Date:

Sun, 17/12/201710:30-16:00

Location:

Hebrew University of Jerusalem, Campus Edmond J. Safra, Manchester building, lecture hall 2

### Organizers: Jasmin Matz (HUJI), Shaul Zemel (HUJI)

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## 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.