# Glossary cards — Chapter ᚒ(U): The crossroads *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#group-T19, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) · [The chapter](ch19.md) *Elliptic curves: the group law, the point at infinity, a rank of zero, and the invariants that name a curve. Each card: its symbol, type, one-line headline, the longer note, the reference entries it cites and the cards it links to.* ### the letter Úr — `ᚒ` {#g-og-ur} *Ogham letter* Ogham letter name meaning ‘earth, clay, soil’; the medieval glossators read it as the heath. Úr is not a tree name. On the evidence of the kennings it means ‘earth, clay, soil’. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give úaraib adbaib, “in cold dwellings”; sílad cland, “propagation of plants”; and forbbaid ambí, “shroud of a lifeless one”—a grave, a seedbed, and a shroud, all of them soil. The Auraicept glosses fraech, heath, which is where the heather of the letters table comes from. Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient. Cites: [2.15](ch02-refs.md#r-2.15), [2.16](ch02-refs.md#r-2.16), [2.17](ch02-refs.md#r-2.17), [2.19](ch02-refs.md#r-2.19) ### the point at infinity, projective X-coordinate — `X(𝒪)` {#g-ptinf} *Integer or rational* The identity of an elliptic curve’s group is the point at infinity, which in projective coordinates is (0 : 1 : 0), so its projective X-coordinate is 0. In projective coordinates a point of the plane is (X : Y : Z) up to scaling; the affine plane is Z = 1 and the points with Z = 0 are at infinity. Every elliptic curve in Weierstrass form passes through exactly one point at infinity, 𝒪 = (0 : 1 : 0), and the group law takes it as the identity. Its projective X-coordinate is 0, unchanged by rescaling, so the numerator is exactly 0, an integer. Cites: [19.1](ch19-refs.md#r-19.1) See also: [the group law](ch19-cards.md#g-grouplaw), [projective coordinates](ch19-cards.md#g-projcoord), [the Weierstrass form](ch19-cards.md#g-weierstrass) ### the discriminant of the curve y² = x³ − x — `Δ(y² = x³ − x)` {#g-discE} *Integer or rational* 64, for the curve y² = x³ − x; a nonzero discriminant is what makes a cubic an elliptic curve at all. The discriminant of y² = x³ + ax + b is Δ = −16(4a³ + 27b²); it is nonzero exactly when the cubic has three distinct roots, which is the definition of an elliptic curve. For y² = x³ − x, a = −1 and b = 0 give Δ = 64, confirmed by LMFDB, where the curve is 32.a3 (Cremona 32a2), with conductor 32, rank 0, and complex multiplication by i. An integer. The curve of conductor 37 in the epilogue is a different curve; this one is chosen for its clean numbers. Cites: [19.5](ch19-refs.md#r-19.5) See also: [the curve of conductor 37](ch28-cards.md#g-E37), [complex multiplication](ch19-cards.md#g-cmult), [the rank of the curve y² = x³ − x](ch19-cards.md#g-rank32), [the Weierstrass form](ch19-cards.md#g-weierstrass) ### the j-invariant of the same curve — `j(y² = x³ − x)` {#g-j1728} *Integer or rational* 1728, the cube of 12: this curve’s value of the invariant that tells two complex curves apart. The j-invariant, 1728·4a³/(4a³ + 27b²), classifies elliptic curves over the complex numbers up to isomorphism: two curves are the same shape if and only if they share a j. For y² = x³ − x it is 1728 = 12³, the value taken by every curve with complex multiplication by i. An integer. Cites: [19.5](ch19-refs.md#r-19.5) See also: [complex multiplication](ch19-cards.md#g-cmult), [the j-invariant](ch14-cards.md#g-jinv), [the Weierstrass form](ch19-cards.md#g-weierstrass) ### the conductor of the curve of conductor 37 — `N(E₃₇)` {#g-cond37} *Integer or rational* 37, the smallest conductor of any elliptic curve of rank one. The conductor of an elliptic curve over the rationals is an integer recording the primes of bad reduction and how bad they are; the modularity theorem, proved for semistable curves such as this one by Wiles and Taylor and in full by Breuil, Conrad, Diamond, and Taylor in 2001, says the curve corresponds to a modular form of that level. The curve 37a1, y² + y = x³ − x, has conductor 37, the smallest conductor of any curve of rank one, found in the tables of Birch, Swinnerton-Dyer, and later Cremona. An integer, and the same curve whose L-value vanishes in the epilogue. Cites: [19.13](ch19-refs.md#r-19.13), [19.8](ch19-refs.md#r-19.8) See also: [the curve of conductor 37](ch28-cards.md#g-E37), [the modularity theorem](ch19-cards.md#g-modularity) ### the rank of the curve y² = x³ − x — `rank(y² = x³ − x)` {#g-rank32} *Integer or rational* Exactly 0: the curve has only its four torsion points over ℚ, which is Fermat’s theorem that 1 is not a congruent number. The Mordell–Weil group of y² = x³ − x is ℤ/2 × ℤ/2 with no points of infinite order, so its rank is 0 (LMFDB 32.a3). A rational right triangle of area 1 would give a rational point with y ≠ 0 on this curve, so the rank being 0 is exactly Fermat’s theorem, proved by infinite descent, that 1 is not a congruent number, and the same descent is the one proof of his last theorem he ever wrote down, for n = 4. Exactly 0, an integer. Cites: [19.18](ch19-refs.md#r-19.18), [19.4](ch19-refs.md#r-19.4) See also: [Fermat’s method of infinite descent](ch19-cards.md#g-descent), [Fermat’s Last Theorem as a bit](ch08-cards.md#g-fermat), [the Mordell–Weil theorem](ch19-cards.md#g-mordellweil), [the rank of an elliptic curve](ch28-cards.md#g-rankE), [the Weierstrass form](ch19-cards.md#g-weierstrass) ### the torsion group of y² = x³ − x — `|E(ℚ)_{tors}|` {#g-tors4} *Integer or rational* 4: the identity, of order 1, and the three points where y = 0, namely (0, 0), (1, 0), and (−1, 0), each of order 2. The points of finite order on y² = x³ − x are 𝒪 and the three points with y = 0, since doubling a point with y = 0 gives 𝒪; the group is ℤ/2 × ℤ/2. Nagell (1935) and Lutz (1937) proved that torsion points on a curve with integer coefficients have integer coordinates and y = 0 or y² dividing the discriminant, which is how such groups are computed by hand. An integer. Cites: [19.21](ch19-refs.md#r-19.21), [19.6](ch19-refs.md#r-19.6) See also: [the Weierstrass form](ch19-cards.md#g-weierstrass) ### the points of y² = x³ − x over the field with five elements — `#E(𝔽_{5})` {#g-pts5} *Integer or rational* 8, counted by hand: seven affine points plus the point at infinity, inside Hasse’s bound |8 − 6| ≤ 2√5. Over 𝔽₅ the curve y² = x³ − x has the points (0, 0), (1, 0), (4, 0), (2,±1), (3,±2) and 𝒪, eight in all. Hasse proved in 1933 that #E(𝔽_p) is within 2√p of p + 1, here |8 − 6| = 2 ≤ 4.47; the differences a_p = p + 1 − #E(𝔽_p), here a₅ = −2, are the numbers the curve’s L-function is built from, and by the modularity theorem that L-function is the L-function of a modular form. Counting points over 𝔽_p is also the arithmetic under elliptic-curve cryptography. An integer. Cites: [19.24](ch19-refs.md#r-19.24), [19.7](ch19-refs.md#r-19.7) See also: [the L-function of an elliptic curve](ch19-cards.md#g-LfuncE), [elliptic-curve cryptography](ch19-cards.md#g-ecc), [the modularity theorem](ch19-cards.md#g-modularity), [the Weierstrass form](ch19-cards.md#g-weierstrass) ### Mazur’s bound on torsion — `16` {#g-mazur16} *Integer or rational* No elliptic curve over ℚ has more than 16 rational points of finite order, and the fifteen possible torsion groups are known exactly. Barry Mazur proved in 1977 and 1978 that the torsion subgroup of an elliptic curve over ℚ is one of exactly fifteen groups: cyclic of order 1 to 10 or 12, or ℤ/2 × ℤ/2n for n = 1 to 4. The largest, ℤ/2 × ℤ/8, has order 16, and the largest cyclic order is 12. Merel proved in 1996 that over every number field the torsion is bounded by a number depending only on the field’s degree. An integer. Cites: [19.20](ch19-refs.md#r-19.20), [19.9](ch19-refs.md#r-19.9) ### the smallest congruent number — `5` {#g-congruent5} *Integer or rational* 5 is the smallest whole number that is the area of a right triangle with rational sides (the triangle 3/2, 20/3, 41/6); 1, 2, 3, and 4 are not. A congruent number is the area of a right triangle with rational sides; 5 is the smallest, then 6 (the 3-4-5 triangle) and 7. Fermat proved that 1 is not congruent, and n is congruent exactly when the curve y² = x³ − n²x has positive rank, which is why this chapter’s rank-0 numerator and this denominator entry are two sides of the same question. Tunnell’s criterion of 1983 decides the question for every n, assuming the Birch and Swinnerton-Dyer conjecture. An integer. Cites: [19.10](ch19-refs.md#r-19.10), [19.24](ch19-refs.md#r-19.24) See also: [the Birch and Swinnerton-Dyer conjecture](ch28-cards.md#g-bsdconj), [the 3–4–5 right triangle](ch12-cards.md#g-pythag345) ### the Weierstrass form — `y² = x³ + ax + b` {#g-weierstrass} *Function or operator* Every elliptic curve over a field of characteristic not 2 or 3 can be written this way; a and b determine Δ and j. Weierstrass’s ℘-function of the 1860s satisfies (℘′)² = 4℘³ − g₂℘ − g₃, which is why cubics in this shape are called Weierstrass form and why every complex elliptic curve is a torus ℂ/Λ. The general form y² + a₁xy + a₃y = x³ + a₂x² + a₄x + a₆, needed over fields of characteristic 2 and 3 and for minimal models such as y² + y = x³ − x, reduces to it by completing the square and cube whenever the characteristic is not 2 or 3. A form, uncolored. Cites: [19.21](ch19-refs.md#r-19.21), [19.5](ch19-refs.md#r-19.5) See also: [the curve of conductor 37](ch28-cards.md#g-E37) ### the group law — `P + Q + R = 𝒪` {#g-grouplaw} *Function or operator* Three points of the curve on one line add to the identity; that rule, with 𝒪 at infinity, makes the rational points an abelian group. Draw the line through P and Q; it meets the cubic in a third point R, and P + Q is defined as the reflection of R in the x-axis, so that any three collinear points sum to 𝒪. The chord-and-tangent construction goes back to Diophantus and Newton; Mordell answered in 1922 a question apparently posed by Poincaré in his memoir of 1901: whether finitely many rational points generate all the others by this construction. Historians differ on how explicitly Poincaré had the group in hand, so the book claims no more than that. The associativity of this addition is the hard part; it can be proved through the ℘-function or by direct computation. A rule, uncolored. Cites: [19.16](ch19-refs.md#r-19.16), [19.19](ch19-refs.md#r-19.19), [19.2](ch19-refs.md#r-19.2) ### the Mordell–Weil theorem — `E(ℚ) ≅ ℤ^{r} ⊕ E(ℚ)_{tors}` {#g-mordellweil} *Function or operator* The rational points of an elliptic curve form a finitely generated group: a finite torsion part and r independent points of infinite order, r the rank. Mordell proved in 1922 that E(ℚ) is finitely generated, answering Poincaré’s 1901 question, and Weil extended it to Jacobians of higher-genus curves over arbitrary number fields in 1928. The rank r is the number the Birch and Swinnerton-Dyer conjecture ties to the L-function; no algorithm is proven to compute it, curves of rank at least 31 are known (Alpöge and Howell, 2026) while the largest rank ever pinned down exactly is 20 (Elkies and Klagsbrun, 2020), and whether ranks are bounded is open. A theorem, uncolored. Cites: [19.17](ch19-refs.md#r-19.17), [19.19](ch19-refs.md#r-19.19), [19.3](ch19-refs.md#r-19.3) See also: [the L-function of an elliptic curve](ch19-cards.md#g-LfuncE), [the Birch and Swinnerton-Dyer conjecture](ch28-cards.md#g-bsdconj), [the rank of an elliptic curve](ch28-cards.md#g-rankE) ### the L-function of an elliptic curve — `L(E, s)` {#g-LfuncE} *Function or operator* An Euler product with one factor for every prime, built from the numbers a_p that come from counting its points modulo each prime; continued past where the product converges, its behavior at s = 1 is what the Birch and Swinnerton-Dyer conjecture is about. L(E, s) = Π_p (1 − a_p p^(−s) + p^(1−2s))^(−1) for primes of good reduction, times a simpler factor (1 − a_p p^(−s))^(−1) at each of the finitely many bad primes, with a_p = p + 1 − #E(𝔽_p); it converges for Re(s) > 3/2 and, by the modularity theorem, continues to the whole plane with a functional equation of its own about s = 1. The conjecture says the order of vanishing at s = 1 equals the rank; for the curve of conductor 37 the value L(E₃₇, 1) = 0 closes the piece. A function, uncolored. Cites: [19.11](ch19-refs.md#r-19.11), [19.21](ch19-refs.md#r-19.21), [28.5](ch28-refs.md#r-28.5) See also: [the curve of conductor 37](ch28-cards.md#g-E37), [the Birch and Swinnerton-Dyer conjecture](ch28-cards.md#g-bsdconj), [the L-function of the rank-one curve at one](ch28-cards.md#g-lE37), [the modularity theorem](ch19-cards.md#g-modularity), [the rank of an elliptic curve](ch28-cards.md#g-rankE) ### the modularity theorem — `E ↔ f ∈ S_{2}(Γ_{0}(N))` {#g-modularity} *Function or operator* Every elliptic curve over ℚ comes from a modular form of weight two and level equal to its conductor: its semistable case, proved by Wiles with Taylor, is what settled Fermat’s last theorem. Conjectured by Taniyama in 1955, sharpened with Shimura by 1957 and given its first serious evidence by Weil in 1967, proved for semistable curves by Wiles with Taylor in 1995 and for all curves by Breuil, Conrad, Diamond, and Taylor in 2001. Ribet showed in the summer of 1986 that a counterexample to Fermat’s last theorem would give a non-modular curve, publishing the proof in 1990, so modularity settled Fermat, the bit in Chapter ᚆ(H). It is also what proved that the L-function has an analytic continuation to the whole plane. A theorem, uncolored. Cites: [19.13](ch19-refs.md#r-19.13), [19.24](ch19-refs.md#r-19.24) See also: [the L-function of an elliptic curve](ch19-cards.md#g-LfuncE), [analytic continuation](ch18-cards.md#g-analcont), [Fermat’s Last Theorem as a bit](ch08-cards.md#g-fermat) ### complex multiplication — `End(E) ⊋ ℤ` {#g-cmult} *Function or operator* A curve with extra endomorphisms beyond multiplication by integers; y² = x³ − x has the map (x, y) ↦ (−x, iy), so its endomorphisms over ℂ are ℤ[i]. Most elliptic curves have only the integer multiplication maps; a curve with complex multiplication has an extra symmetry by an imaginary quadratic order, as y² = x³ − x has by the Gaussian integers, which is why its j-invariant is 1728. Kronecker’s Jugendtraum, realized by the theory of complex multiplication, builds abelian extensions of imaginary quadratic fields from such curves, and CM curves are where the Birch and Swinnerton-Dyer conjecture was first proved in part (Coates and Wiles, 1977). A property, uncolored. Cites: [19.14](ch19-refs.md#r-19.14), [19.24](ch19-refs.md#r-19.24) See also: [the Birch and Swinnerton-Dyer conjecture](ch28-cards.md#g-bsdconj), [the j-invariant of the same curve](ch19-cards.md#g-j1728), [the Weierstrass form](ch19-cards.md#g-weierstrass) ### elliptic-curve cryptography — `kP` {#g-ecc} *Function or operator* Multiplying a point by a large integer k is easy; recovering k from kP is the elliptic-curve discrete logarithm problem, on which much of today’s key exchange rests. Proposed independently by Koblitz and Miller in 1985: on a curve over a large finite field, computing kP by repeated doubling is fast, but no known classical algorithm recovers k from P and kP in fewer than about √n steps, where n is the order of the point. The curve secp256k1, y² = x³ + 7 over a 256-bit prime field, underlies Bitcoin; Shor’s algorithm would break all of it on a sufficiently large fault-tolerant quantum computer. A construction, uncolored. Cites: [19.15](ch19-refs.md#r-19.15), [19.24](ch19-refs.md#r-19.24) ### Fermat’s method of infinite descent — `solution ⇒ smaller solution` {#g-descent} *Function or operator* Assume a solution in positive integers, build a strictly smaller one from it, and repeat: positive integers cannot descend forever, so there was no solution. Fermat coined the name and used the method on Diophantine equations. His one complete surviving proof runs this way: from any Pythagorean triangle whose area is a square he derives a smaller one, so no such triangle exists. That single argument is three statements at once—1 is not a congruent number, the curve y² = x³ − x has rank 0, and Fermat’s last theorem holds for fourth powers. The fourth-power case is the one case of his last theorem he ever wrote down, which is why Chapter ᚆ(H) says the proof he left covers fourth powers only and this chapter uses the same descent to show that no right triangle with rational sides has area 1: they are the same argument. A method, uncolored. Cites: [19.18](ch19-refs.md#r-19.18) See also: [Fermat’s Last Theorem as a bit](ch08-cards.md#g-fermat), [the rank of the curve y² = x³ − x](ch19-cards.md#g-rank32), [the Weierstrass form](ch19-cards.md#g-weierstrass) ### projective coordinates — `(X : Y : Z)` {#g-projcoord} *Function or operator* Three numbers instead of two, with scaling ignored. The ordinary plane is the part with Z ≠ 0, which scaling can always bring to Z = 1, and the points at infinity finally get coordinates of their own. A point of the projective plane is a triple (X : Y : Z), not all zero, and two triples name the same point when one is a nonzero multiple of the other. The ordinary plane sits inside as the part with Z ≠ 0, read off as (X/Z, Y/Z); the triples with Z = 0 are the points at infinity, one for each direction. Möbius introduced the coordinates in Der barycentrische Calcul in 1827, and the payoff is exactly the one this chapter needs: the identity of an elliptic curve’s group is the point at infinity, and in projective coordinates it stops being a figure of speech and becomes the perfectly ordinary triple (0 : 1 : 0), whose X-coordinate is 0. A notation of the framework, uncolored. Cites: [19.23](ch19-refs.md#r-19.23) See also: [the point at infinity, projective X-coordinate](ch19-cards.md#g-ptinf)