Overview of my Research
My research operates at the intersection of Arithmetic and Diophantine Geometry, and Computational Number Theory. A central unifying theme of my work is understanding the distribution, rank, and structural properties of rational points on algebraic varieties (curves, elliptic surfaces, K3 surfaces, abelian varieties, and weighted spaces) over global and function fields
My research program encompasses six main interconnected domains:
I) Arithmetic of Elliptic Curves and $\theta$-Congruent Numbers: Generalizations of classical congruent numbers over $\Q$ and real quadratic/number fields, Fermat-type duplication algorithms, explicit linear independence of power sequences, and rational $\theta$-parallelogram envelopes.
II ) Elliptic K3 Surfaces and Mordell--Weil Lattices:** Determining splitting fields, explicit section generators, and lattice structures for non-trivial families of elliptic K3 surfaces over function fields.
III) Twists of Albanese and Prym Varieties over Higher-Dimensional Function Fields: Bounding Mordell--Weil ranks of Jacobians for cyclic covers $y^s = a x^r + b$, and proving structural isomorphisms for twists of Albanese varieties of cyclic, abelian, and dihedral covers of projective spaces.
IV) Height Theory and Vojta's Conjecture on Weighted Projective Varieties: Formulating local/global heights, stacky height machinery, and generalized weighted greatest common divisors (wgcd) on weighted varieties.
V) Diophantine Approximation and Powerful Values of Polynomials: Bounding polynomial families taking $s$-powerful values assuming quantitative versions of Vojta's conjecture over number fields.
VI) Combinatorial and Cryptographic Applications: Computing zeta functions of Humbert surfaces for post-quantum isogeny-based cryptography, and proving Gaussian limit theorems for linear recurrences with zero coefficients.
In Progress Papers:
3) S. Salami and A. Shamsi Zargar, Quartic elliptic curves with sequences of consecutive powers, In progress
2) S. Salami and A. Shamsi Zargar, Generalization of the congruent number problem in higher dimensions, In progress
1) S. Salami and T. Shaska, Stacky Heights, Entropy, and Zeta Functions of Weighted Hypersurfaces over Finite Fields, In progress.
Submitted Papers
4) S. Salami, The splitting field and generators of the elliptic surfaces, y^2=x^3 +t^360+1, Points-68,
Available at: http://arxiv.org/abs/2512.25009
3) S. Salami and A. Shamsi Zargar, The splitting field and generators of Shioda's elliptic surfaces, y^2=x^3 +t^m+1 (I), Available at: https://arxiv.org/abs/2512.16578
2) S. Salami, T. Shaska, Rational Points in Weighted Projective Spaces over Finite Fields, https://arxiv.org/abs/2511.12812
1) S. Salami, Rank of Jacobian of the curves y^s=x(ax^r+b), Submitted and available at: http://arxiv.org/abs/2511.07680
Accepted and Published Papers
22) S. Salami, Gaussian Behavior and Geometric Gaps in Decompositions from Recurrences with Zero Coefficients, accepted for publication in The Fibonacci Quarterly, and available at: https://arxiv.org/abs/2604.15447
21) S. Salami and A. Shamsi Zargar, Generators and splitting field of certain elliptic K3 surfaces, accepted for publication in Journal de Théorie des Nombres de Bordeaux and available at: http://arxiv.org/abs/2206.05372.
20) S. Salami, On the rank of Jacobian of the curves y^s=ax^r+b, Journal of Number Theory, Vol. 290, pp. 331--346, (2027). DOI: https://doi.org/10.1016/j.jnt.2026.05.011.
(https://www.sciencedirect.com/science/article/pii/S0022314X26001125)
19) S. Salami and A. Shamsi Zargar, On properly θ-congruent numbers over real number fields, Bulletin of the Australian Mathematical Society. 114 (2026), 59–64; DOI: https://doi.org/10.1017/S0004972725100907
18) J. Mello, S. Salami, E. Shaska, and T. Shaska, Rational Points and Zeta Functions of Humbert Surfaces with Square Discriminant, NATO Science for Peace and Security Series - D: Information and Communication Security, Volume 66: Abelian Surfaces and Isogeny-based Cryptography, pp. 92 - 122, (2025), DOI 10.3233/NICSP250004.
17) S. Salami and T. Shaska, Vojta's conjecture on weighted projective varieties, European Journal of Mathematics, Vol. 11, article number 12, (2025), https://link.springer.com/article/10.1007/s40879-024-00804-7
16) S. Salami and T. Shaska, Local and Global heights on Weighted varieties, Houston Journal of Mathematics, Vol. 49, No. 3, pp: 603-636, (2023). Available at: https://arxiv.org/pdf/2204.01624.pdf
15) S. Salami, On the powerful values of polynomials over number fields, Albanian J. Math. Vol. 17 No. 2, pp: 142-165, (2023).
14) A. Mohajer, and S. Salami, Twists of Albanese varieties of abelian and dihedral covers of P^ℓ with large ranks over function fields, Comm. in Algebra 51, no. 8, pp. 3585-3591, (2023).
13) S. Salami and A. Shamsi Zargar, Rational θ-parallelogram envelopes via $\theta$-congruent elliptic curves, Inter. J. of Number Theory, Vol. 19, No. 07, pp.1681-1706, (2023).
12) S. Salami, On special matrices related to Cauchy and Toeplitz matrices, Caderno do IME-UERJ, No. 18, (2022). https://www.e-publicacoes.uerj.br/cadmat/article/view/67104
11) S. Salami and A. Shamsi Zargar, Families of cubic elliptic curves containing sequences of consecutive powers, Rocky Mountain J. Math., Vol. 51, No. 5, pp. 1833–1845, (2021).
10) Corrigendum to “The rational points on certain Abelian varieties over function fields” [J. Number Theory 195 (2019) 330–337; Journal of Number Theory, Vol. 227, pp. 306-307 (2021).
9) S. Salami and A. Shamsi Zargar, The θ-congruent number elliptic curves via Fermat-type algorithms, Bulletin of the Brazilian Mathematical Society, New Series, Vol. 52, pp. 893-908, (2021).
8) S. Salami, Twists of the Albanese varieties of cyclic multiple planes with large ranks over higher-dimensional function fields, Journal de Théorie des Nombres de Bordeaux, Tome 32, No 3, pp. 861-876, (2020).
7) S. Salami, A correction to the paper "On curves with split Jacobians", Comm. in Algebra, Vol. 48, No. 3, pp.1380-1381, (2020).
6) S. Salami, The rational points on certain Abelian varieties over function fields, J. Number Theory, Vol. 195, pp. 330–337, (2019).
5) A. S. Janfada and S. Salami, On the θ-congruent numbers on real quadratic number fields, Kodai Math. J. Vol 38, No.2, 352-364, (2015).
4) A. S. Janfada, S. Salami, A. Dujella, and J. C. Peral, On the high rank 𝝅/3 and 2𝝅/3-congruent number elliptic curves, Rocky Mount. J. Math. Vol. 44, No. 6 (2014).
3) Ali S. Janfada, S. Salami, Congruent numbers problem and the BSD conjecture on the rank of elliptic Curves, Farhang and Andisheh Riazi, 45, (2011) (in Persian).
2) Ali S. Janfada, S. Salami, On the high rank θ-congruent number elliptic curves, 41st Iranian Inter. Math. Conf. 12-15 September (2010).
1) Andrej Dujella, Ali S. Janfada, and Sajad Salami, A search for high rank congruent number elliptic curves, Journal of Integer Sequences, Vol 12, Issue 5, Article 09.5.8, (2009).
Survey Papers:
1) S. Salami, Primitive Pythagorean Triples: A survey, 2025, PDF
2) S. Salami, The Birch and Swinnerton-Dyer Conjecture: A survey, 2025, PDF
Advanced Minicourse Notes:
1) Sajad Salami, Pontos Racionais sobre Curvas Algébricas e Aplicações, apresentado na XII Bienal da Sociedade Brasileira de Matemática (SBM), realizada na Universidade Federal do Rio Grande do Norte (UFRN), em Natal/RN, entre 03 e 07 de agosto de 2026. PDF
My Classes Lecture notes (Brazilian Portuguese)
1) Algebra I, PDF-11-2025
2) Algebra II, PDF-11-2025
3) Algebra III, PDF-03-2026
4) Algebra Linear, PDF-09-2023