Organizers: Alina Cojocaru, Vicentiu Pasol, Alexandru Popa

The talks will take place at IMAR in Amfiteatrul "Miron Nicolescu" (parter). No registration is necessary.

The workshop is partly supported by BitDefender and by CNCS-UEFISCDI grant PN-III-PD-ID-PCE-0157.

**Alina Cojocaru**, University of Illinois at Chicago and IMAR**Yan Hu**, IMAR**Nathan Jones**, University of Illinois at Chicago**George Turcas**, University of Warwick

- 9:30-10:20
**George Turcas** *Diophantine applications of Serre's modularity conjecture*- Successful resolutions of Diophantine equations over Q via Frey elliptic curves and modularity rest on three pillars: Mazur's isogeny theorem, modularity of elliptic curves defined over Q and Ribet's level-lowering theorem. One can replace the last two with Serre's modularity conjecture over Q, now a theorem due to Khare and Wintenberger. For general number fields K, there's no analogue of Mazur's isogeny theorem, but there is a formulation of Serre's modularity conjecture. In this talk, we will show how one can use the latter for showing that certain Diophantine equations do not have solutions in K.
- 10:20-10:40 Coffee break
- 10:40-11:30
**Yan Hu** *A brief introduction to Stark-Heegner points*- Henri Darmon proposes a conjectural p-adic analytic construction of points on elliptic curve, points which are defined over ring class field of real quadratic fields. These points are related to classical Heegner points. For this reason, these points are called Stark-Heegner points. In this talk, I will give a brief introduction to Stark-Heegner points. If time permits, I will give some numerical example and related application.
- 11:35-12:25
**Nathan Jones** *Elliptic curves with missing Frobenius traces*- Let E be an elliptic curve defined over the rational numbers and without complex multiplication. In the 1970s, Lang and Trotter conjectured an asymptotic formula for the number of primes p up to X for which the associated Frobenius trace is equal to a fixed integer m. In this talk, we will consider the following question: For which elliptic curves E and which integers m is it the case that there are only finitely many primes p with Frobenius trace equal to m? We will discuss recent work in progress to resolve the analogous question over the rational function field Q(t). This is based on joint work with Kevin Vissuet.
- 12:30-13:20
**Alina Cojocaru** *Elliptic modules, Frobenius endomorphisms, and CM liftings*- In a 2002 paper, William Duke and Arpad Toth gave a global description of the Frobenius in the division fields of an elliptic curve defined over the rationals. Their description highlighted the important role played by the index of the ring generated by the p-th power Frobenius in the ring of F_p-endomorphisms of the reduction modulo p of the curve. We discuss related constructions and results in the setting of Drinfeld modules. This is joint work with Mihran Papikian.