By Akihiro Munemasa.

Verified locator: Author lecture notes based on 1994 and 1995 lectures (preface). Section 2, initial definition and Proposition 2.1; Theorem 2.11 classifies nondegenerate even-dimensional forms by Witt indices m or m−1.

By Bill Casselman.

Verified locator: Author text revised 24 June 2022. Proposition 2.6 and Corollaries 2.7–2.8 give hyperbolic splitting from a totally isotropic subspace; Corollary 6.8 gives uniqueness of the hyperbolic multiplicity and anisotropic complement.

By Keith Conrad.

Verified locator: Introduction identifies PSL_n as SL_n modulo its center and gives the two exceptional finite fields for n=2; section 3 treats PSL_2.

By Nickolas Andersen, Jeffrey C. Lagarias and Robert C. Rhoades.

Verified locator: Page 1, equation (1.1) and footnotes 1–2: classical Maass forms, moderate growth, sign conventions.

By Hendrik W. Lenstra.

Verified locator: Volume 126 (1987), issue 3, pages 649–673; abstract describes algorithm and dependence on the smallest prime divisor.

IMSc Report116, Madras. Author-hosted notes. Chapter1, Theorem1.7, gives the six-exponentials theorem for independent input families with d times ell greater than d plus ell; Conjecture1.8 states the four-exponentials conjecture.

Primary article text: ramanujan.sirinudi.org/Volumes/published/ram15.pdf . Proceedings of the London Mathematical Society (2) 14, 347–409. Sections 1–8 define record values of the divisor function and describe the prime-exponent structure. The available text is a retypesetting of the original, with editorial remarks distinguished from Ramanujan’s text.

By Annette Huber, Guido Kings and Niko Naumann. Compositio Mathematica147,235–262. Definition2.1.1, printed p236, states filtration and p-valuation axioms; section2.2 gives formal-group examples. Publisher full text: www.cambridge.org/core/services/aop-cambridge-core/content/view/60EA940F508F25A61A13990CF91F27BE/S0010437X10004884a.pdf/some-complements-to-the-lazard-isomorphism.pdf

Original French pamphlet on the impossibility of solving the general quintic equation by radicals.

National Library of Norway scan: www.nb.no/items/3b1b8dc617f27b91783271a0da5e0f30

Primary-work index and verified attribution: abelprize.no/page/work-niels-henrik-abel . The complete original title continues: où on démontre l'impossibilité de la résolution de l'équation générale du cinquième dégré. Cited passage: title and principal impossibility statement; no uninspected proof-page locator is asserted.

Definition 5.1, printed pages 5–6, gives adelic automorphic forms; Proposition 5.5, printed page 7, identifies classical modular forms inside this space.

Version 3.08, 19 July 2020. Chapter 5, Theorem 5.1, gives Dirichlet's unit rank and discusses logarithmic embeddings and real/imaginary quadratic examples.

By Arjen K. Lenstra, Hendrik W. Lenstra, László Lovász. Mathematische Annalen261,515–534(1982).

Journal paper scan: www.math.ucdavis.edu/~deloera/MISC/LA-BIBLIO/trunk/Lovasz/LovaszLenstraLenstrafactor.pdf

Institutional record of preprint IW195/82: ir.cwi.nl/pub/9304 . The record names A.K.Lenstra(Arjen), H.W.Lenstra(Hendrik), and L.Lovász(László), and dates the report1982.

Cited locator: Section1, lattice-basis reduction. The initial reduction section develops the geometric algorithm used by the later polynomial-factorization construction.

MIT 18.783, Lecture 5, Fall 2025: math.mit.edu/classes/18.783/2025/LectureNotes5.pdf

Pages 5–7, Theorem 5.11 and Theorems 5.13/5.15: quotient by a finite subgroup and Vélu coordinate sums and formulas. Pages 10–11, Lemma 5.20 and Theorem 5.21: initial division polynomials, parity and multiplication coordinates. The lecture explicitly distinguishes geometric kernel points from inseparable degree and restricts short Weierstrass formulas away from characteristics two and three. The author is printed on the first-page footer.

Author-hosted second edition: www.jmilne.org/math/Books/EC2.pdf

World Scientific, ISBN 9789811221859 (hardcover) and 9789811221835 (paperback), as stated on the author PDF copyright page. Chapter IV, Section 4, especially Lemmas 4.4–4.6, Theorem 4.7 and Lemma 4.11: canonical-height convergence, torsion criterion and parallelogram identity. Milne uses the normalization without the factor one-half in front of the logarithmic x-height limit; the footnote after Lemma 4.6 explicitly notes the two common normalizations. Chapter II gives the projective Weierstrass group law; Chapter V uses the Weil-pairing determinant argument.

By James S. Milne. Author's course notes, version 1.31, March 22, 2017.

Author's page: www.jmilne.org/math/CourseNotes/mf.html

Primary text: www.jmilne.org/math/CourseNotes/MF.pdf

Section 2 develops modular curves; Section 4 defines modular functions and modular forms and their lattice interpretation; Section 9 relates cusp forms to Dirichlet series and functional equations.

By Terence Tao.

Author's primary text: terrytao.wordpress.com/2015/01/21/254a-notes-4-some-sieve-theory/

Corollary 19 states the fundamental lemma of sieve theory with a summed remainder; equations (26) and Corollary 21 apply it to simultaneous almost prime values. The opening definitions and the quadratic-weight construction describe sieve theory and the Selberg sieve.

By Patrice Philippon, Michel Waldschmidt. Chapter 18 of New Advances in Transcendence Theory, pages 280–312, Cambridge University Press.

Author's primary text: webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/Durham1986PW.pdf

Theorem 1.1, printed pages 280–281, treats algebraic coefficients including the constant term. Theorems 2.1–2.2, printed pages 282–283, distinguish general algebraic coefficients from the homogeneous rational-integer case and give the refined coefficient parameter.

Theorem 15, equations (3.41)–(3.42), printed pp. 71–72, give the Rosser–Schoenfeld totient bound. The coefficient 2.5 has an exception at 223092870; 2.50637 covers that case as well. Section 9 contains the proof.

Summer tutorial notes. Section 2.2, Definition 1 and Theorem 2.3 state primitive positive definite reduction and uniqueness; the following argument bounds the reduced coefficients. The primitive-form hypotheses must be retained when counting a class number of a negative discriminant.

Author lecture notes, lecture 5 of Finite Simple Groups: webspace.maths.qmul.ac.uk/r.a.wilson/FSG/lect5.pdf

Cited passage: Mathieu groups and the coordinate-permutation automorphism group of the Golay code.