidistributed according to a very different law from that of non-CM forms, and the first historical counter examples to the naive generalization of the Ramanujan conjecture were found amongst the theta lifts on the group Sp_{4}. A more recent analogy of these conjectures looks at the L^{p} norms of arithmetic eigenfunctions (with p=∞ corresponding to Ramanujan). The latter are vectors in automorphic representations, realized as functions on a locally symmetric space of congruence type. Their concentration properties, at points or along certain cycles, are of general interest from both an analytic and arithmetic viewpoint. I will describe in this talk a few recent results on the subject, joint with Simon Marshall, which clarify the structure of the problem: the L^{p} norms of a form reflect its functorial origin, in a relative sense (using periods). In particular, in a work in progress, we show the existence of arithmetic eigenfunctions, defined on hyperbolic manifolds and in the image of the theta correspondence from Sp_{4}, which concentrate to some degree along closed geodesics.

### Registration:

Registration is not necessary but if you plan to arrive by car, please contact one of the organizers to get access to the campus.

### Organizers:

Jasmin Matz (jasmin.matz {at} mail.huji.ac.il)

Shaul Zemel (zemels {at} math.huji.ac.il)