pith. sign in

arxiv: 2606.05820 · v1 · pith:FNO2JCLKnew · submitted 2026-06-04 · 🧮 math.NT

Drinfeld modules in rank 2 with CM and S-unit j-invariants

Liam Baker , Fabien Pazuki , Patricio Perez Pina This is my paper

Pith reviewed 2026-06-27 23:56 UTC · model grok-4.3

classification 🧮 math.NT
keywords Drinfeld modulescomplex multiplicationS-unitsj-invariantsfinitenesssingular moduliHilbert modular polynomial
0
0 comments X

The pith

The set of j-invariants of rank-2 CM Drinfeld modules over F_q[T] that are S-units is finite.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that only finitely many j-invariants arise from Drinfeld modules of rank 2 over the polynomial ring F_q[T] when the modules have complex multiplication and the j-invariants are S-units for the infinite set S of primes of even degree. The argument examines the ordinary and supersingular reductions of the modules and the splitting of primes that divide the difference of two Drinfeld singular moduli. An algorithm is constructed that outputs a polynomial over F_q[T] whose roots are exactly the j-invariants with complex multiplication by any prescribed order. For maximal orders the same algorithm produces the Hilbert modular polynomial, and explicit runs of the algorithm yield counterexamples to a conjecture of Dorman.

Core claim

The set of j-invariants of Drinfeld modules of rank 2 over F_q[T] which are CM and S-units, for S the infinite set of primes with even degrees, is finite. The proof rests on the ordinary and supersingular reduction properties of the modules together with the splitting behaviour of primes dividing the difference of two Drinfeld singular moduli. An algorithm computes a polynomial over F_q[T] whose roots are the j-invariants with CM by a given order, and for a maximal order this polynomial is shown to be the Hilbert modular polynomial by a universality argument.

What carries the argument

The splitting behaviour of primes dividing the difference of two Drinfeld singular moduli, used together with the ordinary and supersingular reduction properties of the modules.

If this is right

  • Only finitely many such j-invariants exist.
  • The algorithm produces the minimal polynomial over F_q[T] for the j-invariants with CM by any given order.
  • When the order is maximal the algorithm yields the Hilbert modular polynomial.
  • Explicit computations with the algorithm give counterexamples to Dorman's conjecture.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduction and splitting techniques might apply to Drinfeld modules over other function-field rings if analogous reduction statements hold.
  • Running the algorithm for small q and varying orders would produce tables of CM j-invariants that could be compared with class-number data in function-field class field theory.
  • The counterexamples indicate that any revised version of Dorman's conjecture must impose extra restrictions on the orders or the base ring.

Load-bearing premise

The finiteness argument assumes that ordinary and supersingular reduction types plus the splitting of primes in singular-modulus differences are strong enough to bound the possible CM orders and their j-invariants.

What would settle it

An explicit infinite family of distinct rank-2 CM Drinfeld modules over F_q[T] whose j-invariants are S-units for the even-degree primes would disprove the claim.

Figures

Figures reproduced from arXiv: 2606.05820 by Fabien Pazuki, Liam Baker, Patricio Perez Pina.

Figure 1
Figure 1. Figure 1: Table of PO for q = 3, O = A[ √ D], and D quadratic and squarefree D (q = 3, deg D = 3) PO (with only leading and constant coefficients included and factorised) T 3 + T j 4 + · · · + T 8 (T + 2)16(T 3 + T 2 + 2T + 1)4 T 3 + 2T j 4 + · · · + T 8 (T + 1)8 (T + 2)8 (T 3 + 2T + 1)4 T 3 + 2T + 1 j 7 + · · · + 2(T 3 + T 2 + T + 2)4 (T 3 + T 2 + 2T + 1)4 × (T 3 + 2T + 1)2 (T 3 + 2T + 2)4 (T 3 + T 2 + 2)4 T 3 + 2T… view at source ↗
Figure 2
Figure 2. Figure 2: Table of PO for q = 3, O = A[ √ D], and D cubic and squarefree 14 [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Table of PO for q = 3, O = A[ √ D], and D quartic and squarefree D (q = 3, deg D = 3) PO (with only leading and constant coefficients included and factorised) T 3 (j + 2T 2 (T + 1)4 ) × (j 3 + · · · + 2T 2 (T + 1)16(T 3 + T 2 + 2T + 1)4 ) T 3 + T 2 (j + 2(T + 1)2 (T + 2)4 )×(j 2 +· · ·+ (T + 1)4 (T + 2)8 (T 3 + 2T 2 +T + 1)4 ) T 3 + 2T 2 (j + 2T 4 (T + 2)2 ) × (j 3 + · · · + T 4 (T + 2)8 (T 3 + 2T + 1)4 (T… view at source ↗
Figure 4
Figure 4. Figure 4: Table of PO (all not irreducible) for q = 3, O = A[ √ D], and D cubic and not squarefree D (q = 3, deg D = 4) PO (with leading and constant coefficients included and factorised) 2T 4 + T 2 + 2 = 2(T 2 + 1)2 j × [j 2 + (T 27 + 2T 21 + 2T 19 + 2T 17 + T 15 + 2T 13 + 2T 9 + 2T 7 + T 5 )j + 2T 8 (T + 1)8 (T + 2)8 (T 2 + 1)] [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Example of PO for q = 3, O = A[ √ D], and a selected D quartic and not squarefree D (q = 3) PO (factorised, with leading and constant coefficients included) T j + 2T 2 (T + 1)4 T(T + 2)2 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Table of PA[ √ D] for q = 3 and selected D to illustrate the factorisation in Theorem E 15 [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Table of PO for q = 5, O = A[ √ D], and D quadratic and squarefree D (q = 2, P 2 O (with only leading and constant deg D = 3) coefficients included and factorised) T 3 + 1 j 6 + · · · + T 6 (T + 1)9 (T 3 + T 2 + 1)3 T 3 + T + 1 j 6 + · · · + T 9 (T + 1)6 (T 3 + T + 1)3 [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Table of P 2 O for q = 2, O = A[ √ D], and D cubic and squarefree (inseparable case) D (q = 2, deg D = 5) P 2 O (with only leading and constant coefficients included and factorised) T 5 + 1 j 14 + · · · + T 9 (T + 1)21(T 3 + T 2 + 1)3 (T 3 + T + 1)6 (T 5 + T 2 + 1)3 T 5 + T + 1 j 14 + · · · + T 21(T + 1)9 (T 3 + T + 1)3 (T 3 + T 2 + 1)6 (T 5 + T 4 + T 2 + T + 1)3 T 5 + T 2 + 1 j 14 + · · · + T 9 (T + 1)21(… view at source ↗
Figure 9
Figure 9. Figure 9: Table of P 2 O for q = 2, O = A[ √ D], and D quintic and squarefree (inseparable case) 16 [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Table of PO for q = 2 and O = A[f √ Dξ] (separable case) Acknowledgments – This project emerged under the umbrella of a GandA Research In Pairs visit of FP and PP to LB. This visit was made possible through funding from IRN GandA (CNRS) and the Department of Mathematical Sciences from Stellenbosch University, which the authors heartily thank. The authors thank Valentijn Karemaker and Mihran Papikian for u… view at source ↗
read the original abstract

We prove the finiteness of the set of $j$-invariants of Drinfeld modules of rank 2 over $\mathbb{F}_q[T]$ which are CM and $S$-units, for $S$ the infinite set of primes with even degrees. The proof is based on the study of ordinary reduction and supersingular reduction of Drinfeld modules, and on the splitting behaviour of primes dividing the difference of two Drinfeld singular moduli. We also provide an algorithm to compute a polynomial with coefficients in $\mathbb{F}_q[T]$ and roots the $j$-invariants having CM by a given order, and use it to compute some explicit examples, providing for instance counterexamples to a conjecture of Dorman. For a maximal order $\mathcal{O}$, we prove by a universality argument that our algorithm computes the Hilbert modular polynomial $H_\mathcal{O}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proves the finiteness of the set of j-invariants of rank-2 Drinfeld modules over F_q[T] that have complex multiplication (CM) and are S-units, where S is the infinite set of primes of even degree. The argument relies on ordinary and supersingular reduction properties together with the splitting behavior of primes dividing j1 - j2 for distinct Drinfeld singular moduli. The authors also supply an algorithm that, for a given order, outputs a polynomial over F_q[T] whose roots are the relevant j-invariants; they prove by a universality argument that this algorithm recovers the Hilbert modular polynomial H_O when the order is maximal, and they compute explicit examples that furnish counterexamples to a conjecture of Dorman.

Significance. The finiteness result for an infinite set S is a notable extension of CM theory to the function-field setting. The algorithm, its universality proof for maximal orders, and the explicit counterexamples to Dorman's conjecture constitute concrete computational and theoretical contributions that can be directly verified and built upon.

minor comments (2)
  1. The introduction should explicitly state the base field F_q and the precise definition of the set S of even-degree primes before the main theorem is formulated.
  2. In the description of the algorithm, the input format for the order O (e.g., as a basis or as a conductor) should be clarified to make the procedure reproducible from the text alone.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the recognition of its contributions to CM theory in the function-field setting, and the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity in the finiteness proof or algorithm

full rationale

The paper establishes a finiteness result for CM rank-2 Drinfeld j-invariants over F_q[T] that are S-units (S = even-degree primes) via ordinary/supersingular reduction and splitting of primes dividing differences of singular moduli; these are standard external tools from CM theory for function fields and do not reduce to any fitted parameter or self-definition within the paper. The algorithm for the Hilbert modular polynomial is justified by an independent universality argument for maximal orders, with explicit examples serving only as illustrations (including counterexamples to an external conjecture of Dorman). No load-bearing step equates a claimed output to its input by construction, and the derivation remains self-contained against external number-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are identifiable from the provided text. The argument invokes standard reduction properties of Drinfeld modules presumed known in the field.

pith-pipeline@v0.9.1-grok · 5679 in / 1175 out tokens · 28079 ms · 2026-06-27T23:56:43.532832+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

60 extracted references · 41 canonical work pages

  1. [1]

    , TITLE =

    Thakur, Dinesh S. , TITLE =. 2004 , PAGES =

    2004

  2. [2]

    , TITLE =

    Chen, Yen-Mei J. , TITLE =. Journal of Number Theory , VOLUME =. 2008 , NUMBER =. doi:10.1016/j.jnt.2007.12.011 ,

    work page doi:10.1016/j.jnt.2007.12.011 2008

  3. [3]

    75 years of mathematics of computation , SERIES =

    Caranay, Perlas and Greenberg, Matthew and Scheidler, Renate , TITLE =. 75 years of mathematics of computation , SERIES =. 2020 ,. doi:10.1090/conm/754/15148 ,

    work page doi:10.1090/conm/754/15148 2020

  4. [4]

    Journal of Number Theory , VOLUME =

    Yu, Jiu-Kang , TITLE =. Journal of Number Theory , VOLUME =. 1995 , NUMBER =. doi:10.1006/jnth.1995.1108 ,

    work page doi:10.1006/jnth.1995.1108 1995

  5. [5]

    Mathematische Annalen , VOLUME =

    Gekeler, Ernst-Ulrich , TITLE =. Mathematische Annalen , VOLUME =. 1983 , NUMBER =. doi:10.1007/BF01455309 ,

    work page doi:10.1007/bf01455309 1983

  6. [6]

    Journal f\"ur die Reine und Angewandte Mathematik

    Andr\'e, Yves , TITLE =. Journal f\"ur die Reine und Angewandte Mathematik. [Crelle's Journal] , VOLUME =. 1998 , PAGES =. doi:10.1515/crll.1998.118 ,

    work page doi:10.1515/crll.1998.118 1998

  7. [7]

    Journal of Number Theory , VOLUME =

    Hsia, Liang-Chung and Yu, Jing , TITLE =. Journal of Number Theory , VOLUME =. 1998 , NUMBER =. doi:10.1006/jnth.1997.2211 ,

    work page doi:10.1006/jnth.1997.2211 1998

  8. [8]

    1964 , PAGES =

    Bourbaki, Nicolas , TITLE =. 1964 , PAGES =

    1964

  9. [9]

    Mathematische Zeitschrift , VOLUME =

    Artin, Emil , TITLE =. Mathematische Zeitschrift , VOLUME =. 1924 , NUMBER =. doi:10.1007/BF01181074 ,

    work page doi:10.1007/bf01181074 1924

  10. [10]

    and Thorup, Anders , TITLE =

    Jensen, Christian U. and Thorup, Anders , TITLE =. Journal of Pure and Applied Algebra , VOLUME =. 2015 , NUMBER =. doi:10.1016/j.jpaa.2014.05.013 ,

    work page doi:10.1016/j.jpaa.2014.05.013 2015

  11. [11]

    Journal of Algebra , VOLUME =

    Karemaker, Valentijn and Katen, Jeffrey and Papikian, Mihran , TITLE =. Journal of Algebra , VOLUME =. 2024 , PAGES =. doi:10.1016/j.jalgebra.2023.12.037 ,

    work page doi:10.1016/j.jalgebra.2023.12.037 2024

  12. [12]

    , TITLE =

    Dorman, David R. , TITLE =. Compositio Mathematica , VOLUME =. 1991 , NUMBER =

    1991

  13. [13]

    Algebra & Number Theory , VOLUME =

    Habegger, Philipp , TITLE =. Algebra & Number Theory , VOLUME =. 2015 , NUMBER =. doi:10.2140/ant.2015.9.1515 ,

    work page doi:10.2140/ant.2015.9.1515 2015

  14. [14]

    International Mathematics Research Notices

    Bilu, Yuri and Habegger, Philipp and K\"uhne, Lars , TITLE =. International Mathematics Research Notices. IMRN , YEAR =. doi:10.1093/imrn/rny274 ,

    work page doi:10.1093/imrn/rny274

  15. [15]

    Journal f\"ur die Reine und Angewandte Mathematik

    Breuer, Florian , TITLE =. Journal f\"ur die Reine und Angewandte Mathematik. [Crelle's Journal] , VOLUME =. 2005 , PAGES =. doi:10.1515/crll.2005.2005.579.115 ,

    work page doi:10.1515/crll.2005.2005.579.115 2005

  16. [16]

    Journal f\"ur die Reine und Angewandte Mathematik

    Breuer, Florian , TITLE =. Journal f\"ur die Reine und Angewandte Mathematik. [Crelle's Journal] , VOLUME =. 2012 , PAGES =. doi:10.1515/crelle.2011.136 ,

    work page doi:10.1515/crelle.2011.136 2012

  17. [17]

    Comptes Rendus Math\'ematique

    Breuer, Florian , TITLE =. Comptes Rendus Math\'ematique. Acad\'emie des Sciences. Paris , VOLUME =. 2007 , NUMBER =. doi:10.1016/j.crma.2007.05.008 ,

    work page doi:10.1016/j.crma.2007.05.008 2007

  18. [18]

    Annals of Mathematics

    K\"uhne, Lars , TITLE =. Annals of Mathematics. Second Series , VOLUME =. 2012 , NUMBER =. doi:10.4007/annals.2012.176.1.13 ,

    work page doi:10.4007/annals.2012.176.1.13 2012

  19. [19]

    Mathematical Proceedings of the Cambridge Philosophical Society , VOLUME =

    Bilu, Yuri and Masser, David and Zannier, Umberto , TITLE =. Mathematical Proceedings of the Cambridge Philosophical Society , VOLUME =. 2013 , NUMBER =. doi:10.1017/S0305004112000461 ,

    work page doi:10.1017/s0305004112000461 2013

  20. [20]

    , TITLE =

    Brown, Martin L. , TITLE =. Inventiones Mathematicae , VOLUME =. 1992 , NUMBER =. doi:10.1007/BF01231341 ,

    work page doi:10.1007/bf01231341 1992

  21. [21]

    International Mathematics Research Notices

    Cojocaru, Alina Carmen and Papikian, Mihran , TITLE =. International Mathematics Research Notices. IMRN , YEAR =. doi:10.1093/imrn/rnu178 ,

    work page doi:10.1093/imrn/rnu178

  22. [22]

    arXiv , YEAR =

    Armana, C\'ecile and Berardini, Elena and Caruso, Xavier and Leudi\`ere, Antoine and Nardi, Jade and Pazuki, Fabien , TITLE =. arXiv , YEAR =. 2601.02162 , ARCHIVEPREFIX =

    arXiv

  23. [23]

    On the singular Drinfeld modules of rank 2

    Bae, Sunghan and Koo, Ja Kyung , journal =. On the singular Drinfeld modules of rank 2. , url =

  24. [24]

    , TITLE =

    Silverman, Joseph H. , TITLE =. 1994 , PAGES =. doi:10.1007/978-1-4612-0851-8 ,

    work page doi:10.1007/978-1-4612-0851-8 1994

  25. [25]

    2025 , eprint=

    CM Drinfeld Modules, Self-isogenous Modular Polynomials, and Volcano Structure , author=. 2025 , eprint=

    2025

  26. [26]

    , TITLE =

    Sutherland, Andrew V. , TITLE =. Mathematics of Computation , VOLUME =. 2011 , NUMBER =. doi:10.1090/S0025-5718-2010-02373-7 ,

    work page doi:10.1090/s0025-5718-2010-02373-7 2011

  27. [27]

    and Hayes, David , TITLE =

    Dummit, David S. and Hayes, David , TITLE =. Mathematics of Computation , VOLUME =. 1994 , NUMBER =. doi:10.2307/2153547 ,

    work page doi:10.2307/2153547 1994

  28. [28]

    2010 , PAGES =

    Maciak, Piotr , title =. 2010 , PAGES =

    2010

  29. [29]

    Journal of the Ramanujan Mathematical Society , VOLUME =

    Maciak, Piotr , TITLE =. Journal of the Ramanujan Mathematical Society , VOLUME =. 2011 , NUMBER =

    2011

  30. [30]

    Mathematics of Computation , VOLUME =

    Br\"oker, Reinier , TITLE =. Mathematics of Computation , VOLUME =. 2008 , NUMBER =. doi:10.1090/S0025-5718-08-02091-7 ,

    work page doi:10.1090/s0025-5718-08-02091-7 2008

  31. [31]

    Mathematics of Computation , VOLUME =

    Enge, Andreas , TITLE =. Mathematics of Computation , VOLUME =. 2009 , NUMBER =. doi:10.1090/S0025-5718-08-02200-X ,

    work page doi:10.1090/s0025-5718-08-02200-x 2009

  32. [32]

    Advances in Math

    Armana, C\'ecile and Angl\`es, Bruno and Bosser, Vincent and Pazuki, Fabien , TITLE =. Advances in Math. , VOLUME =. 2026 ,

    2026

  33. [33]

    There are at most finitely many singular moduli that are

    Herrero, Sebasti. There are at most finitely many singular moduli that are. Compositio Mathematica ,. 2024 , language =. doi:10.1112/S0010437X23007704 , keywords =

    work page doi:10.1112/s0010437x23007704 2024

  34. [34]

    Acta Arithmetica , VOLUME =

    Breuer, Florian and Pazuki, Fabien and Razafinjatovo, Mahefason Heriniaina , TITLE =. Acta Arithmetica , VOLUME =. 2021 , NUMBER =. doi:10.4064/aa191029-8-7 ,

    work page doi:10.4064/aa191029-8-7 2021

  35. [35]

    Bombieri, Enrico and Gubler, Walter , TITLE =. 2006 ,. doi:10.1017/CBO9780511542879 ,

    work page doi:10.1017/cbo9780511542879 2006

  36. [36]

    Manuscripta Mathematica , VOLUME =

    Campagna, Francesco , TITLE =. Manuscripta Mathematica , VOLUME =. 2021 , NUMBER =. doi:10.1007/s00229-020-01230-1 ,

    work page doi:10.1007/s00229-020-01230-1 2021

  37. [37]

    Mathematical Proceedings of the Cambridge Philosophical Society , VOLUME =

    Campagna, Francesco , TITLE =. Mathematical Proceedings of the Cambridge Philosophical Society , VOLUME =. 2023 , NUMBER =. doi:10.1017/S0305004122000378 ,

    work page doi:10.1017/s0305004122000378 2023

  38. [38]

    Mathematische Annalen , VOLUME =

    David, Sinnou and Denis, Laurent , TITLE =. Mathematische Annalen , VOLUME =. 1999 , NUMBER =. doi:10.1007/s002080050319 ,

    work page doi:10.1007/s002080050319 1999

  39. [39]

    Li, Yingkun , TITLE =. Compos. Math. , FJOURNAL =. 2021 , NUMBER =. doi:10.1112/S0010437X21007077 ,

    work page doi:10.1112/s0010437x21007077 2021

  40. [40]

    Annali della Scuola Normale Superiore di Pisa

    R\'emond, Ga\"el , TITLE =. Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V , VOLUME =. 2022 , NUMBER =. doi:10.2422/2036-2145.202010_062 ,

    work page doi:10.2422/2036-2145.202010_062 2022

  41. [41]

    Duke Math

    Taguchi, Yuichiro , TITLE =. Duke Math. J. , FJOURNAL =. 1991 , NUMBER =. doi:10.1215/S0012-7094-91-06225-3 ,

    work page doi:10.1215/s0012-7094-91-06225-3 1991

  42. [42]

    Journal of Number Theory , VOLUME =

    Taguchi, Yuichiro , TITLE =. Journal of Number Theory , VOLUME =. 1993 , NUMBER =. doi:10.1006/jnth.1993.1055 ,

    work page doi:10.1006/jnth.1993.1055 1993

  43. [43]

    Journal of Number Theory , VOLUME =

    Taguchi, Yuichiro , TITLE =. Journal of Number Theory , VOLUME =. 1999 , NUMBER =. doi:10.1006/jnth.1998.2316 ,

    work page doi:10.1006/jnth.1998.2316 1999

  44. [44]

    Inventiones Mathematicae , VOLUME =

    Wei, Fu-Tsun , TITLE =. Inventiones Mathematicae , VOLUME =. 2020 , NUMBER =. doi:10.1007/s00222-019-00944-8 ,

    work page doi:10.1007/s00222-019-00944-8 2020

  45. [45]

    International Mathematics Research Notices

    Griffon, Richard and Pazuki, Fabien , TITLE =. International Mathematics Research Notices. IMRN , YEAR =

  46. [46]

    arXiv , year =

    Variation of height in an isogeny class over a function field , author =. arXiv , year =. 2503.14318 , archivePrefix =

    arXiv

  47. [47]

    2023 , PAGES =

    Papikian, Mihran , TITLE =. 2023 , ISBN =. doi:10.1007/978-3-031-19707-9 ,

    work page doi:10.1007/978-3-031-19707-9 2023

  48. [48]

    Goss, David , year =. Basic. doi:10.1007/978-3-642-61480-4 ,

    work page doi:10.1007/978-3-642-61480-4

  49. [49]

    2002 , publisher =

    Algebra , author =. 2002 , publisher =

    2002

  50. [50]

    International Journal of Number Theory , VOLUME =

    Pazuki, Fabien , TITLE =. International Journal of Number Theory , VOLUME =. 2019 , NUMBER =. doi:10.1142/S1793042119500295 ,

    work page doi:10.1142/s1793042119500295 2019

  51. [51]

    , TITLE =

    Faltings, Gerd , TITLE =. Inventiones Mathematicae , VOLUME =. 1983 , NUMBER =. doi:10.1007/BF01388432 ,

    work page doi:10.1007/bf01388432 1983

  52. [52]

    Kronecker limit formula over global function fields , volume =

    Wei, Fu-Tsun , journal =. Kronecker limit formula over global function fields , volume =. doi:10.1353/ajm.2017.0027 , year =

    work page doi:10.1353/ajm.2017.0027 2017

  53. [53]

    Minimal terminal

    Potemine, Igor Yu , journal =. Minimal terminal. 1998 , publisher =

    1998

  54. [54]

    Algebra colloquium (Publications de l'Institut de recherche math

    Robba, Philippe , TITLE =. Algebra colloquium (Publications de l'Institut de recherche math. 1985 ,

    1985

  55. [55]

    Funkcial

    Duval, Anne , TITLE =. Funkcial. Ekvac. , FJOURNAL =. 1983 , NUMBER =

    1983

  56. [56]

    Rosen, Michael , TITLE =. 2002 ,. doi:10.1007/978-1-4757-6046-0 ,

    work page doi:10.1007/978-1-4757-6046-0 2002

  57. [57]

    , TITLE =

    Hayes, David R. , TITLE =. Studies in algebra and number theory , SERIES =. 1979 , ISBN =

    1979

  58. [58]

    V\'elu, Jacques , TITLE =. C. R. Acad. Sci. Paris S\'er. A-B , FJOURNAL =. 1971 , PAGES =

    1971

  59. [59]

    Annales de la Facult

    On singular and supersingular invariants of Drinfeld modules , author=. Annales de la Facult. 1997 , url=

    1997

  60. [60]

    2026 , publisher =

    Liam Baker , title =. 2026 , publisher =

    2026