@misc{ho2014average,
      title={Average size of 2-Selmer groups of elliptic curves over function fields}, 
      author={Q. P. Ho and B. V. Le Hung and B. C. Ngo},
      year={2014},
      eprint={1310.7963},
      archivePrefix={arXiv},
      primaryClass={math.NT},
      howpublished = {\url{https://arxiv.org/abs/1310.7963}}
}

@article{Landesman_2021,
   title={The geometric average size of Selmer groups over function fields},
   volume={15},
   ISSN={1937-0652},
   url={http://dx.doi.org/10.2140/ant.2021.15.673},
   DOI={10.2140/ant.2021.15.673},
   number={3},
   journal={Algebra \& Number Theory},
   publisher={Mathematical Sciences Publishers},
   author={Landesman, Aaron},
   year={2021},
   month={May},
   pages={673–709}
}

@article{basic_hyper,
     author = {L{\o}nsted, Knud and Kleiman, Steven},
     title = {Basics on families of hyperelliptic curves},
     journal = {Compositio Mathematica},
     pages = {83--111},
     publisher = {Sijthoff et Noordhoff International Publishers},
     volume = {38},
     number = {1},
     year = {1979},
     zbl = {0406.14017},
     mrnumber = {523266},
     language = {en},
     url = {http://www.numdam.org/item/CM_1979__38_1_83_0/}
}

@misc{conrad-min,
    author = {Conrad, Brian},
    year = {2005},
    month = {},
    pages = {},
    title = {Minimal models for elliptic curves},
    howpublished = {\url{http://math.stanford.edu/~conrad/papers/minimalmodel.pdf}},
}

@misc{alper,
    author = {Alper, Jarod},
    year = {2022},
    month = {},
    pages = {},
    title = {Notes on Stacks and Moduli},
    howpublished = {\url{https://sites.math.washington.edu/~jarod/moduli.pdf}},
}

@misc{stacks-project,
  author       = {The {Stacks project authors}},
  title        = {The Stacks project},
  howpublished = {\url{https://stacks.math.columbia.edu}},
  year         = {2021},
}

@misc{aaron-geo-cl,
      title={A geometric approach to the Cohen-Lenstra heuristics}, 
      author={Aaron Landesman},
      year={2021},
      eprint={2106.10357},
      archivePrefix={arXiv},
      primaryClass={math.NT},
      howpublished = {\url{https://arxiv.org/abs/2106.10357}}
}

@misc{aaron-thesis,
      title={A thesis of minimal degree: two}, 
      author={Aaron Landesman},
      year={2021},
      howpublished = {\url{https://stacks.stanford.edu/file/druid:fv070dn6727/landesman-thesis-5-30-21-augmented.pdf}}
}

@article {compI,
    AUTHOR = {Altman, Allen B. and Kleiman, Steven L.},
     TITLE = {Compactifying the {P}icard scheme},
   JOURNAL = {Adv. in Math.},
  FJOURNAL = {Advances in Mathematics},
    VOLUME = {35},
      YEAR = {1980},
    NUMBER = {1},
     PAGES = {50--112},
      ISSN = {0001-8708},
   MRCLASS = {14K30 (14C05)},
  MRNUMBER = {555258},
MRREVIEWER = {T. Oda},
       DOI = {10.1016/0001-8708(80)90043-2},
       URL = {https://doi.org/10.1016/0001-8708(80)90043-2},
}
		

@article {compII,
    AUTHOR = {Altman, Allen B. and Kleiman, Steven L.},
     TITLE = {Compactifying the {P}icard scheme. {II}},
   JOURNAL = {Amer. J. Math.},
  FJOURNAL = {American Journal of Mathematics},
    VOLUME = {101},
      YEAR = {1979},
    NUMBER = {1},
     PAGES = {10--41},
      ISSN = {0002-9327},
   MRCLASS = {14K30 (14C05)},
  MRNUMBER = {527824},
MRREVIEWER = {T. Oda},
       DOI = {10.2307/2373937},
       URL = {https://doi.org/10.2307/2373937},
}
		
@incollection {kass,
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     PAGES = {391--427},
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      YEAR = {2013},
   MRCLASS = {14H40 (14C20 14K30)},
  MRNUMBER = {3114949},
MRREVIEWER = {Arnaud Beauville},
}

@inproceedings {de-rap,
    AUTHOR = {Deligne, P. and Rapoport, M.},
     TITLE = {Les sch\'{e}mas de modules de courbes elliptiques},
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     PAGES = {143--316. Lecture Notes in Math., Vol. 349},
      YEAR = {1973},
   MRCLASS = {14K10 (10D05)},
  MRNUMBER = {0337993},
MRREVIEWER = {T. Oda},
}

@book {silverman,
    AUTHOR = {Silverman, Joseph H.},
     TITLE = {The arithmetic of elliptic curves},
    SERIES = {Graduate Texts in Mathematics},
    VOLUME = {106},
   EDITION = {Second},
 PUBLISHER = {Springer, Dordrecht},
      YEAR = {2009},
     PAGES = {xx+513},
      ISBN = {978-0-387-09493-9},
   MRCLASS = {11-02 (11G05 11G20 14H52 14K15)},
  MRNUMBER = {2514094},
MRREVIEWER = {Vasil\cprime  \={I}. Andr\={\i}\u{\i}chuk},
       DOI = {10.1007/978-0-387-09494-6},
       URL = {https://doi.org/10.1007/978-0-387-09494-6},
}

@book {silverman2,
    AUTHOR = {Silverman, Joseph H.},
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    SERIES = {Graduate Texts in Mathematics},
    VOLUME = {151},
 PUBLISHER = {Springer-Verlag, New York},
      YEAR = {1994},
     PAGES = {xiv+525},
      ISBN = {0-387-94328-5},
   MRCLASS = {11G05 (11G07 11G15 11G40 14H52)},
  MRNUMBER = {1312368},
MRREVIEWER = {Henri Darmon},
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@misc{cesnavicius2015selmer,
      title={Selmer groups as flat cohomology groups}, 
      author={Kestutis Cesnavicius},
      year={2015},
      eprint={1301.4724},
      archivePrefix={arXiv},
      primaryClass={math.NT},
      howpublished={\url{https://arxiv.org/abs/1301.4724}}
}

@incollection {dejong,
    AUTHOR = {de Jong, A. J.},
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      NOTE = {Dedicated to Yuri I. Manin on the occasion of his 65th
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   JOURNAL = {Mosc. Math. J.},
  FJOURNAL = {Moscow Mathematical Journal},
    VOLUME = {2},
      YEAR = {2002},
    NUMBER = {2},
     PAGES = {281--311},
      ISSN = {1609-3321},
   MRCLASS = {11G05 (11G25 11G50 14G25)},
  MRNUMBER = {1944508},
MRREVIEWER = {Yuri Tschinkel},
       DOI = {10.17323/1609-4514-2002-2-2-281-311},
       URL = {https://doi-org.libproxy.mit.edu/10.17323/1609-4514-2002-2-2-281-311},
}

@book {mumford,
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     TITLE = {Abelian varieties},
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              Corrected reprint of the second (1974) edition},
 PUBLISHER = {Published for the Tata Institute of Fundamental Research,
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      YEAR = {2008},
     PAGES = {xii+263},
      ISBN = {978-81-85931-86-9; 81-85931-86-0},
   MRCLASS = {14Kxx},
  MRNUMBER = {2514037},
}

@book {neron,
    AUTHOR = {Bosch, Siegfried and L\"{u}tkebohmert, Werner and Raynaud, Michel},
     TITLE = {N\'{e}ron models},
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    VOLUME = {21},
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      YEAR = {1990},
     PAGES = {x+325},
      ISBN = {3-540-50587-3},
   MRCLASS = {14K15 (11G10 14L15)},
  MRNUMBER = {1045822},
MRREVIEWER = {James Milne},
       DOI = {10.1007/978-3-642-51438-8},
       URL = {https://doi-org.libproxy.mit.edu/10.1007/978-3-642-51438-8},
}

@incollection {wpv,
    AUTHOR = {Dolgachev, Igor},
     TITLE = {Weighted projective varieties},
 BOOKTITLE = {Group actions and vector fields ({V}ancouver, {B}.{C}., 1981)},
    SERIES = {Lecture Notes in Math.},
    VOLUME = {956},
     PAGES = {34--71},
 PUBLISHER = {Springer, Berlin},
      YEAR = {1982},
   MRCLASS = {14L32 (14A05 14B05)},
  MRNUMBER = {704986},
       DOI = {10.1007/BFb0101508},
       URL = {https://doi.org/10.1007/BFb0101508},
}

@book {olsson,
    AUTHOR = {Olsson, Martin},
     TITLE = {Algebraic spaces and stacks},
    SERIES = {American Mathematical Society Colloquium Publications},
    VOLUME = {62},
 PUBLISHER = {American Mathematical Society, Providence, RI},
      YEAR = {2016},
     PAGES = {xi+298},
      ISBN = {978-1-4704-2798-6},
   MRCLASS = {14D23 (14D22)},
  MRNUMBER = {3495343},
MRREVIEWER = {Stefan Schr\"{o}er},
       DOI = {10.1090/coll/062},
       URL = {https://doi.org/10.1090/coll/062},
}

@article {ulmer-p,
    AUTHOR = {Ulmer, Douglas L.},
     TITLE = {{$p$}-descent in characteristic {$p$}},
   JOURNAL = {Duke Math. J.},
  FJOURNAL = {Duke Mathematical Journal},
    VOLUME = {62},
      YEAR = {1991},
    NUMBER = {2},
     PAGES = {237--265},
      ISSN = {0012-7094},
   MRCLASS = {11G40 (11G05 11G18)},
  MRNUMBER = {1104524},
MRREVIEWER = {Philippe Satg\'{e}},
       DOI = {10.1215/S0012-7094-91-06210-1},
       URL = {https://doi.org/10.1215/S0012-7094-91-06210-1},
}

@book {hart,
    AUTHOR = {Hartshorne, Robin},
     TITLE = {Algebraic geometry},
    SERIES = {Graduate Texts in Mathematics, No. 52},
 PUBLISHER = {Springer-Verlag, New York-Heidelberg},
      YEAR = {1977},
     PAGES = {xvi+496},
      ISBN = {0-387-90244-9},
   MRCLASS = {14-01},
  MRNUMBER = {0463157},
MRREVIEWER = {Robert Speiser},
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@article {kleiman-duality,
    AUTHOR = {Kleiman, Steven L.},
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   JOURNAL = {Compositio Math.},
  FJOURNAL = {Compositio Mathematica},
    VOLUME = {41},
      YEAR = {1980},
    NUMBER = {1},
     PAGES = {39--60},
      ISSN = {0010-437X},
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  MRNUMBER = {578050},
MRREVIEWER = {Zoghman Mebkhout},
       URL = {http://www.numdam.org/item?id=CM_1980__41_1_39_0},
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@inproceedings {deligne,
    AUTHOR = {Deligne, P.},
     TITLE = {Courbes elliptiques: formulaire d'apr\`es {J}. {T}ate},
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     PAGES = {53--73. Lecture Notes in Math., Vol. 476},
      YEAR = {1975},
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@book {katz-mazur,
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     TITLE = {Arithmetic moduli of elliptic curves},
    SERIES = {Annals of Mathematics Studies},
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 PUBLISHER = {Princeton University Press, Princeton, NJ},
      YEAR = {1985},
     PAGES = {xiv+514},
      ISBN = {0-691-08349-5; 0-691-08352-5},
   MRCLASS = {11G05 (11F11 14G25 14K15)},
  MRNUMBER = {772569},
MRREVIEWER = {Kenneth A. Ribet},
       DOI = {10.1515/9781400881710},
       URL = {https://doi.org/10.1515/9781400881710},
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@misc{tama,
  doi = {10.48550/ARXIV.MATH/0503383},
  
  url = {https://arxiv.org/abs/math/0503383},
  
  author = {Behrend, K. and Dhillon, A.},
  
  keywords = {Number Theory (math.NT), Algebraic Geometry (math.AG), Representation Theory (math.RT), FOS: Mathematics, FOS: Mathematics},
  
  title = {Connected components of moduli stacks of torsors via Tamagawa numbers},
  
  publisher = {arXiv},
  
  year = {2005},
  
  copyright = {Assumed arXiv.org perpetual, non-exclusive license to distribute this article for submissions made before January 2004}
}

@article {bhar-shan,
    AUTHOR = {Bhargava, Manjul and Shankar, Arul},
     TITLE = {Binary quartic forms having bounded invariants, and the
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  FJOURNAL = {Annals of Mathematics. Second Series},
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      ISSN = {0003-486X},
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  MRNUMBER = {3272925},
MRREVIEWER = {John M. Voight},
       DOI = {10.4007/annals.2015.181.1.3},
       URL = {https://doi-org.libproxy.mit.edu/10.4007/annals.2015.181.1.3},
}

@book {liu,
    AUTHOR = {Liu, Qing},
     TITLE = {Algebraic geometry and arithmetic curves},
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      NOTE = {Translated from the French by Reinie Ern\'{e},
              Oxford Science Publications},
 PUBLISHER = {Oxford University Press, Oxford},
      YEAR = {2002},
     PAGES = {xvi+576},
      ISBN = {0-19-850284-2},
   MRCLASS = {14-01 (11G30 14A05 14A15 14Gxx 14Hxx)},
  MRNUMBER = {1917232},
MRREVIEWER = {C\'{\i}cero Carvalho},
}

@book {giraud,
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    SERIES = {Die Grundlehren der mathematischen Wissenschaften, Band 179},
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      YEAR = {1971},
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   MRCLASS = {14F20 (14L99 18D30 18F20 55B99 55F65)},
  MRNUMBER = {0344253},
MRREVIEWER = {R. T. Hoobler},
}

@misc{kleiman2005picard,
      title={The Picard scheme}, 
      author={Steven L. Kleiman},
      year={2005},
      eprint={math/0504020},
      archivePrefix={arXiv},
      primaryClass={math.AG}
}

@book {serre-lf,
    AUTHOR = {Serre, Jean-Pierre},
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    VOLUME = {67},
      NOTE = {Translated from the French by Marvin Jay Greenberg},
 PUBLISHER = {Springer-Verlag, New York-Berlin},
      YEAR = {1979},
     PAGES = {viii+241},
      ISBN = {0-387-90424-7},
   MRCLASS = {12Bxx},
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@article {serre-tate,
    AUTHOR = {Serre, Jean-Pierre and Tate, John},
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      YEAR = {1968},
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  MRNUMBER = {236190},
MRREVIEWER = {M. J. Greenberg},
       DOI = {10.2307/1970722},
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@article {aaron-selmer,
    AUTHOR = {Landesman, Aaron},
     TITLE = {The geometric average size of {S}elmer groups over function
              fields},
   JOURNAL = {Algebra Number Theory},
  FJOURNAL = {Algebra \& Number Theory},
    VOLUME = {15},
      YEAR = {2021},
    NUMBER = {3},
     PAGES = {673--709},
      ISSN = {1937-0652},
   MRCLASS = {11G05 (11N45 11R58)},
  MRNUMBER = {4261097},
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       DOI = {10.2140/ant.2021.15.673},
       URL = {https://doi-org.libproxy.mit.edu/10.2140/ant.2021.15.673},
}

@article {liu:global-weier,
    AUTHOR = {Liu, Qing},
     TITLE = {Global {W}eierstrass equations of hyperelliptic curves},
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  FJOURNAL = {Transactions of the American Mathematical Society},
    VOLUME = {375},
      YEAR = {2022},
    NUMBER = {8},
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      ISSN = {0002-9947},
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       DOI = {10.1090/tran/8672},
       URL = {https://doi-org.libproxy.mit.edu/10.1090/tran/8672},
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@article {sadek:genus-1,
    AUTHOR = {Sadek, Mohammad},
     TITLE = {Counting models of genus one curves},
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    VOLUME = {150},
      YEAR = {2011},
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       URL = {https://doi-org.libproxy.mit.edu/10.1017/S0305004110000666},
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@article {period-index,
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    NUMBER = {2},
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@book {mw-lattice,
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              Surveys in Mathematics]},
    VOLUME = {70},
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      ISBN = {978-981-32-9300-7; 978-981-32-9301-4},
   MRCLASS = {11G05 (11F80 11H06 14J20 14J26 14J27 14J28)},
  MRNUMBER = {3970314},
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       URL = {https://doi-org.libproxy.mit.edu/10.1007/978-981-32-9301-4},
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@article {oort-tate,
    AUTHOR = {Tate, John and Oort, Frans},
     TITLE = {Group schemes of prime order},
   JOURNAL = {Ann. Sci. \'{E}cole Norm. Sup. (4)},
  FJOURNAL = {Annales Scientifiques de l'\'{E}cole Normale Sup\'{e}rieure. Quatri\`eme
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@article {brumer,
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@book {apostol,
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 PUBLISHER = {Springer-Verlag, New York-Heidelberg},
      YEAR = {1976},
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  MRNUMBER = {0434929},
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@article {ogg,
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  FJOURNAL = {American Journal of Mathematics},
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      YEAR = {1967},
     PAGES = {1--21},
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   MRCLASS = {14.49 (14.51)},
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@inproceedings {mum-suo-moduli,
    AUTHOR = {Mumford, David and Suominen, Kalevi},
     TITLE = {Introduction to the theory of moduli},
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              {S}ummer {S}chool in {M}ath.)},
     PAGES = {171--222},
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@article {AOV,
    AUTHOR = {Abramovich, Dan and Olsson, Martin and Vistoli, Angelo},
     TITLE = {Tame stacks in positive characteristic},
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  FJOURNAL = {Universit\'{e} de Grenoble. Annales de l'Institut Fourier},
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    NUMBER = {4},
     PAGES = {1057--1091},
      ISSN = {0373-0956,1777-5310},
   MRCLASS = {14A20 (14L15)},
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MRREVIEWER = {Hsian-Hua\ Tseng},
       URL = {http://aif.cedram.org/item?id=AIF_2008__58_4_1057_0},
}