PUBLICATIONSALEXANDRU BUIUM

1. Research Monographs[1] Differential Function Fields and Moduli of Algebraic Varieties, Lecture Notes
in Math. 1226, Springer 1986.
[2] Differential Algebraic Groups of Finite Dimension, Lecture Notes in Math.
1506, Springer 1992.
[3] Differential Algebra and Diophantine Geometry, Hermann, Paris, 1994.
[4] Arithmetic Differential Equations, Mathematical Surveys and Monographs
118, AMS, 2005.
[5] Foundations of Arithmetic Differential Geometry, Mathematical Surveys andMonographs 222, AMS, 2017.
[6] (with L.E.Miller), Purely Arithmetic PDEs over a p-Adic Field: δ-Charactersand δ-Modular Forms, Memoirs of the EMS., 2023.
2. Textbooks[1] Mathematics: a Minimal Introduction, CRC Press, 2013.
3. Papers published[1] Espaces projectifs anisotropes et diviseurs amples, C. R. Acad. Sci. ParisSér. I Math. 292 (1981), no. 12, 585–586.
[2] Sur les variétés dont le nombre de Picard est égal à 1, C. R. Acad. Sci. Paris
Sr. I Math. 294 (1982), no. 3, 135-137.
[3] Ritt schemes and torsion theory. Pacific J. Math. 98 (1982), no. 2, 281-293.
[4] Degree of the fibres of an elliptic fibration. Ann. Inst. Fourier (Grenoble) 33(1983), no. 1, 269-276.
[5] Sur le nombre de Picard des revêtements doubles des surfaces algébriques, C.
R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 8, 361-364.
[6] Sur le discriminant des sections d’une variété projective singulière par deshypersurfaces de degré élevé, C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), no.
4, 67-70.
[7] Class groups of differential function fields. J. Algebra 89 (1984), no. 1, 56-64.
[8] On surfaces of degree at most 2n + 1 in Pn . Algebraic geometry (Bucharest1982), Lecture Notes in Math., 1056, Springer, Berlin, 1984, 47-67.
[9] Corps différentiels et modules des variétés algébriques, C. R. Acad. Sci. ParisSér. I Math. 299 (1984), no. 19, 983-985.
[10] Killing divisor classes by algebraisation. Ann. Inst. Fourier (Grenoble) 35(1985), no. 2, 107-115.
[11] Corps de définition des variétés algébriques et théorie de Galois pour lescorps différentiels, C. R. Acad. Sci. Paris Sr. I Math. 300 (1985), no. 18, 627-629.
1

2ALEXANDRU BUIUM[12] A note on the class group of surfaces in 3-space. J. Pure Appl. Algebra 37(1985), no. 3, 229-235.
[13] Movable singularities and differential Galois theory. Proceedings of theconference on algebraic geometry (Berlin, 1985), 74-80, Teubner-Texte Math., 92,Teubner, Leipzig, 1986.
[14] Fields of definition of algebraic varieties in characteristic zero. Compositio
Math. 61 (1987), no. 3, 339-352.
[15] (with Takeuchi, M.) Rigidity of maps from Hopf algebras to group algebras.
J. Algebra 118 (1988), no. 1, 14-19.
[16] Birational moduli and nonabelian cohomology. Compositio Math. 68 (1988),
no. 2, 175-202.
[17] Birational moduli and nonabelian cohomology. II. Compositio Math. 71(1989), no. 3, 247-263.
[18] Dérivations sur les groupes algébriques, C. R. Acad. Sci. Paris Sér. I Math.
309 (1989), no. 9, 605-607.
[19] Splitting differential algebraic groups. J. Algebra 130 (1990), no. 1, 97-105.
[20] Intersections in jet spaces and a conjecture of S.Lang, Annals of Math. 136(1992), 583-593.
[21] Geometry of differential polynomial functions I: algebraic groups, Amer J.
Math. 115, 6 (1993), 1385-1444.
[22] Effective bound for the geometric Lang conjecture, Duke Math. J., 71, 2(1993), 475-499.
[23] A finiteness theorem for isogeny correspondences, in: Journes de Geometrie
Algebrique d’Orsay 1992, Asterisque 218, 1993.
[24] (with F.Voloch) Integral points on abelian varieties over function fields ofcharacteristic zero, Math. Ann. 297 (1993), 303-307.
[25] Geometry of differential polynomial functions II: algebraic curves, Amer J.
Math. 116, 4, (1994), 785-819.
[26] The abc theorem for abelian varieties, Intern. Math. Research Notices 5(1994), 219-233.
[27] On a question of B.Mazur, Duke Math. J., 75, 3, (1994), 639-644.
[28] Geometry of differential polynomial functions III: moduli spaces, Amer J.
Math. 117 (1995), 1-73.
[29] (with F.Voloch) Reduction of the Manin map modulo p, Crelle J., 460 (1995),
117-126.
[30] Differential characters of abelian varieties over p-adic fields, Invent. Math.,
122, 2, (1995), 309-340.
[31] Uniform bound for generic points of curves in tori, Crelle J., 469 (1995),211-219.
[32] (with F.Voloch) Mordell conjecture in characteristic p: an explicit bound,Compositio Math., 103, (1996), 1-6.
[33] Geometry of p-jets, Duke Math. J., 82, 2, (1996), 349-367.
[34] An approximation property for Teichmuller points, Math. Research Letters,
3 (1996), 453-457.
[35] Differential characters and characteristic polynomial of Frobenius, Crelle J.,
485 (1997), 209-219.
[36] (with A. Pillay) A gap theorem for abelian varieties over differential fields,
Math. Research Letters 4, 211-219 (1997).

PUBLICATIONS

3[37] Differential algebraic geometry and diophantine geometry: an overview, in:Symposia Math. 37, Cambridge U.P., 1997.
[38] Arithmetic analogues of derivations, J. Algebra, 198, (1997), 290-299.
[39] Intersection multiplicities on abelian varieties, Math. Ann. 310 (1998),
653-659.
[40] Differential subgroups of simple algebraic groups over p-adic fields, Amer.
J. Math. 120 (1998), 1277-1287.
[41] (with P. Cassidy) Differential algebraic geometry and differential algebraic
groups, in: Collected Works of Ellis Kolchin, AMS, 1999.
[42] Differential Modular Forms, Crelle J., 520 (2000), 95-167.
[43] Continuous p-adic functions and p-derivations, J. Number Theory, 84 (2000),
34-39.
[44] Infinitesimal Mordell-Lang, J. Number Theory 90 (2001), 185-206.
[45] (with M. Barcau) Siegel differential modular forms, International Math. Res.
Notices, 2002, No. 28, pp.1459-1503.
[46] Quotients of algebraic varieties by Zariski dense equivalence relations, in:Parshin Festschrift, Zarhin and Vostokov Eds., Contemporary Math., Vol. 300,AMS (2002), 59-97.
[47] (with K. Zimmerman) Homogeneous p-differential polynomials, J. Algebra,269 (2003), 492-507.
[48] Differential modular forms on Shimura curves,I, Compositio Math. 139(2003), 197-237.
[49] Differential modular forms on Shimura curves, II: Serre operators, Compositio Math. 140 (2004), 1113-1134.
[50] Pfaffian equations satisfied by differential modular forms, Math. Res. Letters 11 (2004), 453-466.
[51] Geometry of Fermat Adeles, Trans. AMS 357, 3 (2005), 901-964.
[52] (with K. Zimmerman) Differential orbit spaces of discrete dynamical systems, Crelle J. 580 (2005), 201-230.
[53] Complex dynamics and invariant forms mod p, International Math. Res.
Notices 31 (2005), 1889-1899.
[54] Finiteness results in differential algebraic geometry and diophantine geometry, chapter in: CRM Monograph Series 24, AMS, 2005.
[55] A differential equation satisfied by the arithmetic Kodaira-Spencer class, J.
Number Theory 119 (2006), 297-306.
[56] Correspondences, Fermat quotients, and uniformization, in: Groupes de Galois arithmetiques et differentiels, Seminaires et Congres 13, Société Mathématiquede France, (2007), 70-89.
[57] Differential eigenforms, J. Number Theory 128 (2008), 979-1010.
[58] (with B. Poonen) Independence of points on elliptic curves arising from
special points on modular and Shimura curves, I: global results, Duke Math. J.,
147, 1 (2009), 181-191.
[59] (with B. Poonen) Independence of points on elliptic curves arising from
special points on modular and Shimura curves, II: local results, Compositio Math.,
145 (2009), 566-602.
[60] (with S. Simanca) Arithmetic differential equations in several variables, Annales Inst. Fourier, 59, 7 (2009), 2685-2708 (volume dedicated to B. Malgrange).

4ALEXANDRU BUIUM[61] (with S.R. Simanca) Arithmetic Laplacians, Advances in Math. 220 (2009),
246-277.
[62] (with S. R. Simanca) Arithmetic partial differential equations I, Advancesin Math. 225 (2010), 689-793.
[63] (with S. R. Simanca) Arithmetic partial differential equations II, Advancesin Math., 225 (2010), 1308-1340.
[64] (with C. Ralph and S.R. Simanca) Arithmetic differential operators on Zp ,J. Number Theory 131 (2011), pp. 96-105.
[65] (with J. Borger) Differential forms on arithmetic jet spaces, Selecta Mathematica 17, 2 (2011), pp. 301-335.
[66] Differential characters on curves, in: Number Theory, Analysis and Geometry: In Memory of Serge Lang, D. Goldfeld et al. Editors, Springer, 2011, pp.
111-123.
[67] (with A.Saha) Differential overconvergence, in: Algebraic methods in dynamical systems; volume dedicated to Michael Singer’s 60th birthday, Banach Center Publications, Vol 94, 99-129 (2011).
[68] (with A.Saha) Hecke operators on differential modular forms mod p, J.
Number Theory 132 (2012), 966-997.
[69] (with A. Saha) The ring of differential Fourier expansions, J. of Number
Theory 132 (2012), 896-937.
[70] p-jets of finite algebras, I: p-divisible groups, Documenta Math. 18 (2013)
943–969.
[71] p-jets of finite algebras, II: p-typical Witt rings, Documenta Math. 18 (2013)
971–996.
[72] Galois groups arising from arithmetic differential equations, in: Geometricand differential Galois theories, D. Bertand et. al. Eds., Séminaires et Congrès,Vol. 27, Soc. Math. France, 2013, pp. 13-25.
[73] (with A. Saha) The first p-jet space of an elliptic curve: global functions
and lifts of Frobenius, Math. Res. Letters, Vol. 21, Number 04 (2014), 677-689.
[74] Differential modular forms attached to newforms mod p, J. Number Theory
155 (2015), 111-128.
[75] Transcendental numbers as solutions to arithmetic differential equations,
Bulletin Math. de la Soc. des Sciences Math. de Roumanie, 58 (106) 3, (2015),
245-256. (Special number dedicated to V. Brinzanescu’s 70th birthday).
[76] Arithmetic analogues of some basic concepts from Riemannian geometry,
Revue Roumaine de Mathematiques Pures et Appliquées, Vol. LX, 3, (2015), 257266. (Special number dedicated to L. Badescu’s 70th birthday).
[77] (with Yu. I. Manin) Arithmetic differential equations of Painlevé VI type,
in: Arithmetic and Geometry, London Mathematical Society Lecture Note Series:
420, L. Dieulefait, G. Faltings, D. R. Heath-Brown, Yu. V. Manin, B. Z. Moroz
and J.-P. Wintenberger (eds), Cambridge University Press, 2015, pp. 114-138.
[78] Differential calculus with integers, in: Arithmetic and Geometry, London
Mathematical Society Lecture Note Series: 420, L. Dieulefait, G. Faltings, D. R.
Heath-Brown, Yu. V. Manin, B. Z. Moroz and J.-P. Wintenberger (eds), Cambridge
University Press, 2015, pp. 139-187.
[79] (with T. Dupuy) Arithmetic differential equations on GLn , I: differential
cocycles, arXiv:1308.0748v1, J.Algebra, 454 (2016), 273291.

PUBLICATIONS

5[80] (with T. Dupuy) Arithmetic differential equations on GLn , II: arithmetic
Lie-Cartan theory, arXiv:1308.0744, Selecta Math. 22, 2, (2016), 447-528.
[81] (with T. Dupuy) Arithmetic differential equations on GLn , III: Galois groups,
arXiv:1308.0747, Selecta Math. 22, 2, (2016), 529-552.
[82] (with M. Barrett) Curvature on the integers, I, Journal of Number Theory,167 (2016) pp. 481-508.
[83] Curvature on the integers, II, Journal of Number Theory, 167 (2016) pp.
509-545.
[84] (with E. Previato) Arithmetic Euler top, J. Number Theory, 173 (2017) pp.
37-63.
[85] (with E. Previato) The Euler top and canonical lifts, J. Number Theory 190(2018), 156-168.
[86] Arithmetic Levi-Civita connection, Selecta Math. New Ser. (2019) 25:12,
https://dot.org/10.1007/s00029-019-0464-0, 79 pages.
[87] Arithmetic analogues of Hamiltonian systems, in: Integrable Systems andAlgebraic Geometry, Volume 2, London Math. Soc., Lecture Notes Series 459,
Cambridge University Press, 2020, pp. 13-40.
[88] (with L.E.Miller) Solutions to arithmetic differential equations in algebraicallyclosed fields, Advances in Math. 375 (2020), 107342, 47pp.
[89] (with L.E.Miller) Perfectoid spaces arising from arithmetic differential equations, American J. Math. 145, No. 1, (2023), 41pp.
[90] Lie invariant Frobenius lifts, J. Number Theory, 2024.
4. Papers accepted for publication[91] (with L.E.Miller), A. Buium, L.E.Miller, Arithmetic connections in the arithmetic PDE setting, 40 pages, submitted in 2021 to Algebraic Geometry and Physics,
volume dedicated to Yu.I Manin’s 85th birthday. Accepted September 2024.
5. Preprints[92] (with A. Vasiu), The δ-invariant theory of Hecke correspondences, 200 pages,arXiv., 2022.
[93] (with L.E.Miller), Arithmetic differential geometry and arithmetic PDEs,
243 pages.
6. Other publications[94] Serge Lang’s early work on Diophantine and algebraic geometry, Notices ofthe AMS 54, 4 (2007), 479-484.