Number theory and arithmetic geometry

 

  Project information
Research project no. HRZZ IP-2022-10-5008 funded by the Croatian Science Foundation (HRZZ)
Hosting organization: Faculty of Science (PMF), University of Zagreb
Project duration: 48 months (from December 31, 2023 to December 30, 2027)
Total amount of funding: 166022.97 EUR
 
  Project summary
The project coordinates and supports the research activities of the Croatian number theory and arithmetic geometry group. Research topics are: Arithmetic of modular curves, Diophantine D(n)-m-tuples, Diophantine approximations, Application of deep learning in number theory and Congruences for sporadic sequences and modular forms.
 
  Principal investigator
  • Filip Najman, professor, Department of Mathematics, Faculty of Science, University of Zagreb
  Team members
  • Nikola Adžaga, assistant professor, Faculty of Civil Engineering, University of Zagreb
  • Alen Andrašek, PhD student, Faculty of Civil Engineering, University of Zagreb
  • Marija Bliznac Trebješanin, assistant professor, Department of Mathematics, Faculty of Science, University of Split
  • Sanda Bujačić Babić, assistant professor, Faculty of Mathematics, University of Rijeka
  • Maarten Derickx, postdoc, Faculty of Science, University of Zagreb
  • Goran Dražić , postdoc, Faculty of Food Techology and Biotechonlogy, University of Zagreb
  • Andrej Dujella, professor, Faculty of Science, University of Zagreb
  • Alan Filipin, professor, Faculty of Civil Engineering, University of Zagreb
  • Zrinka Franušić, associate professor, Department of Mathematics, Faculty of Science, University of Zagreb
  • Mirela Jukić Bokun, associate professor, School of Applied Mathematics and Informatics, J. J. Strossmayer University of Osijek
  • Ana Jurasić, assistant professor, Faculty of Mathematics, University of Rijeka, Croatia
  • Matija Kazalicki, professor, Department of Mathematics, Faculty of Science, University of Zagreb
  • Ivan Novak, PhD student, Department of Mathematics, Faculty of Science, University of Zagreb
  • Lukas Novak, PhD student, Department of Mathematics, Faculty of Science, University of Zagreb
  • Petar Orlić, PhD student, Department of Mathematics, Faculty of Science, University of Zagreb
  • Tomislav Pejković, assistant professor, Department of Mathematics, Faculty of Science, University of Zagreb
  • Pavao Radić, PhD student, Department of Mathematics, Faculty of Science, University of Split
  • Ivan Soldo, associate professor, School of Applied Mathematics and Informatics, J. J. Strossmayer University of Osijek
  • Marin Varivoda, PhD student, Department of Mathematics, Faculty of Science, University of Zagreb
  Research publications

  In the 1st project year

[1] F. Najman, On r-isogenies over Q(\zeta_r) of elliptic curves with rational j-invariants, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 118 (2024), 134.
[2] B. S. Banwait, F. Najman, O. Padurariu, Cyclic isogenies of elliptic curves over a fixed quadratic field, Math. Comp. 93 (2024), 841-862.
[3] N. Freitas, F. Najman, Two results on x^r+y^r=dz^p , Proc. Amer. Math. Soc. 152 (2024), 591-598.
[4] N. Adzaga, T. Keller, P. Michaud-Jacobs, F. Najman, E. Ozman, B. Vukorepa, Computing quadratic points on modular curves X_0(N), Math. Comp. 93 (2024), 1371-1397.
[5] F. Najman, P. Orlic, Gonality of the modular curve X_0(N), Math. Comp. 93 (2024), 863-886.
[6] M. Jukic Bokun, I. Soldo, Extensions of D(-1)-pairs in some imaginary quadratic fields, New York Journal of Mathematics 30 (2024), 745-755.
[7] Z. Franusic, A. Jurasic, On nonexistence of D(n)-quadruples, Math. Slovaca 74 (2024), 835-844.
[8] M. Derickx, P. Orlic, Modular curves X0(N) with infinitely many quartic points, Res. Number Theory 10, article number 42, (2024)
[9] Z. Bujanovic, M. Kazalicki, L. Novak, Murmurations of Mestre-Nagao sums, IJDSMS 02 (2024), 51-61.

  In the 2nd project year

[10] M. Derickx, F. Najman, Hyperelliptic and trigonal modular curves in characteristic p, Q. J. Math (Oxford) 76 (2025), 1-18.
[11] P. Orlic, Tetragonal intermediate modular curves, Acta Arith. 220.3 (2025), 263-288.
[12] P. Orlic, Tetragonal modular quotients X_0^+(N), Acta Arith. 217.3 (2025), 261-275.
[13] P. Orlic, Tetragonal modular quotients X_0^{+d}(N), J. Number Theory 276 (2025), 98-114.
[14] A. Bourdon, S. Hashimoto, T. Keller, Z. Klagsbrun, D. Lowry-Duda, T. Morrison, F. Najman, H. Shukla, Towards a Classification of Isolated j-invariants, With an appendix by Maarten Derickx and Mark van Hoeij, Math. Comp. 94 (2025), 447-473.
[15] P. Michaud-Jacobs, F. Najman, Quadratic points on X_0(163), J. Algebra, 666 (2025), 279-288.
[16] T. Pejkovic, Schneider’s p-adic continued fractions of rational numbers, Ramanujan J. 68:65 (2025).
[17] M. Kazalicki, Quadratic twists of genus one curves and Diophantine quintuples, Res. Number Theory 11:72 (2025).
[18] A. Dujella, M. Kazalicki and V. Petricevic, Rational Diophantine sextuples with strong pair, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 119:36 (2025).
[19] N. Adzaga, G. Drazic, A. Dujella and A. Petho, Asymptotics of D(q)-pairs and triples via L-functions of Dirichlet characters, Ramanujan J. 66 (2025), Article 11, (23pp).
[20] A. Filipin, A. Jurasic and L. Szalay, Extensions of the Triangular D(3)-pair {3,6}, Math. Slovaca 75 (2025), 775-790.
[21] Z. Franusic and A. Jurasic, Is there a polynomial D(2X+1)-quadruple?, An. St. Univ. Ovidius Constanta, Vol. 33(2) (2025), 67-88.
[22] M. Kazalicki, B. Naskrecki, Second moments and the bias conjecture for the family of cubic pencils, Math. Z. 309, 72 (2025).
[23] M. Derickx, F. Najman, Classification of torsion of elliptic curves over quartic fields, J. Reine Angew. Math. 829 (2025), 123-156.

  Doctoral dissertations

[1] P. Orlic, Gonality of modular curves and their quotients, 2025.

  Conference and seminar talks

  In the 1st project year

  • F. Najman, Q-curves, sporadic points, and Serre's uniformity conjecture, (invited confernece talk), 31.1.2024, 2024 Workshop on Number Theory and Algebra, Daejeon, South Korea
  • F. Najman, Quadratic points on modular curves, (invited confernece talk), 1.2.2024, 2024 Workshop on Number Theory and Algebra, Daejeon, South Korea
  • P. Orlić, Tetragonal modular quotients X^+_0(N), (confernece talk), 1.2.2024, 2024 Workshop on Number Theory and Algebra, Daejeon, South Korea
  • F. Najman, Elliptic curves, (invited confernece talk), 4.2.2024, 2024 Algebra Camp, Yangpyeong, South Korea
  • P. Orlić, Gonality of the modular curve X_0(N), (invited confernece talk), 4.2.2024, 2024 Algebra Camp, Yangpyeong, South Korea
  • F. Najman, Modular Curves, (invited confernece talk), 5.2.2024, 2024 Algebra Camp, Yangpyeong, South Korea
  • M. Kazalicki, Ranks of elliptic curves and deep neural networks, (invited conference talk), 27.2.2024, Hybrid Conference on AI-MATH, Rio de Janeiro, Brazil
  • M. Kazalicki, Ranks of elliptic curves and neural networks, (invited conference talk), 13.4.2024, Alpe - Adria seminar, Zagreb, Croatia
  • N. Adžaga, Low Degree Points on Modular Curves and Their Quotients, (confernece talk), 18.4.2024, The eighth mini symposium of the Roman Number Theroy Association, Rome, Italy
  • F. Najman, Hyperelliptic and trigonal modular curves, (confernece talk), 18.4.2024, The eighth mini symposium of the Roman Number Theroy Association, Rome, Italy
  • A. Filipin, On the extension of D(4)-triples {q,b,c}, (conference talk), 13.6.2024, Canadian Number Theory Association XVI (CNTA XVI), Toronto, Canada
  • A. Dujella High-rank elliptic curves with given torsion group, (invited conference talk), 25.6.2024, The Quest for the Hidden Simplicity of Noncommutative Harmonic Analysis and Representation Theory - a conference celebrating the 70th birthday of Marko Tadic, Zagreb, Croatia
  • Zrinka Franušić On polynomial D(n)-quadruples in Z[X], (conference talk), 2.7.2024, 8th Croatian Mathematical Congress, Osijek, Croatia
  • Ana Jurasić, On the existence of D(2X+1)-quadruples in Z[X], (conference talk), 2.7.2024, 8th Croatian Mathematical Congress, Osijek, Croatia
  • S. Bujačić Babić Polynomial D(4)-quadruples in R[X] , (conference talk), 3.7.2024, 8th Croatian Mathematical Congress, Osijek, Croatia
  • A. Dujella Rational Diophantine sextuples with strong pair, (conference talk), 3.7.2024, 8th Croatian Mathematical Congress, Osijek, Croatia
  • M. Jukić Bokun, The extendibility of parametric D(-1)-triples, (conference talk), 3.7.2024, 8th Croatian Mathematical Congress, Osijek, Croatia
  • M. Kazalicki, Ranks of elliptic curves and deep neural networks, (invited conference talk), 3.7.2024, 8th Croatian Mathematical Congress, Osijek, Croatia
  • F. Najman, Low degree points on modular curves, (plenary talk), 3.7.2024, 8th Croatian Mathematical Congress, Osijek, Croatia
  • I. Novak, Degrees of cyclic isogenies of elliptic curves with rational j-invariant, (conference talk), 3.7.2024, 8th Croatian Mathematical Congress, Osijek, Croatia
  • L. Novak, Quadratic twists of genus one curves, (conference talk), 3.7.2024, 8th Croatian Mathematical Congress, Osijek, Croatia
  • I. Soldo, Diophantine m- tuples in certain imaginary quadratic rings, (conference talk), 3.7.2024, 8th Croatian Mathematical Congress, Osijek, Croatia
  • P. Radić, New results on D(4)-m-tuples, (poster), 4.7.2024, 8th Croatian Mathematical Congress, Osijek, Croatia
  • M. Kazalicki, Improving Mestre-Nagao heuristics with data science, (invited conference talk), 13.11.2024, Murmurations in Arithmetic Geometry and Related Topics, Stony Brook, NY, USA

  In the 2nd project year

  • T. Pejković, Schneider's p-adic continued fractions of rational numbers, (invited conference talk), 22.6.-28.6.2025, Representation Theory XIX, Dubrovnik, Croatia
  • F. Najman, Ranks of elliptic curves over number fields, (invited conference talk), 22.6.-28.6.2025, Representation Theory XIX, Dubrovnik, Croatia
  • F. Najman, Twist parametrized points on modular curves, (invited conference talk), 23.6.-27.6.2025, Algebraic Points on Curves, Providence, RI, USA
  • F. Najman, Torsion of elliptic curves over quartic fields, (conference talk), 27.7.-2.8.2025, Rational Points 2025, Schney, Germany
  • F. Najman, Isolated points on modular curves, (invited conference talk), 9.9.-12.9.2025, Algebraic and geometric methods for Diophantine problems, Pisa, Italy
  • A. Andrašek, Regular Higher Power Diophantine Triples, (invited conference talk), 22.6.-28.6.2025, Representation Theory XIX, Dubrovnik, Croatia
  • M. Kazalicki, Rational Diophantine m-tuples in the work of Andrej Dujella, (invited conference talk), 22.6.-28.6.2025, Representation Theory XIX, Dubrovnik, Croatia
  • S. Bujačić Babić, Regularity of Diophantine quadruples in Q(i)[X], (invited conference talk), 22.6.-28.6.2025, Representation Theory XIX, Dubrovnik, Croatia
  • A. Jurasić, Extensions of the Triangular D(3)-pair {3, 6}, (invited conference talk), 22.6.-28.6.2025, Representation Theory XIX, Dubrovnik, Croatia
  • I. Novak, Torsion of Q-curves over small odd prime degree number fields, (invited conference talk), 22.6.-28.6.2025, Representation Theory XIX, Dubrovnik, Croatia
  • P. Orlić, Tetragonal intermediate modular curves, (conference talk), 22.6.-28.6.2025, Representation Theory XIX, Dubrovnik, Croatia
  • L. Novak, Twists arising from torsion points, (invited conference talk), 22.6.-28.6.2025, Representation Theory XIX, Dubrovnik, Croatia
  • M. Jukić Bokun, Extensions of D(−1)-pairs in imaginary quadratic fields, (invited conference talk), 22.6.-28.6.2025, Representation Theory XIX, Dubrovnik, Croatia
  • I. Soldo, The extendibility of the triangular D(−1)-pair {1, 2}, (invited conference talk), 22.6.-28.6.2025, Representation Theory XIX, Dubrovnik, Croatia
  • I. Soldo, Diophantine quadruples in Q(i)[X], (conference talk), 8.9.-12.9.2025, 26th Central European Number Theory Conference 2025, Salzburg, Austria
  • S. Bujačić Babić, [Naslov predavanja], (conference talk), 30.6.-4.7.2025, 33emes Journees Arithmetiques, Luxembourg
  • I. Novak, Degrees of cyclic isogenies of rational elliptic curves, (conference talk), 30.6.-4.7.2025, 33emes Journees Arithmetiques, Luxembourg
  • P. Orlić, Gonality of modular curves, (invited conference talk), 30.6.-4.7.2025, 33emes Journees Arithmetiques, Luxembourg
  • M. Derickx, Maps from X0(p) to curves of genus at least 2, (conference talk), 23.6.-27.6.2025, Algebraic Points on Curves, Providence, RI, USA
  • M. Derickx, Answering a question of Krumm about the imaginary quadratic points on Y1(16), (conference talk), 27.7.-2.8.2025, Rational Points 2025, Schney, Germany
  • G. Dražić, Asymptotics of the number of D(q)-pairs and D(q)-triples via L-functions, (conference talk), 30.6.-4.7.2025, 33emes Journees Arithmetiques, Luxembourg
  • P. Radić, On the D(4)-pairs {a, ka}, (poster), 22.6.-28.6.2025, Representation Theory XIX, Dubrovnik, Croatia
  • A. Jurasić, On the Triangular Diophantine m-tuples, (conference talk), 22.6.-28.6.2025, 26th Central European Number Theory Conference 2025, Salzburg, Austria
  • M. Kazalicki, Application of machine learning in the study of the ranks of elliptic curves, (conference talk), 30.6.-4.7.2025, 33emes Journees Arithmetiques, Luxembourg
  Participation in summer schools, workshops, and conferences

  In the 1st project year

  • P. Orlic, A Modern Introduction to Number Theory, 1-6.9.2024, Pisa, Italy
  • I. Novak, A Modern Introduction to Number Theory, 1-6.9.2024, Pisa, Italy
  • P. Orlic, Introduction to Number Theory and Algebraic Curves, 9-20.9.2024, Belgrade, Serbia
  • I. Novak, Introduction to Number Theory and Algebraic Curves, 9-20.9.2024, Belgrade, Serbia
  • A. Andrašek, Introduction to Number Theory and Algebraic Curves, 4-8.11.2024, Groningen, Netherlands

  In the 2nd project year

  • F. Najman, A CIMPA school on Serre's big image theorem for Galois representations associated to elliptic curves, 9-17.1.2025, Valparaiso, Chile
  • I. Novak, A CIMPA school on Serre's big image theorem for Galois representations associated to elliptic curves, 9-17.1.2025, Valparaiso, Chile
  Organization of workshops

  In the 1st project year

  In the 2nd project year

  Open positions

  Postdoc position 1 year, start date April or May 2024: Maarten Derickx joins us.

  Announcements
  December 31, 2023 Project started.