I am a researcher in Mathematics and Cryptography.

I am a researcher in Mathematics and Cryptography working as a Cryptography Researcher (ZKP) at Tokamak Network, working on zkRollups (see here and here) and sybil-resistant algorithms (see here and here).

Before that, I worked as a Cryptographic Engineer at zkFold focusing on zkRollups for the Cardano blockchain. I also held a temporary position as a Cryptographic Researcher at Nethermind.

I have spent a few years in academia, working as a postdoctoral researcher in number theory and algebraic geometry at Università degli Studi di Milano and Università degli Studi di Padova.

I obtained my Ph.D. degree in mathematics at Universität Duisburg-Essen (ESAGA) in September 2021.

You can find my CV here (LinkedIn) and an academic version here (which might be outdated).

Me somewhere in Italy, Dec. 2023.

Contacts

Email: lucadallava.work - at - gmail.com

LinkedIn profile: linkedin.com/in/luca-dall-ava-575817182

GitHub profile: github.com/Luca-DallAva

Google Scholar profile: https://scholar.google.com/citations

Research

Research interests

Cryptography: I have been interested in the application of (algebraic and analytic) number theory and algebraic geometry to cryptography. Lately, I have been working on zero-knowledge protocols, lookup arguments, IVCs, accumulation/folding schemes, and sybil-resistant algorithms.

Mathematics: My research interests lie in the broad area of elliptic curves, special values of p-adic and complex L-functions (mainly triple product L-functions), and automorphic representations. I am particularly interested in the arithmetic aspect of modular forms, and automorphic forms on quaternion algebras, both from a theoretical and a computational point of view. Recently, I became more interested in the geometry of the eigenvarieties together with their relation with quaternion algebras.

Preprints and publications

Publications (Cryptography):

(Albert Garreta, Hendrik Waldner, Katerina Hristova, Dall’Ava, L. (& Cupic M., Klein M., Vlasov I.)) Zinc: Succinct Arguments with Small Arithmetization Overheads from IOPs of Proximity to the Integers (eprint, CryptoDB, PoC implementation), CRYPTO 2025.

Preprints (Mathematics):

(authors appear in alphabetic order)

(Luca Dall'Ava & Aleksander Horawa) Balanced triple product p-adic L-functions and Stark points (ArXiv), submitted.

Here you can look at a poster I presented at the conference in memoriam of Joël Bellaïche in June 2024 regarding this paper, while here you can see Alex giving a talk about our work at the ICMS for the PFAFTA in July 2024.

Publications (Mathematics):

Approximations of the balanced triple product p-adic L-function (ArXiv), Journal of Number Theory, Volume 246 (May 2023), pp. 189-226.

A brief account of this paper can be found in the poster I presented at the RTG meeting in December 2023.

Hida theory for special orders (ArXiv), Int. J. Number Theory, Volume No. 19 (March 2023), Issue No. 02, pp. 347-373.

A brief account of this paper can be found in the poster I presented at the RTG meeting in December 2023.

After this paper got accepted, I discovered I neglected a series of works that I feel must have been acknowledged in Section 4.3. Many thanks go to Kimball Martin and John Voight! These are a few papers that deal with special and, in more generality, Bass orders:

Martin - Exact double averages of twisted L-values.

Martin - The Jacquet–Langlands correspondence, Eisenstein congruences, and integral L-values in weight 2.

Martin - The basis problem revisited.

Martin, Wakatsuki - Mass formulas and Eisenstein congruences in higher rank.

Pacetti, Rodriguez-Villegas - Computing weight 2 modular forms of level p^2.

Pacetti, Sirolli - Computing ideal classes representatives in quaternion algebras.

Ph.D. Thesis

You can find my Ph.D. thesis here (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.

Master Thesis

If you are curious to see what I studied during my ALGANT Master, you can have a look at my thesis. Be aware of mistakes, typos, and grammatical errors.

Code

Computing-Gross-Pizer-Prasad-Eigenspaces

This repository provides algorithms in Magma designed to compute eigenspaces of quaternionic modular forms with level structure given by Special (Bass) order. The focus is on definite quaternion algebras. The repository, available here, includes examples. This repository supports the theoretical results in:

(with Aleksander Horawa) Balanced triple product p-adic L-functions and Stark points (ArXiv).
Approximations of the balanced triple product p-adic L-function (ArXiv), Journal of Number Theory, Volume 246 (May 2023), pp. 189-226.

Characters-of-Dihedral-representations

Given a dihedral weight 1 modular form g and a quadratic (CM) field K, this repository provides tools to output pairs of Hecke characters over K. It demonstrates that the Galois representation of g is isomorphic to the induction of any of these characters. The repository, available here, includes code and examples to illustrate the computations. This repository supports the theoretical results in (with Aleksander Horawa) Balanced triple product p-adic L-functions and Stark points (ArXiv).

Level_Raising_Modular_forms

This project implements a SageMath function to determine if two modular forms are related through level raising. Currently, the initial form must have rational coefficients. The repository, available here, includes examples and documentation.

GGH

A toy implementation of the Goldreich-Goldwasser-Halevi (GGH) cryptosystem in Rust. Based on the description in §7.8 “An Introduction to Mathematical Cryptography” by Hoffstein, Pipher, and Silverman. Available here.

Some geodesics

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

The code considered here is trivial as it amounts to calling the function list_plot on the real and imaginary part of the j-invariant, e.g. for Z_10: p=list_plot( [(elliptic_j(sqrt(10)*cos(t/10000)+sqrt(10)*I*sin(t/10000)).real(), elliptic_j(sqrt(10)*cos(t/10000)+sqrt(10)*I*sin(t/10000)).imag()) for t in range(1, 31415, 1) ], plotjoined=False, color='blue', size=1, aspect_ratio=1 )

Talks

Past research talks

25-26/07/2024 2nd AMS-UMI International Joint Meeting, Automorphic forms, Galois representations, and L-functions Special Session B19, Università degli Studi di Palermo: Balanced triple product p-adic L-functions and classical weight one forms. You can look at the slides of my talk here.

I am glad to announce that for this event I was partially funded by the Journal of Number Theory!

19/06/2024 Poster presentation at Modular Forms, L-functions, and Eigenvarieties, A conference in memoriam of Joël Bellaïche, Paris. You can have a look at the poster here.
19/04/2024 Number Theory Seminar, Heidelberg University: Balanced triple product p-adic L-functions and classical weight one forms.
17/04/2024 Contributed talk at The eighth mini symposium of the Roman Number Theory Association, Università Roma Tre: Balanced triple product p-adic L-functions and classical weight one forms.
19/12/2023 Number Theory Seminar, University of Genoa: Balanced triple product p-adic L-functions and classical weight one forms.
12/12/2023 Poster presentation for RTG Meeting, University of Duisburg-Essen: Triple product p-adic L-functions and weight 1 modular forms. You can have a look at the poster here.
25/05/2023 Number Theory Seminar, University of Oxford (UK): Balanced triple product p-adic L-functions and classical weight one forms.
4/11/2022 Arithmetic Geometry Seminar, Università degli Studi di Milano, Hida theory for Special quaternionic orders.
22/09/2002 Séminaire d'arithmétique à Lyon, Unité de Mathématiques Pures et Appliquées of the École normale supérieure de Lyon, Hida theory for Pizer’s quaternionic orders.
22/07/2021 RTG Seminar, University of Duisburg-Essen, Hida theory for Pizer’s quaternionic orders.

Organized seminars

Past events

2023/2024:

Co-organizer of the Arithmetic Geometry Seminar at Università degli Studi di Milano. You can find more details on the official webpage.

WS 2023/2024:

Co-organizer (with Rodolfo Venerucci) a Working seminar about Lue Pan's work on locally analytic vectors.

2022/2023:

I co-organized the Arithmetic Geometry Seminar at Università degli Studi di Milano. You can find more details on the official webpage.

2021/2022:

I co-organized, together with Matteo Longo, two seminars:

A working seminar about Dasgupta and Kakde's work on Hilbert's 12th Problem and Stark-Heegner points.
A series of number theory seminars, mostly on Zoom, held by young researchers. You can find more details and the schedule hereafter or here.

Outreach

2023/2024:

Co-organizer, together with Carlo Mazza and Alberto Vezzani, the ALGANT Pizza Seminar at Università degli Studi di Milano, a seminar aimed at ALGANT master students, third year bachelor students and young PhDs.

Here and here are the posters, made by Giovanna Domitolo and Alberto Vezzani, for these events.

Teaching

Previous semesters

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.

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.

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