Gauss: the Last EntryFrans Oort

(1) Introduction. Carl Friedrich Gauss (1777-1855) kept amathematical diary (from 1796). The last entry he wrote wason 7 July 1814. A remarkable short statement.

We present one of the shortest examples of a statement with avisionary impact: we discuss this expectation by Gauss. Hisidea preludes developments only started more than a centurylater.

I have two goals:– explain the visionary insight by Gauss and show theconnection with 20-th century mathematics,– with an emphasis on “the flow of mathematics”: early ideasbecoming a theory once understood; this we will see in thesequence:Tagebuch Gauss → The Riemann hypothesis in positivecharacteristic → the Weil conjectures.

In this talk I will explain the diary note by Gauss.

We give a short proof, using methods, developed byHasse, Weil and many others. Of course this ishistory upside down: instead of seeing the LastEntry as a prelude to modern developments, we givea 20-th century proof of this 19-th centurystatement.We will discuss also:Lemniscate transcendental functions → Weierstrass℘ functions → Koebe / Poincare uniformization ofalgebraic curves over C.

The last entry Gauss wrote in his notebook was on7 July 1814.

A remarkable short statement.

Here is the text:Observatio per inductionem facta gravissimatheoriam residuorum biquadraticorum cumfunctionibus lemniscaticis elegantissimenectens. Puta, si a + bi est numerus primus,a − 1 + bi per 2 + 2i divisibilis, multitudoomnium solutionum congruentiae1 = xx + yy + xxyy (mod a + bi) inclusisx =∞, y = ±i , x = ±i , y =∞ fit= (a − 1)2 + bb.

The text of the “Tagebuch” was rediscovered in 1897 and editedand published by Felix Klein (Math. Annalen 1903). In thecollected works by Gauss, in Volume X1 (1917) we find a facsimileof the original text.

A most important observation made byinduction which connects the theory ofbiquadratic residues most elegantly with thelemniscatic functions. Suppose, if a + bi is aprime number, a− 1 + bi divisible by 2 + 2i ,then the number of all solutions of thecongruence1 = xx + yy + xxyy (mod a + bi) includingx =∞, y = ±i ; x = ±i , y =∞, equals(a − 1)2 + bb.

Remarks. In the original we see that Gauss indeed used thenotation xx , as was usual in his time. In his DisquisitionesArithmeticae for example we often see that x2 and x ′x ′ areused in the same formula.

The terminology “Tagebuch” used, with subtitle“Notizenjournal”, is perhaps better translated by “Notebook”in this case. In the period 1796 - 1814 we see 146 entries,and, for example, the Last Entry is the only one in 1814.Gauss wrote down discoveries made. The first entry on 30March 1796 is his famous result that a regular 17-gon can beconstructed by ruler and compass.

Andre Weil remarks that “per inductionem” could beunderstood as, and translated by “emperically”. The word“induction” in our context does not stand for “mathematicalinduction”.

(2) We phrase the prediction by Gauss in other terms. Wewrite Fp = Z/p for (the set, the ring) the field of integersmodulo a prime number p. Suppose p ≡ 1 (mod 4). Once pis fixed we write

N = #((x , y) ∈ (Fp)2 | 1 = x2 + y 2 + x2y 2

)+ 4.

A prime number p with p ≡ 1 (mod 4) can be written as asum of two squares of integers (as Fermat proved). Theseintegers are unique up to sign and up to permutation.Suppose we write

p = a2 + b2, with b even and a − 1 ≡ b (mod 4).

In this case Gauss predicted

N = (a − 1)2 + b2

for every p ≡ 1 (mod 4).

(3) Some history. Herglotz gave in 1921 the firstproof for this expectation by Gauss.

We will see the attempt and precise formulation ofGauss of this problem as the pre-history and aprelude of the Riemann hypothesis in positivecharacteristic as developed by E. Artin, F. K.Schmidt, Hasse, Deuring, Weil and many others.

(4) Some examples. For p = 5 we obtaina = −1, and b = ±2 and N = 8. Indeed,(x = ±1, y = ±1) are the only solutions forY 2 = X (X − 1)(X + 1) (see Proposition 2(a) for an

explanation). p = 5 = (−1)2 + 22, N = 8.

For p = 13 we have (+3)2 + (2)2 = 13 and indeed(+3− 1)2 + 22 = 8 is the number of F13-rationalpoints as can be computed easily;p = 13 = (+3)2 + 22, N = 8.

For p = 17 there are the 12 solutions: the point 0,the three 2-torsion points, and x = 4, 5, 7, 10, 12, 13;the points (x = 4, y = ±3) and (x = 13, y = ±5)we will encounter below as (x = −e, y = 1 + e)

with e2 = −1; p = 17 = (+1)2 + 42, N = 16.

(5) This curve considered by Gauss. We see in thestatement by Gauss four points “at infinity”. Here is hisexplanation. Consider the projective curve

C = Z(−Z 4 + X 2Z 2 + Y 2Z 2 + X 2Y 2) ⊂ P2K

over a field K of characteristic not equal to 2. For Z = 0 wehave points P2 = [x = 0 : y = 1 : z = 0] andP1 = [x = 1 : y = 0 : z = 0]. Around P2 we can use a localchart given by Y = 1, and Z(−Z 4 + X 2Z 2 + Z 2 + X 2); wesee that the tangent cone is given by Z(Z 2 + X 2) (the lowestdegree part); hence we have a ordinary double point, rationalover the base field K and the tangents to the two branches areconjugate if −1 is not a square in K , respectively given byX = ±eZ with e2 = −1 in L. This is what Gauss meant byx =∞, y = ±i . Analogously for P2 and y =∞, x = ±i .

Explanation. Any algebraic curve (an absolutelyreduced, absolutely irreducible scheme of dimensionone) C over a field K is birationally equivalent overK to a non-singular, projective curve C ′, and C ′ isuniquely determined by C . The affine curveZ(−1 + X 2 + Y 2 + X 2Y 2) ⊂ A2

K , over a field K ofcharacteristic not equal to 2 determines uniquely acurve, denoted by EK in this note. This general factwill not be used: we will construct explicit equationsfor EK (over any field considered) and for EL over afield with an element e ∈ L satisfying e2 = −1.

In the present case, we write C ⊂ P2K as above (the

projective closure of the curve given by Gauss), Efor the normalization. We have a morphismh : E → C defined over K . On E we have a set Sof 4 geometric points, rational over any field L ⊃ Kin which −1 is a square, such that the inducedmorphism

E \ S −→ Z(−1 + X 2 + Y 2 + X 2Y 2) ⊂ A2K

is an isomorphism.

(6) A normal form, 1.Proposition 1. Suppose K is a field ofcharacteristic not equal to 2.(a). The elliptic curve E can be given byT 2 = 1− X 4.(b). The elliptic curve E can be given byU2 = V 3 + 4V .(c). There is a subgroup Z/4 → E (K ).

(7) The case p ≡ 3 (mod 4) (not mentioned by Gauss).Theorem 2. The elliptic curve E over Fp with p ≡ 3(mod 4) has:

# (E (Fp)) = p + 1.

Proof. The elliptic curve E can be given by the equationY 2 = X 3 + 4X . We define E ′ by the equation−Y 2 = X 3 + 4X . We see:

# (E (K )) + # (E ′(K )) = 2p + 2; E ∼=K E ′.

Indeed, any x ∈ P1(K ) giving a 2-torsion point contributes +1to both terms; any possible (x ,±y) with y 6= 0 contributes +2to exactly one of the terms (we use the fact that −1 is not asquare in Fp in this case). The substitution X 7→ −X showsthe second claim. Hence # (E (K )) = (2p + 2)/2.

(8) A normal form, 2.Proposition 3. Suppose L is a field ofcharacteristic not equal to 2. Suppose there is anelement e ∈ L with e2 = −1.(a). The elliptic curve EL can be given byY 2 = X (X − 1)(X + 1).(b). There is a subgroup (Z/4× Z/2) → E (L).Note: x = e gives e(e − 1)(e + 1) = (e − 1)2.We will study this in case either L = Q(

√−1) or

L = Fp with p ≡ 1 (mod 4) (as Gauss did in hisLast Entry).

(9) An aside: a historical line, uniformizationof algebraic curves. We work over C.(a) The circle given by X 2 + Y 2 = 1 can beparametrized by X = sin(z), and Y = cosin(z) (auniformization by complex functions). However, asany conic is a rational curve it can also beparametrized by rational functions, in this case as

(t2 − 1

t2 + 1)2 + (

2t

t2 + 1)2 = 1.

(b) The quartic equation given by Gauss in his LastEntry originates in the theory of the lemniscatefunctions. At the time of Gauss it was known thatthe curve given by x2 + y 2 + x2y 2 = 1 can beuniformed by the lemniscate functions

t 7→ (x = cl(t), y = sl(t)) .

In the notation of Gauss: x = sinlemn u,y = coslemn u. These functions are analogous ofthe usual sine and cosine functions, with the circlereplaced by the lemniscate of Bernouilli. Howeverfor this curve (of genus one) there is no rationalparametrization.

(c) Abel, Jacobi and Weierstrass proved that thisparametrization of this particular elliptic curve canbe generalized to any elliptic curve (a curve ofgenus one) in a parametrization by transcendentalfunctions: write an equation for the curve in aspecial form, and show thatx = ℘(z), y = (∂/∂z)(℘(z)) satisfies thisdifferential equation.

(d) Such a uniformization for elliptic curves wasgeneralized by Koebe and Poincare (1907) to curvesof arbitrary genus at least two (mapping thePoincare upper half plane onto the set of complexpoint of the curve considered, again bytranscendental functions).

In the 20-th century we observe the problem thatarithmetic properties are difficult to read off fromtranscendental functions. How do we determinerational points (say over a number field) on a givenalgebraic curve (e.g. as we like to do in FLT) ?

(e) The great breakthrough was in theShimura-Taniyama-Weil conjecture: elliptic curvesover Q are rationally parametrized (not by P1, but)by modular curves (and these curves we know verywell).

We see the genesis of mathematics, thedevelopments from the roots to final theory(uniformizations of algebraic curves).

We come to a proof of the expectation by Gauss inhis “Last Entry”:

Let a + bi be a prime element in Z[i ] such thata − 1 + bi is divisible by 2 + 2i ; then

#(E (Fp)) = (a − 1)2 + b2.

We conclude that a2 + b2 = p, a (rational) primenumber with p ≡ 1 (mod 4).

A version of he claim by Gauss:N = N(p) :=#(x , y) ∈ Fp | y 2 = X (X + 1)(X − 1)+ 1;p = a2 + b2, a, b ∈ Z;we see that b is even, and a is odd anda − 1 ≡ b (mod 4). Claim by Gauss:

N = (a − 1)2 + b2.

We write π = a + bi , and see N = Norm(π − 1).Reminder, examples:p = 5, and π = −1 + 2i ; N(5) = 8;p = 13, and π = 3 + 2i ; N(13) = 8;p = 17, and π = +1 + 4i ; N(17) = 16;p = 29, and π = −5 + 2i ; N(29) = 32; etc.We discuss a proof given in 20-th century language and notation.

(10) An aside: a historical line, from Gaussinto the 20-th century. We work over a finitefield.(a) The Last Entry by Gauss, giving a (possible)interpretation of the number of rational points asthe norm of an endomorphism.

(b) The Riemann Hypothesis for the zeros of thezeta function.

(c) The PhD thesis (1921, 1924) by Emil Artin,where a characteristic p analog of RH is launched,with many proofs, and checking the pRH in 40different cases.

(d) Proofs by Hasse of pRH for elliptic curves(1933, 1934, 1936).

(e) Proofs by F. K. Schmidt, Deuring and manyothers trying to understand the zeta function of analgebraic curve over a finite field.

(f) Proofs by Weil of the pRH for curves of arbitrarygenus and for abelian varieties (1940, 1941, 1948).Algebraic geometry over an arbitrary field had to bedescribed in great generality.

(g) Weil gives an interpretation of the pRH as ananalogue of the Fixed Point Theorem of Lefschetz,formulation of the Weil conjectures.

(h) Foundations of algebraic geometry necessary forproving the Weil conjectures (Weil 1946, Serre,1955, Grothendieck ≥ 1958). Proofs of manyaspects of the Weil conjectures (Grothendieck andmany others).

(i) Proof of pRH in full generality by Deligne (1974).

(j) “Standard conjectures” (Grothendieck), stillwide open.

A comment. Gauss considered solutions of this equationmodulo p. Only much later EFp was considered as anindependent mathematical object, not necessarily a set ofmodulo p solutions of a characteristic zero polynomial. In thebeginning as the set of valuations of a function field, as in thePhD-thesis by Emil Artin, 1921/1924. For elliptic curves thiswas an accessible concept, but for curves of higher genus(leave alone for varieties of higher dimensions) this wascumbersome. A next step was to consider instead a geometricobject over a finite field; a whole now aspect of (arithmetic)algebraic geometry had to be developed before we couldproceed. Each of these new insights was not easily derived;however, as a reward we now have a rich theory, and athorough understanding of the impact of ideas as in the LastEntry of Gauss.

(11) Frobenius maps and formulas. We recallsome theory developed by Emil Artin, F.K. Schmidt,Hasse, Deuring, Weil and many others, nowwell-known, and later incorporated in the generaltheory concerning “the Riemann Hypothesis inpositive characteristic”. For simplicity we onlydiscuss the case of the ground field Fp.

Easy, but crucial observation.Let Fp ⊂ k . The map x 7→ xp is a fieldhomomorphism k → k . Moreover the set of fixedpoints is:

Fp = x ∈ k | xp = x.This simple-minded fact, via a geometricinterpretation of higher dimensional analogues, isthe basis of our proof (and of the proof of theRiemann Hypothesis in positive characteristic forcurves and for abelian varieties).

The Frobenius morphism. For a variety V overthe field Fp we construct a morphism

FrobV = F : V → V ,

defined by “raising all coordinates to the power p”.Note that if (x) is a zero of f =

∑aαX α, then

indeed (xp) is a zero of∑

apαX α, because apα = aα,

f (x)p = (∑

aαxα)p =∑

apα(xα)p = f (xp).

This morphism was considered by Hasse in 1930.

(Side remark. An analogous morphism can bedefined for any variety over Fq for any q = pn.)

Easy fact: the set invariants of FrobV : V → V isexactly V (Fp), the set of Fp-rational points on V .

A little warning. The morphism π : V → V induces abijection π(k) : V (k)→ V (k) for every algebraically closedfield k ⊃ Fp; however (in case the dimension of V is at leastone) π : V → V is not an isomorphism.

We illustrate this in one example. Let V = A1Fp

, the affineline. The morphism FrobV : V → V on points is given byx 7→ xp; on coordinate rings it is given by K [T ]← K [S ] withT p ← [ S . We see this is not an isomorphism. However notethat V (x) = x if and only if x ∈ A1

Fp(Fp) = Fp. Moreover not

that for any perfect field Ω ⊃ Fp (e.g. a finite field, or analgebraically closed field) we obtain a bijection

Frob(Ω) : Ω = V (Ω) Ω = V (Ω),

injective because yp = zp implies (y − z)p = 0 hence y = z ;surjective because p

√z 7→ z .

We are going to apply this to our elliptic curve E .We write π = FrobE . Using addition on E we canwrite, and see

Ker(π − 1 : E → E ) = E (Fp).

It turns to that we can consider π ∈ End(E ) as acomplex number. (For specialists: End(E ) is acharacteristic zero domain.) A small argumentshows that

Norm(π − 1) = #(E (Fp)) =: N .

(For specialists: π − 1 is separable, hence all fixedpoints have multiplicity one.)

Moreover for the complex conjugate π we haveπ·π = p. Write β := π + π, the trace of π. We seethat π is a zero of

T 2 − β·T + p; |π| =√

p;

N = Norm(π − 1) = (π − 1)(π − 1) = 1− β + p.

Crucial observation by Andre Weil. Stare at theequality

#(E (Fp)) = 1− Trace(π) + p,

where #(E (Fp)) is the number of fixed points ofπ : E → E . What is the interpretation (and possiblegeneralization) of this?

The statements usually indicated by “the Riemann hypothesisin positive characteristic” I tend to indicate by pRH, in orderto distinguish this from the classical Riemann hypothesis RH.For any elliptic curve C over a finite field Fq one can define itszeta function (as can be done for more general curves, andmore general varieties over a finite field). As E. Artin and F.K. Schmidt showed, for an elliptic curve over Fq we have

Z (E ,T ) =(1− ρT )(1− ρT )

(1− T )(1− qT ).

As is usual, the (complex) variable s is defined by T = q−s .The theorem proved by Hasse is

|ρ| =√

q = |ρ|; this translates into s =1

2(pRH),

and we see the analogy with the classical RH, which explainsthe terminology pRH.

A comment. Gauss considered solutions of this equationmodulo p. Only much later EFp was considered as anindependent mathematical object, not necessarily a set ofmodulo p solutions of a characteristic zero polynomial. In thebeginning as the set of valuations of a function field, as in thePhD-thesis by Emil Artin, 1921/1924. For elliptic curves thiswas an accessible concept, but for curves of higher genus(leave alone for varieties of higher dimensions) this wascumbersome. A next step was to consider instead a geometricobject over a finite field; a whole now aspect of (arithmetic)algebraic geometry had to be developed before we couldproceed. Each of these new insights was not easily derived;however, as a reward we now have a rich theory, and athorough understanding of the impact of ideas as in the LastEntry of Gauss.

(12) A proof for the statement by Gauss in hisLast Entry. We have p = a2 + b2, where 2 + 2idivides a − 1 + bi . Hence 8 divides (a − 1)2 + b2.We conclude:p ≡ 1 (mod 4), with a odd and b even;the sign of a is determined by

a − 1 ≡ b (mod 4), explicitly,p ≡ 1 (mod 8) then a ≡ 1 (mod 4)p ≡ 5 (mod 8) then a ≡ 3 (mod 4);proof: 8 divides(a − 1)2 + b2 = a2 − 2a + 1 + b2 = p + 1− 2a;either p ≡ 1 (mod 8) and 2a ≡ 2 (mod 8)or p ≡ 5 (mod 8) and 2a ≡ 6 (mod 8).

Expectation / Theorem (Gauss 1841, Herglotz1921).

# (E (Fp)) = Norm(π − 1) = (a − 1)2 + b2.

Proof (assuming our knowledge for π ∈ End(E )).We know π·π = p and

Norm(π − 1) = #(E (Fp)) =: N

andN = 1− Trace(π) + p.

Hence

N = 1− 2a + a2 + b2 = (a − 1)2 + b2.

Thank you for this opportunityto talk to you,

and thank you for yourattention.