@article{AKKM,
  author  = {Atanasov, Stanislav and Kaplan, Nathan and Krakoff, Benjamin and Menzel, Julia H.},
  title   = {Counting finite index subrings of {$\mathbb Z^n$}},
  journal = {Acta Arith.},
  volume  = {197},
  number  = {3},
  year    = {2021},
  pages   = {221--246},
  doi     = {10.4064/aa180201-29-7}
}

@article{Voll2010,
  author  = {Voll, Christopher},
  title   = {Functional equations for zeta functions of groups and rings},
  journal = {Ann. of Math. (2)},
  volume  = {172},
  number  = {2},
  year    = {2010},
  pages   = {1185--1222},
  doi     = {10.4007/annals.2010.172.1185}
}

@article{GrigorchukNagnibeda1997,
  author  = {Grigorchuk, Rostislav and Nagnibeda, Tatiana},
  title   = {Complete growth functions of hyperbolic groups},
  journal = {Invent. Math.},
  volume  = {130},
  number  = {1},
  year    = {1997},
  pages   = {159--188},
  doi     = {10.1007/s002220050181}
}

@book{LubotzkySegal2003,
  author    = {Lubotzky, Alexander and Segal, Dan},
  title     = {Subgroup Growth},
  series    = {Progress in Mathematics},
  volume    = {212},
  publisher = {Birkh{\"a}user},
  address   = {Basel},
  year      = {2003},
  doi       = {10.1007/978-3-0348-8965-0}
}

@article{LarsenLubotzky2008,
  author  = {Larsen, Michael and Lubotzky, Alexander},
  title   = {Representation growth of linear groups},
  journal = {J. Eur. Math. Soc.},
  volume  = {10},
  number  = {2},
  year    = {2008},
  pages   = {351--390},
  doi     = {10.4171/JEMS/113}
}

@incollection{Voll2011,
  author    = {Voll, Christopher},
  title     = {A newcomer's guide to zeta functions of groups and rings},
  booktitle = {Lectures on Profinite Topics in Group Theory},
  editor    = {Segal, Dan},
  series    = {London Mathematical Society Student Texts},
  volume    = {77},
  publisher = {Cambridge University Press},
  address   = {Cambridge},
  year      = {2011},
  pages     = {99--144},
  doi       = {10.1017/CBO9780511793837.005}
}

@article{IshamLowerBounds,
  author        = {Isham, Kelly},
  title         = {Lower bounds for the number of subrings in
                   {$\mathbb Z^n$}},
  journal       = {Journal of Number Theory},
  volume        = {234},
  year          = {2022},
  pages         = {363--390},
  doi           = {10.1016/j.jnt.2021.06.026},
  eprint        = {2010.09123},
  archiveprefix = {arXiv},
  primaryclass  = {math.NT}
}

@misc{IshamKaplan,
  author        = {Isham, Kelly and Kaplan, Nathan},
  title         = {Most subrings of {$\mathbb Z^n$} have large corank},
  year          = {2024},
  note          = {With an appendix by Gautam Chinta},
  eprint        = {2412.18692},
  archiveprefix = {arXiv},
  primaryclass  = {math.NT}
}

@incollection{Voll2015,
  author    = {Voll, Christopher},
  title     = {Zeta functions of groups and rings -- recent developments},
  booktitle = {Groups St Andrews 2013},
  publisher = {Cambridge Univ. Press},
  year      = {2015},
  pages     = {469--492},
  doi       = {10.1017/cbo1316227343.031}
}
@book{KatzMazur,
  author    = {Katz, Nicholas M. and Mazur, Barry},
  title     = {Arithmetic Moduli of Elliptic Curves},
  series    = {Annals of Mathematics Studies},
  volume    = {108},
  publisher = {Princeton University Press},
  address   = {Princeton, NJ},
  year      = {1985}
}

@book{SGA4III,
  editor    = {Artin, Michael and Grothendieck, Alexander
               and Verdier, Jean-Louis},
  title     = {Th\'eorie des topos et cohomologie \'etale
               des sch\'emas. Tome 3},
  series    = {Lecture Notes in Mathematics},
  volume    = {305},
  publisher = {Springer-Verlag},
  address   = {Berlin},
  year      = {1973}
}

@book{MilneAlgebraicGroups,
  author    = {Milne, J. S.},
  title     = {Algebraic Groups: The Theory of Group Schemes
               of Finite Type over a Field},
  series    = {Cambridge Studies in Advanced Mathematics},
  volume    = {170},
  publisher = {Cambridge University Press},
  year      = {2017}
}

@incollection{EisenbudHarris1987,
  author    = {Eisenbud, David and Harris, Joe},
  title     = {On varieties of minimal degree
               (a centennial account)},
  booktitle = {Algebraic Geometry, Bowdoin, 1985, Part 1},
  series    = {Proceedings of Symposia in Pure Mathematics},
  volume    = {46},
  pages     = {3--13},
  publisher = {American Mathematical Society},
  address   = {Providence, RI},
  year      = {1987}
}

@article{BjornerEkedahl,
  author  = {Bj{\"o}rner, Anders and Ekedahl, Torsten},
  title   = {Subspace arrangements over finite fields:
             cohomological and enumerative aspects},
  journal = {Advances in Mathematics},
  volume  = {129},
  number  = {2},
  year    = {1997},
  pages   = {159--187},
  doi     = {10.1006/aima.1997.1647}
}

@book{FultonIntersection,
  author    = {Fulton, William},
  title     = {Intersection Theory},
  edition   = {2},
  series    = {Ergebnisse der Mathematik und ihrer Grenzgebiete.
               3. Folge},
  volume    = {2},
  publisher = {Springer-Verlag},
  address   = {Berlin},
  year      = {1998}
}

@article{FisherInvariants,
  author  = {Fisher, Tom},
  title   = {The invariants of a genus one curve},
  journal = {Proc. Lond. Math. Soc. (3)},
  volume  = {97},
  number  = {3},
  year    = {2008},
  pages   = {753--782},
  doi     = {10.1112/plms/pdn021},
  eprint  = {math/0610318},
  archiveprefix = {arXiv}
}

@article{KMT,
  author  = {Kaplan, Nathan and Marcinek, Jake
             and Takloo-Bighash, Ramin},
  title   = {Distribution of orders in number fields},
  journal = {Research in the Mathematical Sciences},
  volume  = {2},
  year    = {2015},
  note    = {Article 6, 57 pp.},
  doi     = {10.1186/s40687-015-0027-8}
}

@book{AdkinsWeintraub,
  author    = {Adkins, William A. and Weintraub, Steven H.},
  title     = {Algebra: An Approach via Module Theory},
  series    = {Graduate Texts in Mathematics},
  volume    = {136},
  publisher = {Springer-Verlag},
  address   = {New York},
  year      = {1992},
  doi       = {10.1007/978-1-4612-0923-2}
}

@misc{MilneLEC,
  author = {Milne, James S.},
  title  = {Lectures on \'{E}tale Cohomology},
  year   = {2013},
  note   = {Version 2.21},
  url    = {https://www.jmilne.org/math/CourseNotes/LEC.pdf}
}

@incollection{Deligne,
  author    = {Deligne, Pierre},
  title     = {Formes modulaires et repr\'{e}sentations {$\ell$}-adiques},
  booktitle = {S\'{e}minaire Bourbaki 1968/69},
  series    = {Lecture Notes in Math.},
  volume    = {179},
  publisher = {Springer},
  address   = {Berlin},
  year      = {1971},
  pages     = {139--172}
}

@article{DeligneHodgeII,
  author  = {Deligne, Pierre},
  title   = {Th\'{e}orie de {H}odge. {II}},
  journal = {Inst. Hautes \'{E}tudes Sci. Publ. Math.},
  volume  = {40},
  year    = {1971},
  pages   = {5--57},
  doi     = {10.1007/BF02684692}
}

@article{Getzler99,
  author  = {Getzler, Ezra},
  title   = {Resolving mixed {H}odge modules on configuration spaces},
  journal = {Duke Math. J.},
  volume  = {96},
  number  = {1},
  year    = {1999},
  pages   = {175--203},
  doi     = {10.1215/S0012-7094-99-09605-9}
}

@misc{Gorinov,
  author        = {Gorinov, Alexey G.},
  title         = {Rational cohomology of the moduli spaces of pointed genus {$1$} curves},
  year          = {2013},
  howpublished  = {\href{https://arxiv.org/abs/1303.5693}{arXiv:1303.5693}},
  eprint        = {1303.5693},
  archiveprefix = {arXiv},
  primaryclass  = {math.AG}
}

@book{Hartshorne,
  author    = {Hartshorne, Robin},
  title     = {Algebraic Geometry},
  series    = {Graduate Texts in Mathematics},
  volume    = {52},
  publisher = {Springer-Verlag},
  address   = {New York},
  year      = {1977}
}

@article{Isham,
  author  = {Isham, Kelly},
  title   = {Is the number of subrings of index {$p^e$} in {$\mathbb Z^n$} polynomial in {$p$}?},
  journal = {Comm. Algebra},
  volume  = {51},
  number  = {12},
  year    = {2023},
  pages   = {5127--5156},
  doi     = {10.1080/00927872.2023.2225600},
  eprint  = {2203.00646},
  archiveprefix = {arXiv}
}

@article{Liu,
  author  = {Liu, Ricky Ini},
  title   = {Counting subrings of {$\mathbb Z^n$} of index {$k$}},
  journal = {J. Combin. Theory Ser. A},
  volume  = {114},
  number  = {2},
  year    = {2007},
  pages   = {278--299},
  doi     = {10.1016/j.jcta.2006.05.002}
}

@book{MilneEtale,
  author    = {Milne, James S.},
  title     = {\'{E}tale Cohomology},
  series    = {Princeton Mathematical Series},
  volume    = {33},
  publisher = {Princeton University Press},
  address   = {Princeton, NJ},
  year      = {1980}
}

@misc{MishraRay,
  author        = {Mishra, Hrishabh and Ray, Anwesh},
  title         = {On the number of subrings of {$\mathbb Z^n$} of prime power index},
  year          = {2026},
  howpublished  = {\href{https://arxiv.org/abs/2211.16595}{arXiv:2211.16595}},
  eprint        = {2211.16595},
  archiveprefix = {arXiv},
  primaryclass  = {math.NT},
  note          = {Version 2}
}

@book{PetersSteenbrink,
  author    = {Peters, Chris A. M. and Steenbrink, Joseph H. M.},
  title     = {Mixed {H}odge Structures},
  series    = {Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge},
  volume    = {52},
  publisher = {Springer-Verlag},
  address   = {Berlin},
  year      = {2008},
  doi       = {10.1007/978-3-540-77017-6}
}

@article{Petersen,
  author  = {Petersen, Dan},
  title   = {Cusp form motives and admissible {$G$}-covers},
  journal = {Algebra Number Theory},
  volume  = {6},
  number  = {6},
  year    = {2012},
  pages   = {1199--1221},
  doi     = {10.2140/ant.2012.6.1199}
}

@book{SGA4half,
  author    = {Deligne, Pierre},
  title     = {Cohomologie \'{e}tale: {S\'{e}minaire de G\'{e}om\'{e}trie Alg\'{e}brique du Bois-Marie} ({SGA} 4{$\frac12$})},
  series    = {Lecture Notes in Mathematics},
  volume    = {569},
  publisher = {Springer-Verlag},
  address   = {Berlin},
  year      = {1977},
  doi       = {10.1007/BFb0091516}
}

@article{Scholze13,
  author  = {Scholze, Peter},
  title   = {{$p$}-adic {H}odge theory for rigid-analytic varieties},
  journal = {Forum Math. Pi},
  volume  = {1},
  year    = {2013},
  pages   = {e1, 77},
  doi     = {10.1017/fmp.2013.1}
}

@book{SerreAbelianLadic,
  author    = {Serre, Jean-Pierre},
  title     = {Abelian {$\ell$}-adic Representations and Elliptic Curves},
  series    = {Research Notes in Mathematics},
  volume    = {7},
  publisher = {A K Peters},
  address   = {Wellesley, MA},
  year      = {1998},
  note      = {Revised reprint of the 1968 original}
}

@book{SilvermanAEC,
  author    = {Silverman, Joseph H.},
  title     = {The Arithmetic of Elliptic Curves},
  edition   = {2},
  series    = {Graduate Texts in Mathematics},
  volume    = {106},
  publisher = {Springer},
  address   = {New York},
  year      = {2009},
  doi       = {10.1007/978-0-387-09494-6}
}

@incollection{Denef1991,
 author = {Denef, Jan},
 title = {Report on {Igusa}'s local zeta function},
 booktitle = {S\'eminaire Bourbaki, Vol. 1990/91},
 series = {Ast\'erisque},
 number = {201--203},
 year = {1991},
 pages = {359--386},
 note = {Expos\'e No. 741},
 publisher = {Soci\'et\'e math\'ematique de France},
 url = {https://numdam.org/item/SB_1990-1991__33__359_0/}
}

@article{duSautoyGrunewald2000,
 author = {du Sautoy, Marcus and Grunewald, Fritz},
 title = {Analytic properties of zeta functions and subgroup growth},
 journal = {Ann. of Math. (2)},
 volume = {152},
 number = {3},
 year = {2000},
 pages = {793--833},
 doi = {10.2307/2661355}
}

@article{MirandolaZemor2015,
 author = {Mirandola, Diego and Z\'emor, Gilles},
 title = {Critical pairs for the {Product Singleton Bound}},
 journal = {IEEE Trans. Inform. Theory},
 volume = {61},
 number = {9},
 year = {2015},
 pages = {4928--4937},
 doi = {10.1109/TIT.2015.2450207},
 eprint = {1501.06419},
 archiveprefix = {arXiv}
}


@article{Hironaka1964,
  author  = {Hironaka, Heisuke},
  title   = {Resolution of singularities of an algebraic variety over a field of characteristic zero. {I}},
  journal = {Ann. of Math. (2)},
  volume  = {79},
  number  = {1},
  year    = {1964},
  pages   = {109--203},
  doi     = {10.2307/1970486}
}

@article{Hironaka1964II,
  author  = {Hironaka, Heisuke},
  title   = {Resolution of singularities of an algebraic variety over a field of characteristic zero. {II}},
  journal = {Ann. of Math. (2)},
  volume  = {79},
  number  = {2},
  year    = {1964},
  pages   = {205--326},
  doi     = {10.2307/1970547}
}

@article{Ribet1975,
  author  = {Ribet, Kenneth A.},
  title   = {On {$\ell$}-adic representations attached to modular forms},
  journal = {Invent. Math.},
  volume  = {28},
  number  = {3},
  year    = {1975},
  pages   = {245--275},
  doi     = {10.1007/BF01425561},
  url     = {https://math.berkeley.edu/~ribet/Articles/invent_28.pdf}
}

@article{Scholl1990,
  author  = {Scholl, Anthony J.},
  title   = {Motives for modular forms},
  journal = {Invent. Math.},
  volume  = {100},
  year    = {1990},
  pages   = {419--430},
  doi     = {10.1007/BF01231194}
}

@article{KaplanPetrow2017,
  author={Kaplan, Nathan and Petrow, Ian},
  title={Elliptic curves over a finite field and the trace formula},
  journal={Proc. Lond. Math. Soc. (3)},
  volume={115}, number={6}, year={2017}, pages={1317--1372},
  doi={10.1112/plms.12069},
  eprint={1510.03980}, archiveprefix={arXiv}, primaryclass={math.NT}
}
@article{LetellierRodriguezVillegas2023,
  author={Letellier, Emmanuel and Rodriguez-Villegas, Fernando},
  title={{E}-series of character varieties of non-orientable surfaces},
  journal={Annales de l'Institut Fourier},
  volume={73}, number={4}, year={2023}, pages={1385--1420},
  doi={10.5802/aif.3540}
}

@article{DeligneHodgeIII,
  author  = {Deligne, Pierre},
  title   = {Th\'eorie de {H}odge. {III}},
  journal = {Inst. Hautes \'Etudes Sci. Publ. Math.},
  volume  = {44},
  year    = {1974},
  pages   = {5--77},
  doi     = {10.1007/BF02685881},
  url     = {https://www.numdam.org/item/PMIHES_1974__44__5_0/}
}

@misc{StacksProject,
  author       = {{The Stacks Project Authors}},
  title        = {The Stacks Project},
  howpublished = {\url{https://stacks.math.columbia.edu}},
  note         = {Lemma 10.32.6, Tag 00J9}
}

% REVIEW-ADD-BEGIN new-bibliography
% New entries only. Verified against publisher records and the cited papers.
% These entries supplement references.bib; no existing entry is altered.

@article{GSS1988,
  author  = {Grunewald, F. J. and Segal, D. and Smith, G. C.},
  title   = {Subgroups of finite index in nilpotent groups},
  journal = {Invent. Math.},
  volume  = {93},
  number  = {1},
  year    = {1988},
  pages   = {185--223},
  doi     = {10.1007/BF01393692},
  url     = {https://link.springer.com/article/10.1007/BF01393692}
}

@article{duSautoy2002,
  author  = {du Sautoy, Marcus},
  title   = {Counting subgroups in nilpotent groups and points on elliptic curves},
  journal = {J. Reine Angew. Math.},
  volume  = {549},
  year    = {2002},
  pages   = {1--21},
  doi     = {10.1515/crll.2002.063},
  url     = {https://marcusdusautoy.com/wp-content/uploads/2016/05/duSautoyPublication6.pdf}
}

@article{Birch1968,
  author  = {Birch, B. J.},
  title   = {How the number of points of an elliptic curve over a fixed prime field varies},
  journal = {J. London Math. Soc.},
  volume  = {43},
  number  = {1},
  year    = {1968},
  pages   = {57--60},
  doi     = {10.1112/jlms/s1-43.1.57},
  url     = {https://academic.oup.com/jlms/article-abstract/s1-43/1/57/934352}
}

% REVIEW-ADD-END new-bibliography