## AWM at JMM 2015

### Special Session Abstracts

### Contents

- AMS-AWM Special Session on Recent Developments in Algebraic Number Theory, I
- AMS-AWM Special Session on Recent Developments in Algebraic Number Theory, II
- AWM Workshop on Homotopy Theory, I
- AWM Workshop on Homotopy Theory, II

### Tuesday January 13, 2015, 8:00am-10:50am Room 007B, Convention Center

### AMS-AWM Special Session on Recent Developments in Algebraic Number Theory, I

### 8:00 am Visualising the arithmetic of quadratic imaginary fields

#### Katherine E Stange*, University of Colorado, Boulder (1106-11-1783)

##### We study the orbit of ℝ under the Bianchi group PSL_{2}(?_{K}), where K is an imaginary quadratic field. The orbit, called a Schmidt arrangement SK, is a geometric realisation, as an intricate circle packing, of the arithmetic of K. This paper

presents several examples of this phenomenon. First, we show that the curvatures of the circles are integer multiples √−Δ and describe the curvatures of tangent circles in terms of the norm form of ?_{K}. Second, we show that the circles themselves are in bijection with certain ideal classes in orders of OK, the conductor being a certain multiple of the curvature. This allows us to count circles with class numbers. Third, we show that the arrangement of circles is connected if and only if OK is Euclidean if and only if the tangency graph contains loops. These results are meant as foundational for a study of a new class of thin groups generalising Apollonian groups.

### 8:30 am Derivatives of p-adic L-functions of Hilbert modular forms

#### Daniel Barrera, University of Montreal;

Mladen Dimitrov, University of Lille;

Andrei Jorza*, University of Notre Dame (1106-11-2418)

##### P-adic L-functions are analogues of classical complex L-functions where the variable is a p-adic number instead of a com- plex one. Just like the Birch and Swinnerton-Dyer conjectures (and their generalizations) study the first Taylor coefficient of complex L-functions of Galois representations, so-called exceptional zero conjectures relate the first Taylor coefficients of p-adic L-functions to arithmetic information. This relationship encodes congruences between Hilbert modular forms and has wide-ranging applications. I will present recent results in the case of Hilbert modular forms.

### 9:00 am Moduli Interpretations for Noncongruence Modular Curves

#### William Y Chen*, Pennsylvania State University (1106-11-1877)

##### Let Γ be a subgroup of SL_{2}(ℤ), and let ℋ be the upper half plane. If Γ is a congruence subgroup, then it’s well known that the quotient Γ \ ℋ is a coarse moduli space for isomorphism classes of elliptic curves equipped with some level structure. We will generalize the standard level structures and show that for most noncongruence subgroups Γ, the quotient Γ \ ℋ has a natural interpretation as the coarse moduli space classifying isomorphism classes of elliptic curves together with a generalized level structure. In this generalization the standard level structures associated to congruence subgroups should be considered “abelian”, while those corresponding to noncongruence subgroups should be considered “nonabelian”. We will also discuss applications to the arithmetic of noncongruence modular forms.

### 9:30 am p-adic q-expansions and families of automorphic forms

#### Ellen Eischen*, The University of North Carolina at Chapel Hill (1106-11-1801)

##### One approach to p-adically interpolating special values of certain L-functions relies on constructing p-adic families of automorphic forms. I will explain how to produce such p-adic families for certain unitary groups. In particular, this talk will focus on the q-expansions (and certain analogues of q-expansions) of these forms. I will also mention some applications to number theory and beyond.

### 10:00 am Local points of supersingular elliptic curves on ℤ_{p}-extensions

#### Mirela Ciperiani*, University of Texas at Austin

(1106-11-2572)

##### By work of Kobayashi and Iovita-Pollack we know that local points of supersingular elliptic curves on ramified ℤ_{p}– extensions of ℚ_{p} split into two strands of even and odd points. We will discuss a generalization of this result to ℤ/p-extensions that are localizations of anticyclotomic ℤ_{p}-extensions over which the elliptic curve has non-trivial CM points.

### 10:30 am On the restriction of F-crystalline p-adic Galois representations

Bryden Cais*, University of Arizona

Tong Liu, Purdue University (1106-11-1225)