Luca Dall'Ava

Since December 2021, I am part of Matteo Longo's group. Previously, I obtained my Ph.D. degree at Universität Duisburg-Essen (ESAGA) under the supervision of Massimo Bertolini. Contacts Via Trieste, 63, 35121 Padova (PD) - Torre Archimede, Room 639. You can reach me at: luca.dallava -at- math.unipd.it. You can find my CV here. I also have a Google webpage (Which might be, from time to time, more up-to-date).

Research interests

My research interests lie in the arithmetic of quaternion algebras and its relation with the arithmetic of elliptic curves and modular forms. In particular, special values and the behavior of p-adic and complex L-functions at critical points. I am also interested in computational approaches to the study of these number-theoretic questions.

Preprints

• Hida theory for special orders (ArXiv), submitted.


• Approximations of the balanced triple product p-adic L-function (ArXiv), submitted.

Geodesics

Hereafter you can have a look at some interesting drawings that I made with the Software SageMath. Some of them appear in Special curves in modular surfaces by Matteo Tamiozzo.

Theses

• Ph.D. Thesis: Quaternionic Hida families and the triple product p-adic L-function (Successfully defended on September 30th, 2021).
  
  Comments are still welcome!
  
  The primary purpose of my thesis is to provide an algorithm for approximating the values of the balanced p-adic L-function, as constructed by Hsieh, at the limit point (2,1,1); you can read about it in Section 2. The first section is instead dedicated to the study of families of quaternionic modular forms arising from orders defined by Pizer and Hijikata-Pizer-Shemanske; the main result is a control theorem in the spirit of Hida, in which the novelty lies in the rank of the Hecke-eigenspaces being 2 (and no more 1 as in the classical case of Eichler orders). The motivation for the first section, as well as its relation with the second one, is explained in Section 3.


• ALGANT Master Thesis: Arithmetic of special values of triple product L-functions (Be aware of mistakes, typos, and grammatical errors!).

2021/2022 SS:

Together with Maria Rosaria Pati, I am teaching the Ph.D. minicourse "Basics on Hida Theory" (at Università degli Studi di Padova). You can find the program here.

Previous semesters:

2021/2022 WS: No teaching.

2020/2021 SS: Teaching assistant for Modular Forms 2 at Universität Duisburg-Essen. Lecturer: Jie Lin.

2020/2021 WS: Teaching assistant for Modular Forms 1 at Universität Duisburg-Essen. Lecturer: Jie Lin.

2018/2019 WS: Teaching assistant for Modular Forms 1 at Universität Duisburg-Essen. Lecturer: Rodolfo Venerucci. (Here are some old exercise sheets.)

Working seminar: Dasgupta-Kakde and Stark-Heegner points (co-organized with Matteo Longo)

A working seminar about Dasgupta and Kakde's work on Hilbert's 12th Problem and Stark--Heegner points. If you are interested in giving a talk, feel free to write me an email.

Research seminar (co-organized with Matteo Longo)

A series of number theory talks, mostly on Zoom, by young researchers. You can find more details and the schedule here or here. If you would like to attend one or more talks, write me or Matteo Longo an e-mail.