# References — Chapter ᚒ(U): The crossroads *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ref-ch-19, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) · [The chapter](ch19.md) *Each entry: one plain sentence saying exactly what it is cited for, its status, then every source with a note saying what that source actually carries. "Own computation" / "Own reading" entries are the author's own work, not outside sources.* ### 19.1 {#r-19.1} A nonsingular projective Weierstrass cubic has the unique point at infinity (0 : 1 : 0), and this point is the identity for the elliptic-curve group law; its homogeneous X-coordinate is 0. *Status: confirmed* - [RFC 6090, Fundamental Elliptic Curve Cryptography Algorithms](https://www.rfc-editor.org/rfc/rfc6090) — Section 3 names the point at infinity as the identity (“a special point (@,@) that represents the identity element (which is called the “point at infinity”)”) and says “The point at infinity is its own inverse”; Appendix F.2 says it is “regarded as equivalent to the homogenous coordinates (0, 1, 0).” - [Elliptic curve, projective group law](https://en.wikipedia.org/wiki/Elliptic_curve) — the projective equation has the unique point O = [0:1:0] at infinity, which is the identity of the group. ### 19.2 {#r-19.2} Elliptic curves in short Weierstrass form have the chord-and-tangent group law: three collinear intersection points sum to the identity, and the rational points form an abelian group. *Status: confirmed* - [Ben Lynn, Elliptic Curves: Group of Points (Stanford crypto notes)](https://crypto.stanford.edu/pbc/notes/elliptic/group.html) — describes the projective point at infinity, the line-through-two-points construction, and the abelian group of points: in (projective) Weierstrass form “an elliptic curve always contains exactly one point of infinity, (0, 1, 0) (‘the point at the ends of all lines parallel to the y-axis’)”; “every line intersects the curve at exactly three points (counting multiplicities)”; and “It can be verified that P + Q = Q + P, P + O = P, (−P) + P = O,” with O taken to be the point at infinity (0, 1, 0). - [RFC 6090, Fundamental Elliptic Curve Cryptography Algorithms](https://www.rfc-editor.org/rfc/rfc6090) — gives the affine addition and inverse formulas and the exceptional point-at-infinity cases. ### 19.3 {#r-19.3} The Mordell–Weil theorem says E(Q) is a finitely generated abelian group, decomposing into a finite torsion subgroup and a free part of rank r. *Status: confirmed* - [Elliptic curve, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Elliptic_curve) — (§ Elliptic curves over non-closed fields): “If k is finitely generated over its prime subfield, then X(k) is a finitely-generated group (the Mordell–Weil theorem),” and “The group X(ℚ) is isomorphic to F ⊕ X(ℚ)ₜ, where X(ℚ)ₜ is a finite Abelian group and F is a free Abelian group of a certain finite rank r.” - [Elliptic curve rank](https://en.wikipedia.org/wiki/Rank_of_an_elliptic_curve) — writes E(Q) as Z^r plus its finite torsion group and defines r as the rank. ### 19.4 {#r-19.4} For E: y² = x³ − x, the rational rank is 0 and E(Q) has torsion Z/2 × Z/2, namely the identity and the three points (0, 0), (1, 0), and (−1, 0); this is equivalent to 1 not being a congruent number. *Status: confirmed* - [Poonen, Rational points on varieties, section 8.3](https://math.mit.edu/~poonen/papers/Qpoints.pdf) — identifies 32A2 as y² = x³ − x, states rank 0 and torsion Z/2 × Z/2, and concludes that there are four rational points. - [Keith Conrad, The Congruent Number Problem (PDF)](https://kconrad.math.uconn.edu/articles/congruentnumber.pdf) — shows that n is not congruent exactly when the only rational points of y² = x³ − n²x have y = 0, namely (0, 0), (n, 0) and (−n, 0); so for n = 1 the curve y² = x³ − x has only those three affine points (with the point at infinity, four). ### 19.5 {#r-19.5} For y² = x³ + ax + b, the discriminant is −16(4a³ + 27b²); for y² = x³ − x it is 64, and the j-invariant is 1728 = 12³, reflecting complex multiplication by i. For y² = x³ + ax + b, j = 1728·4a³/(4a³ + 27b²); j(i) = 1728. *Status: confirmed* - [LMFDB 32.a3 (Cremona 32a2), the curve y² = x³ − x](https://www.lmfdb.org/EllipticCurve/Q/32a2/) — gives conductor 32, discriminant 64, j-invariant 1728 = 2⁶ · 3³, rank 0 and torsion Z/2 × Z/2 for this curve. - [Elliptic curve invariants](https://en.wikipedia.org/wiki/Elliptic_curve) — gives the short Weierstrass discriminant −16(4a³ + 27b²), writes j as 1728·g′₂³/(g′₂³ − 27g′₃²) in lattice invariants (the a, b form is on the j-invariant line), and says the j-invariant specifies isomorphism classes of elliptic curves; the curve “is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition 4a³ + 27b² ≠ 0, that is, being square-free in x.),” and “Since the curve is smooth, hence continuous, it can be shown that this point at infinity is the identity element of a group structure.” - **Own computation** (the author's own work) — Substitute a = −1 and b = 0: Δ = −16(4(−1)^3) = 64 and j = 1728·4(−1)^3/(4(−1)^3) = 1728 = 12^3. - [j-invariant, Wikipedia](https://en.wikipedia.org/wiki/J-invariant) — read as raw wikitext: “In case of a short Weierstraß elliptic curve y² = x³ + ax + b, this formula simplifies to” j = 1728·4a³/(4a³ + 27b²). - [Complex multiplication, Wikipedia](https://en.wikipedia.org/wiki/Complex_multiplication) — read as raw wikitext: CM “is the theory of elliptic curves E that have an endomorphism ring larger than the integers”; curves Y² = 4X³ − aX have “two conjugate order-4 automorphisms” sending Y → ±iY, X → −X; “Kronecker conjectured – in what became known as the Kronecker Jugendtraum – that every abelian extension of K could be obtained by the (roots of the) equation of a suitable elliptic curve with complex multiplication”; “j(Z[i]) = j(i) = 1728.” - [LMFDB, elliptic curve 37.a1 (Cremona label 37a1)](https://www.lmfdb.org/EllipticCurve/Q/37a1/) — minimal Weierstrass equation y² + y = x³ − x; “This elliptic curve is semistable. There is only one prime p of bad reduction”; analytic rank 1, so L(E, 1) = 0. ### 19.6 {#r-19.6} The torsion points of y² = x³ − x are the identity and the three roots of x³ − x, each of order 2. *Status: confirmed* - [Poonen, Rational points on varieties, section 8.3](https://math.mit.edu/~poonen/papers/Qpoints.pdf) — states that the curve 32A2 has torsion Z/2 × Z/2 and four rational points. - [RFC 6090, Fundamental Elliptic Curve Cryptography Algorithms](https://www.rfc-editor.org/rfc/rfc6090) — states that a point with y = 0 doubles to the point at infinity and that the point at infinity is the identity. ### 19.7 {#r-19.7} The curve y² = x³ − x has eight points over F5, and Hasse’s bound is |#E(F5) − 6| ≤ 2√5. a_p = p + 1 − #E(F_p). *Status: confirmed* - [Hasse’s theorem for elliptic curves](https://en.wikipedia.org/wiki/Hasse%27s_theorem_on_elliptic_curves) — states |#E(Fq) − (q+1)| ≤ 2√q. - **Own computation** (the author's own work) — For x = 0,1,4 modulo 5, x³−x = 0 and y = 0; for x = 2 the value is 1, giving y = ±1; for x = 3 the value is 4, giving y = ±2. Thus there are 7 affine points plus O, so #E(F5)=8 and |8−6|=2≤2√5. - [Elliptic curve, Wikipedia](https://en.wikipedia.org/wiki/Elliptic_curve) — read as raw wikitext: a_p = p + 1 − #E(F_p); the curve’s L-function “is defined as an Euler product, with one factor for every prime number p,” built from these a_p, and the proof of the Shimura–Taniyama–Weil conjecture “implies that its L-function is the L-function of a modular form whose analytic continuation is known.” ### 19.8 {#r-19.8} The curve y² + y = x³ − x has conductor 37 and rank 1, the smallest known conductor for a rank-one elliptic curve; its L-value at 1 vanishes. 37a1 is semistable with analytic rank 1; the Antwerp tables of curves of conductor up to 200 came from Swinnerton-Dyer’s search, later extended by Cremona. *Status: confirmed* - [Elkies, Best known conductors for elliptic curves of given rank](https://people.math.harvard.edu/~elkies/womack2001.html) — lists rank 1, curve [0 0 1 −1 0], conductor 37, and marks it as the smallest conductor found by exhaustive search for that rank. - [Poonen, Rational points on varieties, section 8.3](https://math.mit.edu/~poonen/papers/Qpoints.pdf) — Poonen’s survey of rational points on curves; read again on 23 September 2026, it does not display the 37a1 model, which the LMFDB 37a1 line carries. - [Christopher Skinner, A converse to a theorem of Gross, Zagier, and Kolyvagin (Annals of Mathematics, 2020)](https://annals.math.princeton.edu/2020/191-2/p01) — for a semistable curve with suitable reduction, proves that algebraic rank 1 and finite Ш give analytic rank 1, a converse to the Gross–Zagier/Kolyvagin theorem; the page does not discuss 37a1, whose L(1) = 0 is on the LMFDB lines. - [LMFDB, elliptic curve 37.a1 (Cremona label 37a1)](https://www.lmfdb.org/EllipticCurve/Q/37a1/) — minimal Weierstrass equation y² + y = x³ − x; “This elliptic curve is semistable. There is only one prime p of bad reduction”; analytic rank 1, so L(E, 1) = 0; its Mordell-Weil generators table lists the point (0, 0), of order ∞. - [John Cremona, The elliptic curve database for conductors to 130000 (ANTS VII)](https://johncremona.github.io/papers/ecdb_ants.pdf) — read from the PDF: the Antwerp Table 1 lists “All elliptic curves of conductor N ≤ 200”; “Swinnerton-Dyer searched for curves with small coefficients and kept those with conductor N ≤ 200”; the Antwerp tables are in “Antwerp IV” … “(edited by Birch and Kuyk).” ### 19.9 {#r-19.9} Mazur’s theorem gives exactly fifteen possible torsion subgroups for elliptic curves over Q; the largest possible order is 16. *Status: confirmed* - [Mazur theorem lecture notes, IMPA](https://w3.impa.br/~hossein/myarticles/EllipticCurvesCourse.pdf) — states the fifteen groups in Mazur’s classification of E(Q)_tors. - [Mazur torsion theorem overview](https://wstein.org/projects/kleinerman_senior_thesis.pdf) — lists the cyclic groups of orders 1–10 and 12 and the four products Z/2 × Z/2n, whose largest order is 16. ### 19.10 {#r-19.10} Five is the smallest congruent number, represented by the rational right triangle with sides 3/2, 20/3, and 41/6; 1 through 4 are not congruent. 6 is the area of the 3-4-5 triangle; Fibonacci found 7 congruent; Fermat proved 1 is not (1640); n is congruent exactly when y² = x³ − n²x has a point of infinite order. *Status: confirmed* - [OEIS A268602, first congruent number triangle](https://oeis.org/A268602) — gives the triangle 3/2, 20/3, 41/6, verifies its area is 5, and identifies 5 as the first congruent number. - [The congruent number problem](https://www.johndcook.com/blog/2021/12/17/congruent-number/) — defines congruent numbers and independently verifies the area-5 triangle and the fact that 5 is the smallest. - [Keith Conrad, The congruent number problem (University of Connecticut)](https://kconrad.math.uconn.edu/articles/congruentnumber.pdf) — read from the PDF: 6 is the area of a (3, 4, 5) right triangle; “Fibonacci discovered in the 13th century that 7 is congruent”; “Theorem 2 (Fermat, 1640). The number 1 is not congruent.” - [Elliptic curve, Wikipedia](https://en.wikipedia.org/wiki/Elliptic_curve) — read as raw wikitext: “n is a congruent number if and only if the elliptic curve y² = x³ − n²x has a rational point of infinite order.” ### 19.11 {#r-19.11} Birch and Swinnerton-Dyer conjectured that the rank equals the order of vanishing of L(E, s) at s=1; this remains a Millennium Problem. *Status: confirmed* - [Birch–Swinnerton-Dyer conjecture, MathConjectures](https://www.mathconjectures.com/conjectures/NT-ARITH-002) — states ord_{s=1} L(E, s) = r and labels the conjecture open. - [Clay Mathematics Institute, Millennium Prize Problems](https://www.claymath.org/millennium-problems/) — lists Birch and Swinnerton-Dyer among the Millennium Prize Problems. ### 19.12 {#r-19.12} Birch and Swinnerton-Dyer’s early computational evidence came from counting points modulo primes, and Gross–Zagier and Kolyvagin proved the relevant rank-one implication in the late twentieth century. *Status: confirmed* - [Birch–Swinnerton-Dyer conjecture](https://en.wikipedia.org/wiki/Birch%E2%80%93Swinnerton-Dyer_conjecture) — “In the early 1960s, Swinnerton-Dyer used the EDSAC-2 computer at the University of Cambridge Computer Laboratory to calculate the number N​p of points modulo p for a large number of primes p on elliptic curves whose rank was known”; and Gross and Zagier in 1986 and Kolyvagin in 1989 proved the rank-zero and rank-one implications, not the vanishing of any particular L-value. - [Christopher Skinner, A converse to a theorem of Gross, Zagier, and Kolyvagin (Annals of Mathematics, 2020)](https://annals.math.princeton.edu/2020/191-2/p01) — proves a converse to the theorem of Gross, Zagier and Kolyvagin: for certain semistable curves, algebraic rank 1 and finite Ш give analytic rank 1. ### 19.13 {#r-19.13} Every elliptic curve over Q is modular: semistable curves were handled by Wiles and Taylor–Wiles, and the full theorem was proved by Breuil, Conrad, Diamond, and Taylor in 2001; this was the bridge to Fermat’s Last Theorem via Ribet’s theorem, which Ken Ribet proved in the summer of 1986 and published in 1990. The conductor of the curve is the level of the corresponding modular form. Hasse’s conjecture that L(E, s) continues to the whole plane was shown in 1999 to follow from the modularity theorem. *Status: confirmed* - [Breuil, Conrad, Diamond and Taylor, On the modularity of elliptic curves over Q](https://www.math.u-psud.fr/~breuil/PUBLICATIONS/STW.pdf) — the four authors’ preprint: “Theorem A. If E/Q is an elliptic curve, then E is modular.”; it is dated 2000, “and in revised form” January 2001; the 2001 journal publication is on the Edixhoven line. - [Edixhoven, Rational elliptic curves are modular](https://www.numdam.org/article/SB_1999-2000__42__161_0.pdf) — records the semistable Wiles/Taylor–Wiles result, the full Breuil–Conrad–Diamond–Taylor theorem, and the consequence for Fermat’s Last Theorem; its bibliography gives “On the modularity of elliptic curves over Q: wild 3-adic exercises. J. Amer. Math. Soc. 14 (2001), no. 4, 843-939.” - [Ribet’s theorem, Wikipedia](https://en.wikipedia.org/wiki/Ribet%27s_theorem) — read as raw HTML: “In the summer of 1986, Kenneth Alan Ribet proved the epsilon conjecture, thereby proving that the Modularity theorem implied FLT”, and its bibliography gives the publication as Ribet, Ken (1990), “On modular representations of Gal(Q/Q) arising from modular forms”, Inventiones Mathematicae 100 (2), pp. 431–476. - [Modularity theorem, Wikipedia](https://en.wikipedia.org/wiki/Modularity_theorem) — read as raw wikitext: “Weil also showed that the conductor of the elliptic curve should be the level of the corresponding modular form.”; the parametrization “may be defined in terms of a mapping generated by a particular kind of modular form of weight two and level N.” - [Elliptic curve, Wikipedia](https://en.wikipedia.org/wiki/Elliptic_curve) — read as raw wikitext: “Hasse’s conjecture affirms that the L-function admits an analytic continuation to the whole complex plane … In 1999 this was shown to be a consequence of the proof of the Shimura–Taniyama–Weil conjecture.” ### 19.14 {#r-19.14} The curve y² = x³ − x has complex multiplication by i, with the extra map (x, y) ↦ (−x, iy) and endomorphism ring Z[i]. *Status: confirmed* - [LMFDB 32.a3 (Cremona 32a2), the curve y² = x³ − x](https://www.lmfdb.org/EllipticCurve/Q/32a2/) — records complex multiplication with discriminant −4 and geometric endomorphism ring Z[√−1]. - [Complex multiplication, Wikipedia](https://en.wikipedia.org/wiki/Complex_multiplication) — read as raw wikitext: CM “is the theory of elliptic curves E that have an endomorphism ring larger than the integers”; curves Y² = 4X³ − aX have “two conjugate order-4 automorphisms” sending Y → ±iY, X → −X; “Kronecker conjectured – in what became known as the Kronecker Jugendtraum – that every abelian extension of K could be obtained by the (roots of the) equation of a suitable elliptic curve with complex multiplication”; “To this day this remains one of the few cases of Hilbert’s twelfth problem which has actually been solved.”; “j(Z[i]) = j(i) = 1728.” ### 19.15 {#r-19.15} Elliptic-curve cryptography uses fast scalar multiplication kP while relying on the assumed hardness of recovering k from P and kP; secp256k1 is the curve used by Bitcoin, and Shor’s algorithm threatens this classical security model. Double-and-add computes kP quickly; the fastest known classical attacks need O(√n) steps; secp256k1 is y² = x³ + 7 over a 256-bit prime field and is Bitcoin’s ECDSA curve; Shor’s algorithm would break ECC on a large fault-tolerant quantum computer. *Status: confirmed* - [RFC 6090, Fundamental Elliptic Curve Cryptography Algorithms](https://www.rfc-editor.org/rfc/rfc6090) — defines elliptic-curve point multiplication and the discrete-logarithm setting used by ECC. - [NIST, Post-Quantum Cryptography FAQ](https://csrc.nist.gov/projects/post-quantum-cryptography/faqs) — explains that Shor’s algorithm is the kind of “dramatic quantum attack” for which classical security measures stop being relevant; the FAQ does not itself name discrete logarithms, which the Elliptic-curve cryptography line carries. - [Elliptic curve point multiplication, Wikipedia](https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication) — read as raw wikitext: “The simplest method is the double-and-add method.” - [Elliptic-curve cryptography, Wikipedia](https://en.wikipedia.org/wiki/Elliptic-curve_cryptography) — read as raw wikitext: “all the fastest known algorithms that allow one to solve the ECDLP (baby-step giant-step, Pollard’s rho, etc.), need O(√n) steps”; “Shor’s algorithm can be used to break elliptic curve cryptography by computing discrete logarithms on a sufficiently large fault-tolerant quantum computer.” - [SECG, SEC 2: Recommended Elliptic Curve Domain Parameters, version 2.0](https://www.secg.org/sec2-v2.pdf) — §2.4.1 secp256k1: p = 2²⁵⁶ − 2³² − 2⁹ − 2⁸ − 2⁷ − 2⁶ − 2⁴ − 1; “The curve E: y² = x³ + ax + b over Fp is defined by” a = 0, b = 7. - [Bitcoin Developer Guide, Transactions](https://developer.bitcoin.org/devguide/transactions.html) — “Bitcoin uses the Elliptic Curve Digital Signature Algorithm (ECDSA) with the secp256k1 curve.” - [Cloudflare blog, Automatic key exchange for origins](https://blog.cloudflare.com/automatic-key-exchange-for-origins/) — “over 95% of origins support X25519”; the post describes the move to post-quantum hybrid key agreement. ### 19.16 {#r-19.16} The historical attributions in this chapter compress a long development: Diophantus, Newton, Poincaré, Fermat, Birch, Swinnerton-Dyer, Frey, Ribet, and Wiles are associated with the milestones stated in the prose. The tangent step was used by Diophantus and the chord-tangent construction by Newton in the 1670s; associativity is the only non-trivial part of the group law. *Status: judgement* - [Elliptic curve, Wikipedia](https://en.wikipedia.org/wiki/Elliptic_curve) — points to “elliptic integral for the origin of the term,” gives the group law “by the tangent and secant method,” and says the curves “were used in Andrew Wiles’s proof of Fermat’s Last Theorem”; the historical attributions are on the other lines. - [Breuil, Conrad, Diamond and Taylor, On the modularity of elliptic curves over Q](https://www.math.u-psud.fr/~breuil/PUBLICATIONS/STW.pdf) — confirms the modern modularity attribution and dates. - [Fermat’s right triangle theorem](https://en.wikipedia.org/wiki/Fermat%27s_right_triangle_theorem) — “It is the only complete proof given by Fermat”; Fermat wrote it in the margin of his copy of Diophantus and his son Clement-Samuel published it in 1670. - [Fermat’s Last Theorem](https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem) — “In 1984, Gerhard Frey noticed an apparent link between these two previously unrelated and unsolved problems”; Ribet completed the link in 1986; Wiles lectured 21–23 June 1993 and the corrected proof filled the May 1995 Annals of Mathematics. - [Araceli Bonifant and John Milnor, On real and complex cubic curves, arXiv:1603.09018](https://arxiv.org/abs/1603.09018) — read from the PDF: “The “tangent” part of the chord-tangent map was used by Diophantus of Alexandria in the third century”; “in the 1670’s Newton used the “chord-tangent construction” to interpret the solutions of Diophantine equations given by Diophantus and Fermat.” - [Sander Zwegers, On the group law on elliptic curves, arXiv:2401.02346](https://arxiv.org/abs/2401.02346) — read from the PDF: “The only non-trivial part is the associativity. One approach to this is to concentrate on elliptic curves over C and use the theory of elliptic functions, in particular the Weierstraß ℘-function”; the paper gives “a simple and elementary proof of the associativity.” Editorial note: We found authoritative support for the major milestones, but the chapter compresses several independent historical threads into a single narrative and the exact priority wording for Diophantus, Newton, and Poincaré is not uniquely documented in the fetched sources. We retain the chapter’s broad chronology while treating the compressed priority language as historical judgement. The Frey, Ribet and Wiles dates given in the prose are the ones the Fermat’s Last Theorem record uses; other accounts date Frey’s observation to his 1986 paper and Ribet’s publication to 1990, which are different milestones rather than a different chronology. Settled on 17 September 2026 and not to be reopened: they are the proof and the printing of one result. Ribet proved the epsilon conjecture in the summer of 1986 and published it as “On modular representations of Gal(Q/Q) arising from modular forms,” Inventiones Mathematicae 100 (1990), pp. 431–476; both dates are now carried by 19.13 and stated on the #modularity card. No source we found fixes a year for Fermat’s own proof, so the prose names no decade for it. ### 19.17 {#r-19.17} The largest known rank of an elliptic curve over Q is at least 31 (Alpöge and Howell, 2026), after rank 29 (Elkies and Klagsbrun, 2024) and rank 28 (Elkies, 2006); the largest rank known exactly and unconditionally is 20 (Elkies and Klagsbrun, 2020), and whether ranks are bounded is open. Mordell’s theorem gives no effective method for computing the rank, and it is unknown whether the numerical methods are effective for all curves. *Status: confirmed* - [Dujella, History of elliptic curve rank records](https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html) — re-read raw on 17 September 2026 and unchanged: the table lists rank 28 in 2006 (Elkies), rank 29 in 2024 (Elkies–Klagsbrun) and ranks 30 and 31 in 2026 (Alpöge–Howell), the running text says “The current record is an example of elliptic curve with rank ≥ 31, found by Alpöge and Howell in 2026,” and the highest rank known unconditionally and exactly is still 20, found by Elkies–Klagsbrun in 2020. - [Rank of an elliptic curve](https://en.wikipedia.org/wiki/Rank_of_an_elliptic_curve) — as of August 2026 the largest known rank is at least 31, exactly 31 under the generalized Riemann hypothesis and the Birch and Swinnerton-Dyer conjecture, and there is no consensus among experts on whether ranks are bounded. - [Birch and Swinnerton-Dyer conjecture, Wikipedia](https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture) — read as raw wikitext: “Although Mordell’s theorem shows that the rank of an elliptic curve is always finite, it does not give an effective method for calculating the rank of a given curve. The rank can sometimes be calculated using numerical methods, but it is currently unknown if these methods are effective for all curves.” ### 19.18 {#r-19.18} Fermat’s one surviving complete proof is his right triangle theorem, proved by infinite descent: from any Pythagorean triangle whose area is a square one can derive a smaller one. That theorem is equivalent to the statement that 1 is not a congruent number, and Fermat’s Last Theorem for exponent 4 is an immediate consequence of it. *Status: confirmed* - [Fermat’s right triangle theorem, Wikipedia](https://en.wikipedia.org/wiki/Fermat%27s_right_triangle_theorem) — carries all four halves of the claim in so many words — “It is the only complete proof given by Fermat”; “Fermat’s proof is a proof by infinite descent. It shows that, from any example of a Pythagorean triangle with square area, one can derive a smaller example”; “the existence of a square congruum is equivalent to the statement that the number 1 is not a congruent number”; and “An immediate consequence of the last of these formulations is that Fermat’s Last Theorem is true in the special case that its exponent is 4.” - [Proof by infinite descent, Wikipedia](https://en.wikipedia.org/wiki/Proof_by_infinite_descent) — defines the method — “a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction” — and records that Fermat coined the term and often used it for Diophantine equations. Editorial note: Added in the V14 prose pass for the new card `descent` and to settle cross-chapter conflict C1. Chapter ᚆ(H) calls the descent “the one proof of that kind he left behind … covers fourth powers only” and Chapter ᚒ(U) called it “the only complete proof he left”; both sentences are about this single argument, and this entry is the source that says so. The “only complete proof” wording was already quoted under 19.16; it is repeated here so the card can stand on one entry. ### 19.19 {#r-19.19} Mordell’s theorem, that the rational points of an elliptic curve over Q form a finitely generated abelian group, was proved by Louis Mordell in 1922, answering a question posed by Henri Poincare around 1901 in the memoir Sur les proprietes arithmetiques des courbes algebriques, Journal de Mathematiques Pures et Appliquees, serie 5, tome 7 (1901), pp. 161-233. Andre Weil generalised it to Jacobians of higher-genus curves over arbitrary number fields in his doctoral dissertation, published in 1928 and dated 1929 by some accounts. *Status: confirmed* - [Mordell–Weil theorem, Wikipedia](https://en.wikipedia.org/wiki/Mordell%E2%80%93Weil_theorem) — states “The case with A an elliptic curve E and K the field of rational numbers is Mordell’s theorem, answering a question apparently posed by Henri Poincaré around 1901; it was proven by Louis Mordell in 1922,” and gives the general statement that A(K) is a finitely-generated abelian group. Re-read raw on 17 September 2026 for Weil: “Some years later André Weil took up the subject, producing the generalisation to Jacobians of higher genus curves over arbitrary number fields in his doctoral dissertation published in 1928,” while the same page’s infobox gives “First proof by André Weil, First proof in 1929” — the dissertation and its publication, not two claims. - [H. Poincaré, Sur les propriétés arithmétiques des courbes algébriques, Journal de Mathématiques Pures et Appliquées, série 5, tome 7 (1901)](https://www.numdam.org/item/JMPA_1901_5_7__161_0/) — the memoir itself, free in full text, pp. 161–233, publisher Gauthier-Villars. This is the primary document behind the 1901 date. Editorial note: Added in the V14 prose pass. Chapter ᚒ(U)’s “In 1922 Louis Mordell proved” and the `mordellweil` card’s 1922 and 1901 dates were footnoted only at 19.3, whose two sources state the theorem but name no year for it and no questioner. This entry carries both dates in one sentence, so it also stands behind the chapter’s “stated clearly by Poincaré in 1901” in the first paragraph. Extended 17 September 2026. The Wikipedia source hedges — “a question apparently posed by Henri Poincaré around 1901” — and names no paper, so the identification of the memoir is the book’s own, made from its NUMDAM record; the memoir is now cited directly so a reader can check it. The entry was also attached to #grouplaw’s “Poincaré stated in 1901” clause, which until now carried no citation for the date at all: its only reference, 19.2, describes the chord-and-tangent construction and names neither a person nor a year. ### 19.20 {#r-19.20} Barry Mazur proved the full torsion conjecture for elliptic curves over the rationals in two papers, of 1977 and 1978; the fifteen groups are the cyclic groups of order 1 to 10 and 12 together with C2n x C2 for n = 1 to 4; Loïc Merel proved in 1996 that torsion is uniformly bounded over every number field. *Status: confirmed* - [Torsion conjecture, Wikipedia](https://en.wikipedia.org/wiki/Torsion_conjecture) — states “Barry Mazur (1977, 1978) proved the full torsion conjecture for elliptic curves over the rationals,” and lists the groups as C_n with 1 ≤ n ≤ 10, C_12, and C_2n × C_2 with 1 ≤ n ≤ 4. Re-read raw on 17 September 2026 for Merel: “Finally, Loïc Merel (1996) proved the conjecture for elliptic curves over any number field,” after Kamienny (1992) and Kamienny and Mazur (1995) had reached quadratic fields and fields of degree at most 8; the page adds that Merel proved “that there is a bound on the order of the torsion group depending only on the degree.” Editorial note: Added in the V14 prose pass to carry the year the chapter uses. “Mazur proved in 1977” was footnoted 19.9 alone, and neither of 19.9’s two sources gives a date for the theorem; they state the classification only. The source here dates it to two papers rather than one, which is what the `mazur16` card already says. ### 19.21 {#r-19.21} The L-function of an elliptic curve over Q is the Euler product of (1 - a_p p^-s + p^(1-2s))^-1 over the primes of good reduction, times (1 - a_p p^-s)^-1 at the primes dividing the conductor, with a_p = p + 1 - #E(F_p); the product converges only for Re(s) > 3/2, and modularity gives it an analytic continuation to the whole plane and a functional equation relating L(E, s) to L(E, 2 - s). The Nagell-Lutz theorem is named for two independent discoverers, Trygve Nagell, who published it in 1935, and Elisabeth Lutz, in 1937. Weierstrass introduced the symbol for his elliptic function in his lectures of 1862-1863. *Status: confirmed* - [Elliptic curve, Wikipedia (§ L-functions)](https://en.wikipedia.org/wiki/Elliptic_curve) — read as raw HTML: the L-function is “defined as an Euler product, with one factor for every prime number p,” with “a_p = p + 1 − #E(F_p),” and given as L(E(Q), s) = ∏ over p not dividing N of (1 − a_p p^−s + p^(1−2s))^−1 times ∏ over p dividing N of (1 − a_p p^−s)^−1. “This product converges for Re(s) > 3/2 only. Hasse’s conjecture affirms that the L-function admits an analytic continuation to the whole complex plane and satisfies a functional equation relating, for any s, L(E, s) to L(E, 2 − s). In 1999 this was shown to be a consequence of the proof of the Shimura–Taniyama–Weil conjecture.” - [Nagell–Lutz theorem, Wikipedia](https://en.wikipedia.org/wiki/Nagell%E2%80%93Lutz_theorem) — read as raw HTML: “The result is named for its two independent discoverers, the Norwegian Trygve Nagell (1895–1988) who published it in 1935, and Élisabeth Lutz (1937)”; for y² = x³ + ax² + bx + c “with integer coefficients a, b, c,” and D “the discriminant of the cubic polynomial on the right side”: “If P = (x, y) is a rational point of finite order on E, for the elliptic curve group law, then: x and y are integers; either y = 0, in which case P has order two, or else y divides D, which immediately implies that y² divides D” (formulae as typeset on the page). - [Weierstrass elliptic function, Wikipedia](https://en.wikipedia.org/wiki/Weierstrass_elliptic_function) — read as raw HTML: the function “is usually written with a rather special, lower case script letter ℘, which was Weierstrass’s own notation introduced in his lectures of 1862–1863,” which is the 1860s the #weierstrass card gives. - [LMFDB, elliptic curve 37.a1 (Cremona label 37a1)](https://www.lmfdb.org/EllipticCurve/Q/37a1/) — minimal Weierstrass equation y² + y = x³ − x; “This elliptic curve is semistable. There is only one prime p of bad reduction”; analytic rank 1, so L(E, 1) = 0. Editorial note: Added 17 September 2026 for the B6 pass. Three separate gaps in one chapter, all of the same shape: a card asserting something its cited entry does not carry. #LfuncE’s whole quick line is the Euler product and the only entry behind it was r-19.11, which states the Birch and Swinnerton-Dyer conjecture and its Millennium-Problem status; no entry anywhere in references.html carried an elliptic-curve Euler product, the six “Euler product” hits all being the Riemann zeta function in the prime-mileposts chapter. #tors4’s Nagell 1935 and Lutz 1937 stood on r-19.6, which is the torsion of one curve. #weierstrass’s “℘-function of the 1860s” stood on r-19.5, which is Δ = 64 and j = 1728. Not sourced and therefore not asserted: the #j1728 card said the 1728 normalisation “goes back to Klein’s work on the modular function in the 1870s,” and no source found in this pass dates Klein’s work, so the decade was removed from the card rather than left standing. ### 19.22 {#r-19.22} The chapter epigraph is the opening line of the preface to Serge Lang’s Elliptic Curves: Diophantine Analysis, Grundlehren der mathematischen Wissenschaften 231, Springer-Verlag, 1978. *Status: confirmed* - [Serge Lang, Wikipedia](https://en.wikipedia.org/wiki/Serge_Lang) — read as raw HTML: its bibliography gives “Lang, Serge, Elliptic curves: Diophantine analysis, Grundlehren der Mathematischen Wissenschaften, volume 231, Springer-Verlag, Berlin–New York, 1978, ISBN 3-540-08489-4, doi 10.1007/978-3-662-07010-9,” with Alan Baker’s review in the Bulletin of the AMS (N.S.) 2 (1980), 352–354. Editorial note: Added 17 September 2026 to source the chapter epigraph, which carried no marker. The book, the series, the publisher and the year are sourced here; that the sentence is the preface’s opening line is the author’s own reading of the printed book and is offered as such. ### 19.23 {#r-19.23} Homogeneous or projective coordinates were introduced by August Ferdinand Möbius in Der barycentrische Calcul of 1827. A point is a triple (X : Y : Z), not all zero, with triples counted as equal when they differ by a nonzero common scale; the ordinary affine plane is the part with Z ≠ 0, recovered as (X/Z, Y/Z), and the points with Z = 0 are the points at infinity. The advantage is that points at infinity get finite coordinates. The point at infinity, the identity of the curve’s group, has projective coordinates (0 : 1 : 0). *Status: confirmed* - [Homogeneous coordinates, Wikipedia](https://en.wikipedia.org/wiki/Homogeneous_coordinates) — read as raw HTML: “homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul… They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates,” with the scaling equivalence and the Z = 0 line at infinity. - [RFC 6090, Fundamental Elliptic Curve Cryptography Algorithms](https://www.rfc-editor.org/rfc/rfc6090) — a special point “represents the identity element (which is called the “point at infinity”)”; “The point at infinity (@,@) is regarded as equivalent to the homogenous coordinates (0, 1, 0).” ### 19.24 {#r-19.24} Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Coates and Wiles (1977): first general result towards BSD, for CM curves; Tunnell’s criterion dates from 1983; Koblitz and Miller independently proposed elliptic-curve cryptography in 1985; Taniyama stated the first version of the conjecture in 1955; Taniyama and Shimura worked on the conjecture until 1957, and Weil gave it its first serious evidence in 1967; Weil showed in 1967 that the modularity conjecture would follow from conjectured functional equations; Wiles, with Taylor, proved the semistable case in 1995; Hasse proved the bound |#E(F_p) - (p+1)| <= 2 sqrt(p) in 1933. Coates and Wiles proved their theorem in 1976 and published it in 1977; it was the first general result toward the conjecture. *Status: confirmed* - [Birch and Swinnerton-Dyer conjecture, Wikipedia](https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture) — read raw: “Coates, J.; Wiles, A. (1977). “On the conjecture of Birch and Swinnerton-Dyer.” Inventiones Mathematicae 39 (3): 223–251..” - [Tunnell’s theorem, Wikipedia](https://en.wikipedia.org/wiki/Tunnell%27s_theorem) — read raw: “The theorem is named for Jerrold B. Tunnell, a number theorist at Rutgers University, who proved it in Tunnell (1983).”; “Tunnell’s theorem gives a partial resolution to the congruent number problem, and under the Birch and Swinnerton-Dyer conjecture, a full resolution.” - [Elliptic-curve cryptography, Wikipedia](https://en.wikipedia.org/wiki/Elliptic-curve_cryptography) — read raw: “The use of elliptic curves in cryptography was suggested independently by Neal Koblitz and Victor S. Miller in 1985..” - [Modularity theorem, Wikipedia](https://en.wikipedia.org/wiki/Modularity_theorem) — read raw: “Yutaka Taniyama stated a preliminary (slightly incorrect) version of the conjecture at the 1955 international symposium on algebraic number theory in Tokyo and Nikkō as the twelfth of his set of 36 unsolved problems.”; “Goro Shimura and Taniyama worked on improving its rigor until 1957. André Weil rediscovered the conjecture, and showed in 1967 that it would follow from the (conjectured) functional equations for some twisted L-series of the elliptic curve”; “In 1995, Andrew Wiles, with some help from Richard Taylor, proved the Taniyama–Shimura–Weil conjecture for all semistable elliptic curves..” - [Hasse’s theorem on elliptic curves, Wikipedia](https://en.wikipedia.org/wiki/Hasse%27s_theorem_on_elliptic_curves) — read raw: “It was proven by Hasse in 1933, with the proof published in a series of papers in 1936..” - [Andrew Wiles, The Birch and Swinnerton-Dyer Conjecture (Clay Mathematics Institute problem description)](https://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdf) — read from the PDF: “The first general result proved was for elliptic curves with complex multiplication … This theorem was proved in 1976 and is due to Coates and Wiles.” Editorial note: Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.