Moustapha Camara
email: m.camaraedu@gmail.com

Moussa Fall
email: moussafalls@yahoo.fr

Oumar Sall
email: oumarsfr@yahoo.fr

Algebraic points on the hyperelliptic curves y 2 = x 5 + n 2 1

Keywords: Hyperelliptic curves, Mordell-Weil groups, Rational points, Linear system, Galois conjugates

We give an algebraic description of the set of algebraic points of degree at most d over Q on hyperelliptic curves y 2 = x 5 + n 2 .

Introduction and result

Let Q be the eld of rational numbers and Q a algebraic closure of Q. Let C be an algebraic curve of genus g ≥ 2 dened over Q, and J C its jacobian variety. A celebrated theorem of Mordell-Weil states that the group J C (Q) of rational points of the jacobian J C is a abelian group of nite type, e.g., J C (Q) ∼ = Z r × J C (Q) tors , where the integer r is called the rank of the variety J C and J C (Q) tors the torsion subgroup. In this note, we study the algebraic points of degree at most d on hyperelliptic curves C k of genus 2 of ane equations C k : y 2 = x 5 + k for some integer k.

The degree of an algebraic point on C k is the degree of its eld of denition over Q. Note that the case k = 1 goes back to Schaefer ([8]), Fall ([3]) and Sall,et al. ([7]). The purpose of this note is to settle the case k = n 2 with n ∈ {4, 5, 8, 10, 12, 16, 20, 27, 36, 144, 162, 216, 400, 432, 625, 648, 1250, 1296, 5000}. Let η be a primitive 10 -th root of unity in Q and we put

A n k = ( 5 √ n 2 η 2k+1 , 0) with 0 ≤ k ≤ 4. Also let P n = (0, n), P n = (0, -n) and P ∞ the point at innity on C n 2 . The AbelJacobi map associated to P ∞ is the embedding j : C n 2 -→ J C n 2 , P -→ [P -P ∞ ]
where [P -P ∞ ] denotes the class of the divisor P -P ∞ . Various works study these curves (see [START_REF] Stoll | On the arithmetic of the curves y 2 = x l + A and their Jacobians[END_REF], [START_REF] Stoll | On the arithmetic of the curves y 2 = x l + A, II[END_REF], [START_REF] Stoll | On the L-functions of the curves y 2 = x l + A[END_REF]). Combining the results given by Mulholland ( [START_REF] Mulholland | Elliptic curves with rational 2-torsion and related ternary Diophantine equations[END_REF], p. 178) and Bruni ( [START_REF] Anthony | Twisted Extensions of Fermat's Last Theorem[END_REF], p. 142), we obtain the following theorem : Theorem 1. The Q-rational points on the curve C n 2 are given by

C (1) n 2 (Q) = {P n , P n , P ∞ }.
It is also known since Faltings ([2]), for a number eld K, the set C n 2 (K) of K-rational points on C n 2 is nite. We are interested mostly in this note in describing this set. More precisely, we give an algebraic description of the set of algebraic points of degree at most d over Q on the curve C n 2 . Our main result is the following theorem : Theorem 2. For all number eld K such that

[K : Q] ≤ d, we have C n 2 (K) ⊆ A n 0 ∪ A n 1 ∪ A n 2 with A n 0 =                      x, - i≤ d 2 a i x i j≤ d-5 2 b j x j     | a d 2 ̸ = 0 if d is even, b d-5 2 ̸ = 0 if d is odd and x root of B n 0 (x) = i≤ d 2 a i x i 2 - j≤ d-5 2 b j x j 2 (x 5 + n 2 )                  , A n 1 =                      x, - i≤ d+1 2 a i x i j≤ d-4 2 b j x j     | a 0 = ±nb 0 , a d+1 2 ̸ = 0 if d is odd, b d-4 2 ̸ = 0 if d is even and x root of B n 1 (x) = i≤ d+1 2 a i x i 2 - j≤ d-4 2 b j x j 2 (x 5 + n 2 )                  , A n 2 =                      x, - i≤ d+2 2 a i x i j≤ d-3 2 b j x j     | a 0 = ±nb 0 , a 1 = ±nb 1 , a d+2 2 ̸ = 0 if d is even, b d-3 2 ̸ = 0 if d is odd and x root of B n 2 (x) = i≤ d+2 2 a i x i 2 - j≤ d-3 2 b j x j 2 (x 5 + n 2 )                 
.

Particular case :

1. The algebraic points of degree 2 over Q on C n 2 are described by

C (2) n 2 (Q) = x, ± x 5 + n 2 | x ∈ Q * .
2. The algebraic points of degree 3 over Q on C n 2 are described by

C (3) n 2 (Q) = (x, ±n -λx 2 ) | λ ∈ Q * and x root of x 3 -λ 2 x 2 ± 2λn = 0 .
3. The algebraic points of degree 4 over Q on C n 2 are described by C (4)

n 2 (Q) = A n 0 ∪ A n 1 ∪ A n 2 with A n 0 = x, ± x 5 + n 2 | [Q(x) : Q] = 2 , A n 1 = x, ±n -λx -µx 2 | λ ∈ Q * , µ ∈ Q and x root of B n 1 (x) = x 4 -µ 2 x 3 -2λµx 2 + (-λ 2 ± 2µn)x ± 2λn , A n 2 = x, ±n -λx 2 -µx 3 | λ, µ ∈ Q * and x root of B n 2 (x) = µ 2 x 4 + (2λµ -1)x 3 + λ 2 x 2 ∓ 2µnx ∓ 2λn
.

Fundamental lemmas

Let D be a divisor on C n 2 . The vector space L(D) is dened to be the set of rational functions

L(D) = {f ∈ Q(C n 2 ) * : div(f ) ≥ -D} ∪ {0}.
The dimension of L(D) as a Q-vector space is denoted by l(D). Let x and y denote the rational functions on C n 2 given by x(X, Y, Z) = X/Z and y(X, Y, Z) = Y /Z. The projective equation of the curve

C n 2 is C n 2 : Y 2 Z 3 = X 5 + n 2 Z 5 .
The following lemma gives the structure of the Mordell-Weil group J C n 2 (Q) and that the niteness of the latter group is essential for this work :

Lemma 1. J C n 2 (Q) ∼ = Z/5Z.
Proof : Using of MAGMA for 2-descent on jocabians of hyperelliptic curves we obtain the desired result (For more detaits, we refer to [START_REF] Stoll | Implementing 2-descent for Jacobians of hyperelliptic curves[END_REF], [START_REF] Mulholland | Elliptic curves with rational 2-torsion and related ternary Diophantine equations[END_REF], [START_REF] Anthony | Twisted Extensions of Fermat's Last Theorem[END_REF]).

□ Lemma 2. • div(y -n) = 5P n -5P ∞ ; div(y + n) = 5P n -5P ∞ • div(x) = P n + P n -2P ∞ ; div(y) = A n 0 + • • • + A n 4 -5P ∞ Proof : It suces to apply the following relation div(y -α) = (Y -αZ = 0).C n 2 -(Z = 0).C n 2 ,
with α ∈ Z and Γ.C n 2 is the intersection cycle of a algebraic curve Γ dened over Q and the curve C n 2 . □

From Lemma 2, we see that 5j(P n ) = 5j(P n ) = 0, and j(P n ) + j(P n ) = 0. Thus, j(P n ) and j(P n ) generate the same group

J C n 2 (Q) which is isomorphic to Z/5Z. Lemma 3. A Q-basis for L(dP ∞ ) is given by B d = x i | i ∈ N et i ≤ d 2 ∪ yx j | j ∈ N et j ≤ d -5 2 .
Proof :

▷ It is clear that l(P ∞ ) = 1. But L(P ∞ ) certainly contains the constant functions, thus L(P ∞ ) =< 1 >.

▷ Since the genus of C n 2 is equal to 2, then 2P ∞ is a canonical divisor on C n 2 , so l(2P ∞ ) = 2, thus {1, x} provides a basis for L(2P ∞ ).

▷ For d ≥ 3, we can see that the elements of B d are linearly independent and are in L(dP ∞ ). Thus, it stays to show that the cardinal of B d is equal to l(dP ∞ ). According to the Riemann-Roch theorem (see [START_REF] Hindry | Diophantie geometry, an introduction, springer verlage[END_REF], p. 71), we have l(dP ∞ ) = d -1. Two cases arise :

⋄ 1 er case : if d is even, then by setting d = 2h, we have i ≤ d 2 = h; j ≤ d -5 2 = 2h -5 2 ⇔ j ≤ h-3.
Therefore, we have B d = {1, x, . . . , x h } ∪ {y, yx, . . . , yx h-3 } and hence card(

B d ) = (h + 1) + (h -3 + 1) = 2h -1 = d -1. ⋄ 2 er case : if d is odd, then by putting d = 2h + 1, we have i ≤ d 2 = 2h + 1 2 ⇔ i ≤ h; j ≤ d -5 2 = 2h -4 2 = h -2. Thus, we have B d = {1, x, . . . , x h } ∪ {y, yx, . . . , yx h-2 } and therefore card(B d ) = (h + 1) + (h -2 + 1) = 2h = d -1. □ 3 Proof of Theorem 2
Let R be an algebraic point on C n 2 of degree d over Q; if d = 1 these points are given by theorem 1., so we can assume that d ≥ 2. Let R 1 , • • • , R d be the Galois conjugates of R. Then none of the points R i is equal to the algebraic points on

C n 2 of degree ≤ d -1 over Q. We have [R 1 + • • • + R d -dP ∞ ] ∈ J Cn (Q) and Lemma 1 gives [R 1 + • • • + R d -dP ∞ ] = mj(P n ) with 0 ≤ m ≤ 4. ( * ) Case m = 0 : The formula ( * ) becomes [R 1 + • • • + R d -dP ∞ ] = 0.
The Abel-Jacobi theorem (see [START_REF] Griths | Introduction to algebraic curves[END_REF], p. 155) implies the existence of a rational function

f dened over Q such that div(f ) = R 1 + • • • + R d -dP ∞ . Therefore f ∈ L(dP ∞ ), hence f = i≤ d 2 a i x i + y j≤ d-5 2 b j x j with a d 2 ̸ = 0 if d is even or b d-5 2 ̸ = 0 if d is odd. At points R i , we have i≤ d 2 a i x i + y j≤ d-5 2 b j x j = 0, hence y = - i≤ d 2 a i x i j≤ d-5 2 b j x j . Replacing the expression of y in y 2 -x 5 -n 2 = 0, we obtain i≤ d 2 a i x i 2 - j≤ d-5 2 b j x j 2 (x 5 + n 2 ) = 0.
Thus, we obtain a family of points

A n 0 = x, - i≤ d 2 a i x i j≤ d-5 2 b j x j | x root of B n 0 (x) = i≤ d 2 a i x i 2 - j≤ d-5 2 b j x j 2 (x 5 + n 2 ) .
Cases m = 1 and m = 4 :

-For m = 1 : the formula

( * ) becomes [R 1 + • • • + R d + P n -(d + 1)P ∞ ] = 0. There exists a function f such that div(f ) = R 1 + • • • + R d + P n -(d + 1)P ∞ . Therefore f ∈ L((d + 1)P ∞ ), hence f = i≤ d+1 2 a i x i + y j≤ d-4 2 b j x j with a d+1 2 ̸ = 0 if d is odd or b d-4 2 ̸ = 0 if d is even. The function f is of order 1 in P n , hence a 0 = nb 0 . At points R i , we have i≤ d+1 2 a i x i + y j≤ d-4 2 b j x j = 0, which implies that y = - i≤ d+1 2 a i x i j≤ d-4 2 b j x j . The substitution y in y 2 -x 5 -n 2 = 0 gives i≤ d+1 2 a i x i 2 - j≤ d-4 2 b j x j 2 (x 5 + n 2 ) = 0.
Thus, we obtain a family of points

A n 1,1 = x, - i≤ d+1 2 a i x i j≤ d-4 a i x i 2 - j≤ d-4 2 b j x j 2 (x 5 +n 2 )
-For m = 4 : by similar reasoning to the case m = 1, we obtain a family points

A n 1,4 = x, - i≤ d+1 2 a i x i j≤ d-4 2 b j x j | a 0 = -nb 0 and x root of B n 1,4 (x) = i≤ d+1 2 a i x i 2 - j≤ d-4 2 b j x j 2 (x 5 +n 2 ) Finally, we put A n 1 = A n 1,1 ∪ A n 1,4 and B n 1 = B n 1,1 ∪ B n 1,4 . Cases m = 2 et m = 3 : -For m = 2 : the formula ( * ) becomes [R 1 + • • • + R d + 2P n -(d + 2)P ∞ ] = 0.
According to the Abel-Jacobi theorem, there exists a function f such that

div(f ) = R 1 + • • • + R d + 2P n -(d + 2)P ∞ . Therfore f ∈ L((d + 2)P ∞ ), hence f = i≤ d+2 2 a i x i + y j≤ d-3 2 b j x j with a d+2 2 ̸ = 0 if d is even or b d-3 2 ̸ = 0 if d is odd. The function f is of order 2 in P n , so a 0 = nb 0 and a 1 = nb 1 . At points R i , we have i≤ d+2 2 a i x i + y j≤ d-3 2 b j x j = 0, which leads to y = - i≤ d+2 2 a i x i j≤ d-3 2 b j x j . The substitution of y in y 2 -x 5 -n 2 = 0 gives i≤ d+2 2 a i x i 2 - j≤ d-3 2 b j x j 2 (x 5 + n 2 ) = 0.
Thus, we nd a family of points -For m = 3 : by analogous reasoning to the case m = 2, we nd a family of points Remark : The result obtained remains true for any integer n for which J C n 2 (Q) ∼ = Z/5Z and that the set of Q-rational points on C n 2 is given by {P n , P n , P ∞ }.

A n 2,2 = x, -
A n 2,3 = x, -

i≤ d+2 2 a i x i j≤ d- 3 2b 2 a i x i 2 - j≤ d- 4 2b j x j 2 (x 5 +

 2322425 j x j | a 0 = nb 0 , a 1 = nb 1 and x root of B n 2,2 (x) with B n 2,2 (x) = i≤ d+1 n 2 ).

i≤ d+2 2 a i x i j≤ d-3 2 b 2 a i x i 2 - j≤ d- 4 2b j x j 2 (x 5 +

 2222425 j x j | a 0 = -nb 0 , a 1 = -nb 1 and x root of B n 2,3 (x) n 2 ).

Finally, we put A n 2

 2


Pierre Schneider
email: pierre.schneider@mom.fr

P Pu ub bl li is sh he ed d i in n K Kl li io o 9 93 3/ /1 1 ( (2 20 01 15 5) ) 1 13 35 5--1 15 56 6 1 Quod nunc Rubrum ad mare patescit : the mare Rubrum as a frontier of the Roman Empire

Summary: This article deals with an excerpt from Tacitus' Annals which has often been commented on.

The famous account of Germanicus' Egyptian tour contains a puzzling remark about Syene (claustra olim Romani imperii quod nunc Rubrum ad mare patescit) : the adverb nunc refers in an allusive way to a recent expansion of the Roman Empire. Several scholars have rightly pointed to the Nabataean kingdom, a conclusion which seems to be the best one. The mare Rubrum therefore was not seen by Tacitus and his public as a limit of Roman Egypt. In contrast, from Tacitus' point of view, it is the southern part of Arabia, bordered by the mare Rubrum, that became the new frontier of the Roman Empire after the annexation of the Nabataean kingdom. How this event was considered as a major expansion of the imperium Romanum is examined in the last section of the present paper.

Which part of the mare Rubrum was incorporated into the imperium Romanum after 19 AD ?

The simplified map of the south-eastern boundary of the imperium Romanum in the early 1 st century AD appended to this article will help to put Tacitus' comment in its geographical and historical context : when Germanicus toured Egypt in 19 AD, he says, the Roman border was located at Syene ; but later (= nunc), when he was composing the Annals (say in the early 2 nd century AD) the mare Rubrum had become the new boundary of the Roman Empire in this part of the inhabited word. To quote Goodyear: "And what does (imperium) patescit ad denote ? Presumably the present extent of direct Roman control, as distinct from mere influence, in other words the area of a province or provinces (...)The frontier had been extended." 3 This means that Rome had managed to take control of a country bordered by the mare Rubrum.

Unfortunately, the name mare Rubrum (Red Sea) had several meanings in Greco-Roman times. Broadly, mare Rubrum -translated from the Greek name Erythra thalassa -would refer to three maritime areas, like its Greek equivalent.

The sea lying between East Africa and India was commonly called mare Rubrum : it roughly corresponds to our Western Indian Ocean -an alternative name was "Indian sea" 4 . Instead of producing a series of examples, it suffices to mention the Periplus of the Erythraean Sea, whose title is particularly relevant to the point. As such, the name mare Rubrum may refer to the easternmost part of the ocean, in prose as well as in poetry, as shown by the following two examples : "… Alexander, in the thirteen years of his reign, first brought under his sway all the well-nigh boundless empire that had belonged to the Persians, and then traversed Arabia and India, where the mare Rubrum embraces the uttermost ends of the earth (… Arabas hinc Indiamque, qua terrarum ultimos fines Rubrum mare amplectitur, peragravit) The Arabicus sinus / Ἀράβιος κόλπος (what we call today the Red Sea) and the Persicus sinus / Περσικὸς κόλπος (viz. the Arab-Persian Gulf) were both regarded as parts of the mare Rubrum, being formed by the penetration of this body of water into the Asian land mass 7 . As a consequence, they were commonly designated by the name mare Rubrum (or Erythra thalassa in Greek sources). As to the former, see, among many examples, Pliny the Elder or Strabo: (i) "Travellers set sail from India on the return voyage at the beginning of the Egyptian month Tybis, which is our December (…).They set sail from India with a southeast wind, and after entering the mare Rubrum 8 , continue the voyage with a south-west or south wind (navigant autem ex India vento volturno et, cum intravere Rubrum mare, Africo vel austro)" ; (ii)"There is another canal also, which empties itself into the Red Sea, or Arabian Gulf (εἰς τὴν Ἐρυθρὰν καὶ τὸν Ἀράβιον κόλπον), near the city Arsinoë, which some call Cleopatris." 3 Rubrum" (tandem per Babyloniorum fines in Rubrum mare inrumpunt)

10

. There seems to be no difference between these designations, which explains why more than one author equally employs Arabicus (or Persicus) sinus and mare Rubrum.

.

If ancient document give no clue, which is the case with the passage in question 12 , the name mare Rubrum may become a puzzle to the modern historian. Indeed, because of its fundamental ambiguity 13 , this passage has been endlessly interpreted in different ways . This was, however, to be challenged. In particular, O. Clason interpreted mare Rubrum as Red Sea (in its modern sense) [START_REF] Clason | Est-il vraisemblable que Tacite, ayant rédigé cette allusion triomphante pendant les quelques mois où l'Empire toucha le golfe Persique, ne l'ait pas effacée quand les précaires conquêtes de Trajan furent évacuées[END_REF] . Later J. Beaujeu argued that Tacitus would hardly refer to an achievement which actually ended in a failure 17 . b) An alternative option is the annexation of the Nabataean kingdom, whose south-western part is bordered by the north-east extremity of the mare Rubrum (= our Red Sea), i.e. the Gulf of Aqaba (see below). This event took place in 106 AD, after the death of the Nabataean king Rabbel II (105 AD). The annexation was immediately followed by the creation of the province of Arabia, and the establishment of the Roman administration

18

. This interpretation was first suggested by J. Aspach in 1896 19 . Later G. Bowersock offered fresh and good arguments in favour of this theory which definitely seems to be the less debatable. His conclusions are based on an inscription found in Saudi Arabia: "The problem has long seemed to me to be resolved decisively in favor of Princeton, 1993, 3-10, 4-5, 5. Bowersock rightly objects to Syme's view that the coastal part of Roman Arabia was small (Syme [n. 2] 768 : "Virgil on Marcus Antonius (Aen. 8.686 : litore rubro) would be enough to discount speculation about small tracts of the Red Sea coastland, whether Egyptian or Nabatean"). N.b. : the Mesene, the annexation of which supports Lipsius' theory, encompassed a very small part of the Arab-Persian Gulf. 22 Three other interpretations are known to me. (i) Meister (n. 14)113-116, argues that in the 2 nd century AD, the Eastern desert between the right bank of the Nile and the Red Sea, and particularly the areas of Berenice and Myos Hormos, were incorporated in the administrative territory of Egypt. This assertion is allegedly proven by the disappearance of the praefectura praesidiorum et montis Berenicidis in the late 1 st cent. AD, and the "Neueinrichtung des Gaues Berenike" in the early 2 nd cent. Volkmann (n. 21) and K. Wellesley, The date of composition of Ann. II, RhM 98,1955, 135-149, have proven Meister wrong: "it seems reasonable to conclude that Roman Egypt reached the Red Sea, in theory from 30 BC, and in fact shortly afterwards, and that Tacitus has misled us in suggesting an extension of the frontier in this direction between AD 19 and the time at which he wrote Annals II." (Wellesley 146). Sage (n. 13) 958 also objected to Meister's explanation, claiming that "the event is too minor for the emphatic nature of Tacitus' statement (also see Beaujeu [n. 14] 213 ; Syme [n. 2] 768 : "an opinion less plausible"; Goodyear [n. 4] 392). (ii) E. Laube, Die Abfassungszeit von Tacitus Annalen, (Diss., Freie Universität Berlin, Philos. Fakult.) Berlin 1955, unpublished (non vidi-known to me only from what Beaujeu reports) has suggested a novel solution of this vexed question. He has written that the growth of commerce with India, following the discovery of the monsoon in the mid 1 st century AD (sic), is the true Roman expansion in the mare Rubrum. Actually Rome's Eastern trade developed soon after the annexation of Egypt in 30 BC, which demolishes Laube's argument. In addition, as Beaujeu (n. 14) 214-215, rightly points out, "il est difficile d'admettre que Tacite ait eu en vue seulement une expansion commerciale, sans extension territoriale (…), le verbe patescere s'appliquant à une extension territoriale, chez Tacite, lorsqu'il a pour sujet un nom de pays." (J. Beaujeu has been followed by Sage [n. 13] 958). The final word goes to Syme [n. 2] 768: "… it is all misconception. The context and the colour of the Tacitean text seem to have been neglected." (iii) J. Desanges, Arabes et Arabie en terre d'Afrique, in: T. Fahd (ed.), L'Arabie préislamique et son environnement historique et culturel. Actes du Colloque de Strasbourg, 24-27 juin 1987, Leiden 1989, 413-429, 426-427: "A la différence de G. Bowersock, nous ne croyons donc pas que la remarque de Tacite soit une allusion précise à l'annexion de la Nabatène. Sans doute Tacite avait-il (...) une vision globale d'une présence arabe qui s'étendait de Syène à Petra, et entendait-il seulement dire que sur ces vastes espaces enserrant la mer Rouge, l'imperium Romanum était désormais solidement établi." 23 Meister (n. 14) wrongly thought that the Roman Empire expanded from Syene. That is why he logically rejected the two other interpretations. As Volkmann (n. 21) 237, rightly says in his thorough critical review of Meister's theory, "Tacitus nicht von den claustra Aegypti, sondern von den claustra Romani imperii spricht". 24 Res Gest. Div. Aug. 27 (cf. also the Egyptian obelisk bearing the inscription: Aegypto in potestatem populi Romani redacta, which had been erected in Rome [Nicolet [n. 3] 30]). 25 Beaujeu (n. 14) 206: "L'Empire touchait à la mer Rouge depuis que la conquête de l'Égypte lui en avait ouvert l'accès, en 29 av. . Was Tacitus completely wrong, or at least dramatically inaccurate ? Some have thought so : "It seems reasonable to conclude that Roman Egypt reached the Red Sea, in theory from 30 BC, and in fact shortly afterwards, and that Tacitus has misled us in suggesting an extension of the frontier in this direction between AD 19 and the time at which he wrote Annals II. (...)Tacitus' rather sketchy knowledge of Egyptian and Arabian topography has led him to suppose that in that year (i.e. 106 AD) the Roman Empire for the first time reached the Red Sea" 27 . I believe, on the contrary, that this text is clear and coherent 28 . The present article aims to prove this.

Aegyptus and the mare Rubrum

To sum up the first problem: Tacitus claims that the Roman Empire did not reach the mare Rubrum until Arabia was annexed in 106 AD, which brings us to the conclusion that the modern Red Sea was neither the boundary of the "province" of Egypt, nor that of the imperium Romanum. In reality this statement reflects a common geographical conception of Egypt: basically ancient Egypt covered only the Nile valley and the Delta. This conception goes back to the Ionian geography, as this passage quoted from Herodotus shows: "Beyond and above Heliopolis, Egypt is a narrow land. For it is bounded on the one side by the mountains of Arabia, which run north to south, (…); on the side of Libya, Egypt is bounded by another range of rocky mountains. One could object, however, that Tacitus lived in a period when the Eastern trade of Rome flourished. The bulk of exotic goods reached the Mediterranean world via the Egyptian gates of the Indian Ocean trade, that is to say ports lying on the Red Sea coast, such as Myos Hormos and Berenice. These harbours, founded in Ptolemaic times mainly for trade purposes, fell under the power of Rome as soon as Egypt was incorporated into the Empire 32 . Both were connected to the Nile valley (mostly to the city of Coptos) by several tracks crossing the Eastern desert. Since the Roman annexation of Egypt in 30 BC, an increasing flow of Indian, Arabian and east African commodities (spices, aromatics, ivory, gemstones, pearls etc.) was imported into the Mediterranean markets through these places. Thus it would be reasonable to assume that the Red Sea coast, being used for commerce purposes by Rome, had been included in Roman Egypt. Here it is worth recalling this significant passage of Strabo's Geography : "The ancients gave the name of Egypt to that country only which was inhabited and watered by the Nile, and the extent they assigned to it was from the neighbourhood of Syene to the [Mediterranean] sea. But later writers [= Hellenistic chorography ?], to the present time, have included 26 These "difficultés géographiques" (Beaujeu [n. 14] 222) have been really discussed only by Meister (n. 14) )". Must this -actually isolated -text be regarded as a decisive piece of evidence to establish that the Egyptian mare Rubrum was, contrary to Tacitus' words, the boundary of Rome ? In my opinion, such is not the case.

In the prevalent Greco-Roman representation of space, Myos Hormos and Berenice were actually regarded as "excrecenses", or "offshoots" protruding from, or very intimately connected to the Nile Valley 34 . The real centres of the Eastern trade of Rome were, in fact, Coptus / Koptos [today's Qift] and above all Alexandria, rather than those places. Pliny the Elder, eg., points out: "(…) notable, as is Coptos, the market near the Nile for Indian and Arabian goods (Coptos, Indicarum Arabicarumque mercium Nilo proximum emporium); to take another example, Aelius Aristide writes: "At Coptos, the Indian and Arabian trading center, it (= the Nile) [rises] twenty-one cubits." 35 ". Interestingly Strabo claims that Berenikê lies at the end of an "isthmus" which begins at Coptos, this "isthmus» being an extension of the Nile valley towards the Red Sea shore 36 . Thus, despite the presence of the Roman administration in the coastal area 37 , the true core of Roman Egypt remained the Nile valley. To put it in other words, even if Rome controlled the above mentioned places, which were harbours rather than emporia, the mare Rubrum was, to some extent, disconnected from the Egyptian territory. As a consequence, Myos Hormos and Berenike were not important enough to form the boundary of the Empire in this region 38 . As a matter of fact, this conclusion is, as I think, reinforced by a further argument.

The Eastern Egyptian desert was occupied by several tribes, as we are told by Strabo: "The lower parts of the country towards the Erythra thalatta (= the Arabian Gulf) along the Nile are occupied by Megabaroi and Blemmyes, who are subjects to the Ethiopians, and border upon the Egyptians; about the sea are Trôglodytai.

39

" Strabo adds:" These are nomads, and not numerous nor warlike, but accounted so by the ancients, because frequently, like robbers, they attacked defenceless persons.

"

There is no doubt that this passage reflects Roman ideas of nomadic tribes 41 . Such peoples had to be controlled by Rome; in particular, they had to be prevented from attacking merchants in the desert 33 Strab. 17,1,5 : οἱ μὲν οὖν ἀρχαῖοι τὸ οἰκούμενον αὐτὸ καὶ ποτιζόμενον ὑπὸ τοῦ Νείλου μόνον Αἴγυπτον ἐκάλουν ἀπὸ τῶν περὶ Συήνην τόπων ἀρξάμενοι μέχρι τῆς θαλάττης, οἱ δ´ ὕστερον μέχρι νῦν προσέλαβον ἐκ μὲν τῶν πρὸς ἕω μερῶν {τὰ} μεταξὺ τοῦ Ἀραβίου κόλπου καὶ τοῦ Νείλου σχεδόν τι πάντα (οἱ δ´ Αἰθίοπες οὐ πάνυ χρῶνται τῇ Ἐρυθρᾷ θαλάττῃ). (trans. H. C. Hamilton). As far as I understand, Strabo wants to say that indigenous tribes called "Ethiopians" did not really take part in long distance commerce: only the Greeks, followed by the Romans were able to make use the Erythra thalassa. In fact Adoulis, a port gradually controlled by Axum, had not yet emerged as a major place of the Indian Ocean traffic. But was Strabo really referring to the Aithiopes living in modern Eritrea and Ethiopia (eg., Diod. 3,18,5 = Agatharch. Erythr. 5,41)? I rather think that he alludes to the "Ethiopian" neighbours of Egypt (Blemyes, Trogodytes etc.-see Strab. 17,1,53 : λοιπὰ δὲ τὰ πρὸς νότον Τρωγλοδύται {καὶ} Βλέμμυες καὶ Νοῦβαι καὶ Μεγάβαροι οἱ ὑπὲρ Συήνης Αἰθίοπες), who did not engage much in this trade. 34 36 Strab. 17,1,45 : "Then follows the isthmus, extending to the Red Sea near Berenice, which has no harbour, but good landingplaces, because the isthmus is conveniently situated (ἐντεῦθέν ἐστιν ἰσθμὸς εἰς τὴν Ἐρυθρὰν κατὰ πόλιν Βερενίκην, ἀλίμενον μὲν τῇ δ´ εὐκαιρίᾳ τοῦ ἰσθμοῦ καταγωγὰς ἐπιτηδείους ἔχουσαν). Philadelphus is said to be the first person that opened, by means of his army, this road, which had no supply of water, and to have provided stations. This he did because the navigation of the Red Sea was difficult, particularly to those who set out from the recess of the bay. Experience showed the great utility of this plan, and at present all the Indian, Arabian, and such Ethiopian merchandise as is imported by the Arabian Gulf is carried to Coptos, which is the mart for such commodities (ἐμπόριον τῶν τοιούτων φορτίων). Not far from Berenice is Myos Hormus, a city with a naval station for vessels which navigate this sea; at no great distance from Coptos is the city of Apollo, so that two cities are the boundaries of the isthmus, one on each side (ὥστε καὶ αἱ διορίζουσαι τὸν ἰσθμὸν δύο πόλεις ἑκατέρωθέν εἰσιν)." (transl. H. C. . Such a conception is also expressed by Appianus (flor. under Trajan and Hadrian) : "Possessing (sc. the Romans) the best part of the earth and sea they have, on the whole, aimed to preserve their empire by the exercise of prudence, rather than to extend their sway indefinitely over poverty stricken and profitless tribes of Barbarians, some of whom I have seen at Rome offering themselves, by their ambassadors, as its subjects ; but the emperor would not accept them because they would be of no use to him.

43

" Similarly, the destitute tribes of Trogodytice were not judged important and prestigious enough to be incorporated into the Empire. This is the reason why I believe that the Egyptian side of the mare Rubrum was not perceived as a frontier of the Roman Empire.

Thus, in terms of political geography, the only well-defined border (claustra 44) was located at Syene, which marked the southern end of the Roman Empire. Beyond this point lay Aethiopia -viz the kingdom of Meroe [START_REF][END_REF] -an organized state to which some of the above mentioned nomadic tribes were subject (see n. 40). Some scholars, however, have accused Tacitus of being inaccurate, since, at the time of Germanicus' trip, the border was situated 120 km south of Syene, at Hiera Sykaminos (today's Wadi Maharraqah), in a district called the Dodecaschenus

46

. In reality, the Dodecaschenus was just a kind of buffer area under the joint control of Romans and Ethiopians, or "une sorte de principauté romano-méroïtique" 47 . That is why, according not only to Tacitus, but also Strabo, Pliny or Pomponius Mela, Syene represented the actual limit between the imperium Romanum and Aethiopia To conclude, the sentence exim ventum Elephantinen ac Syenen, claustra olim Romani imperii seems to reflect the imperial representation of space: the Red Sea area, although it was under the control of the Roman administration, was not a source of glory to the Empire; accordingly, it seems to have been ignored in the political geography, the only one in which Tacitus is interested -at least in this passage. Thus, from this point of view, Syene instead of the mare Rubrum was the boundary of imperium Romanum on its south-eastern fringe. . However, if the reader accepts the idea that Tacitus echoes an imperial representation of space, which implies all kinds of manipulations and distortions 81 , things become different. The point is that the mare Rubrum (= the Red Sea) was regarded by the Greco-Romans as a gulf of the Outer Ocean, like the Arab-Persian Gulf and the Caspian Sea. Accordingly, any imperator or king who had managed to conquer a country lying on such a gulf could claim that he had reached the ocean itself, that is to say, the edge of the oikoumenê: reaching the open ocean was by no means necessary; controlling the bottom of the Arabicus or Persicus sinus sufficed to achieve this. It is not an accident that, in the following three texts, Arabia and / or the Erythra thalassa ( = mare Rubrum) is the link between Rome and the Outer Ocean 82 :

 "But the dire evil which had been slumbering for a long time, namely, his (sc. Mark Antony, in 36 BC, during the Eastern campaign) passion for Cleopatra, (…) blazed up again with renewed power as he drew near to Syria (…). And when she was come, he made her a present of no slight or insignificant addition to her dominions, namely, Phoenicia, (…) and all that part of Arabia Nabataea which slopes toward the Outer Sea (= the Ocean -καὶ τῆς Ναβαταίων Ἀραβίας ὅση πρὸς τὴν ἐκτὸς ἀποκλίνει θάλασσαν) 83 .



"Moreover, a great and eager passion possessed him (sc. Pompey the Great, ca 65 BC) to recover Syria, and march through Arabia to the Erythra thalassa, in order that he might bring his victorious career into touch with the Ocean which surrounds the world on all sides (διὰ τῆς Ἀραβίας ἐπὶ τὴν ἐρυθρὰν ἐλάσαι θάλασσαν, ὡς τῷ περιϊόντι τὴν οἰκουμένην πανταχόθεν Ὠκεανῷ προσμίξειε νικῶν). For in Africa he had been the first to carry his conquests as far as the Outer Sea (πρῶτος ἄχρι τῆς ἐκτὸς θαλάσσης κρατῶν προῆλθε), and again in Spain he had made the Atlantic Ocean the boundary of the Roman dominion (…). In order, therefore, that he might connect the circuit of his military expeditions with the Erythra thalassa, he put his army in motion (ὡς οὖν συνάψων τῇ ἐρυθρᾷ τὴν περίοδον τῆς στρατείας ἀνίστατο).

84 "  "Again, the duration of the Assyrians, Medes, and Persians taken together (the three greatest empires before Alexander), does not amount to nine hundred years, a period which that of Rome has already reached and the size of their empire, I think, was not half that of the Romans, whose boundaries extend from the setting of the sun and the Western Ocean to Mount Caucasus and the river Euphrates, and through Egypt to Ethiopia, and through Arabia as far as the Eastern Ocean, so that their boundary is the ocean both where the sun-god rises and where he sinks, while they control the entire Mediterranean, and all its islands as well as Britain in the ocean ( These three excerpts show how space was shaped and distorted for political and ideological purposes. The process consists in a combination of simple ideas: Arabia is connected to the mare Rubrum ; the mare Rubrum is connected to the ocean ; the ocean is the edge of the world ; thus Arabia is connected to the edge of the world. A more developed form of this reasoning appears in Appian's Preface: on the one hand, he says, the Roman Empire is limited to the east by the Euphrates and to the south by Aethiopia (= Syene) ; on the other hand (the province of) Arabia and its southern boundary, the Red Sea, allow Rome to reach the eastern ocean. Thus, the imperium Romanum reaches the extreme east and west edges of the inhabited world, namely the Ocean.

No doubt, in this representation of space which enhances the Emperor's glory, the scientific tools of geography and cartography (orientations, measurements, angles 86 etc.) were absolutely irrelevant.

Tacitus' text was similarly more influenced by political geography than cartographic accuracy. Celebrating Trajan's recent achievement, he emphasized that in the southeastern part of the oikoumenê, the Roman Empire had reached the ocean, viz. the end of the world 87

. From this point of view Rome really expanded far beyond Syene (compare with Appianus). Incidentally, it is worth noticing that Tacitus uses the verb patescit : whereas clausta refers to a limit, patescit implies an unlimited expansion 88 .

Of course, as many scholars have observed, the word oceanus (as well as mare Rubrum "im Sinn des Sudostmeeres") was connected to the name of Alexander the Great 89 . At the end of the Indian expedition in 325 BC, the young king sailed on a short way off the mouths of the Indus. The political message was obvious: he was the first to reach the Outer Ocean and the edge of the world

Conclusion

The sentence exim ventum Elephantinen ac Syenen, claustra olim Romani imperii quod nunc Rubrum ad mare patescit seems to me to be clear and unambiguous. When the Nabataean kingdom was annexed by Trajan, for the first time, the Roman Empire was really bordered by the external ocean, in this part of the world 94 . This noticeable expansion must have been widely celebrated. The traces of some official records may have been preserved in Tacitus' Annals. This passage also discloses how the Roman power used certain names -in this case mare Rubrum -for political purposes. It also reveals how the Roman power defined the boundaries and shaped the imperial space for its own interest without cartographic accuracy: the Trogodytes were more or less ignored, even if Rome controlled their country. On the other hand, the annexation of Arabia widened the Roman Empire far beyond the actual limits of the province 95 . As Carrié says, "dans la situation d'hégémonie où se trouve l'empire, la frontière a moins de signification par elle-même que par le territoire de souveraineté qu'elle délimite. L'extension de ce territoire fait honneur à Rome portant loin et haut son nomen, tout ensemble nom et renom. Horizon quasi abstrait de cette gloire, les frontières se trouvent incontestablement investies de toute une charge idéologique qui comptait sans doute pour beaucoup dans la décision de les porter plus avant ou de les maintenir dans l'extension qu'elles avaient déjà atteinte." 96

  or thou (sc. Liber Pater) art rushing in mad and flaring triumph by swelling Ganges and the farthest confines of red Tethys and the Eastern lands etc. (… aut tumidum Gangen aut claustra novissima Rubrae /Tethyos Eoasque domos flagrante triumpho / perfuris …) 6 .

  J.-C." This geographical fact provided a good argument to those who rejected the equation mare Rubrum = Red Sea (Beaujeu 213). P Pu ub bl li is sh he ed d Saleh, which is not as remote in latitude as Syene 26

29"

  Later on, Strabo, Pliny the Elder and other authorities shared the same conception. Thus their geographical accounts of Egypt do not deal with the Red Sea coastal area, which was regarded as the domain of various tribes, and particularly of the Trog(l)odytes 30 . That is why in Greek and Roman chorography, Egypt and Trogodytice were separately treated 31 .

  28 Contra, Beaujeu (n. 14) 208: "Faute de pouvoir prendre ce texte à la lettre, il reste à l'interpréter, c'est-à-dire à déterminer ce que l'auteur avait en vue, et à dégager soit le sens caché de son affirmation apparemment injustifiable, soit les raisons qui expliquent son erreur, si erreur il y a." 29 Hdt. 2,8 (ἀπὸ δὲ Ἡλίου πόλιος ἄνω ἰόντι στεινή ἐστι Αἴγυπτος · τῇ μὲν γὰρ τῆς Ἀραβίης ὄρος παρατέταται, φέρον ἀπὸ ἄρκτου πρὸς μεσαμβρίην τε καὶ νότον (...) Tὸ δὲ πρὸς Λιβύης τῆς Αἰγύπτου ὄρος ἄλλο πέτρινον τείνει -transl. A.D. Goodley). Egypt comprises also the Nile Delta (Hdt. 2, 5-7). 30 See, eg., Plin. nat. 6,164 (… Artemidorus Arabiae latere | V II| L , Trogodytico, vero I | L V I I D p. Ptol<e>maida usque …); Curt. 4,7,18 (… Trogodytis cognomen est ; horum regio usque ad Rubrum mare); Plut. Defect. Or. 2. This fact has been rightly stressed by Meister (n. 14) 401: "Er (sc. Strabo) betrachtet das Troglodytenland, das sich westlich von Ägypten und Äthiopien längs der Küste des Arabischen Golfes erstreckte, nicht als Teil, sondern als Nachbarland von Ägypten." 31 See, in particular: (i) Strabo: Egypt = 17,1 (in extenso); the west side of the Red Sea = 16,4,1-13. (ii) Pliny: Egypt = 5,48-64 ; the west side of the R. S. = 6,163-176. (iii) Mela: Egypt = 1,49-60; the west side of the R. S. = 3,80-81. Note that the Troglodyte country is included by some authorities in Arabia (eg., Str., loc. cit.; Plin. nat. 6.168). 32 About the Red Sea ports of trade, see G. K. Young, Rome's Eastern Trade. International Commerce and Imperial Policy (31 BC-AD 305), London-New York 2001, 38-45; about the desert roads, id. 45-47 ; about Coptos, id. 47-51.PPu ub bl li is sh he ed d i in n side almost all the tract between the Arabian Gulf and the Nile (the Aethiopians however do not make much use of the Erythra thalatta 33

48 . 50 toponym

 4850 Here, three cohorts in charge of protecting the Egyptian borders were garrisoned 49 . In addition, the famous Syene was likely to be more familiar to the general public than the obscure name Hiera Sykaminos 51 .

14 .

 14 But two interpretations stand out from the rest, for these only are linked to a precise historical event (the other ones are summarized below, n. 23). a) In 113 AD Trajan waged a war against the Parthian empire, and two years later he conquered Mesopotamia. This region was incorporated in the Roman Empire as a province in 115 (Mesopotamia). One year later, in 116, Trajan and his armies headed further south. He reached the bottom of the Arab-Persian Gulf (= mare Rubrum) which was a remarkable military success. The king of Mesenia (or Characene, a kingdom whose capital city was Spasinou), Attambelos V, surrendered to Trajan, who then created the province of Assyria. However, the whole territory was recovered by the Parthians in 117, and the death of Trajan in the same year put an end to the imperialistic ambitions of the optimus Princeps. This interpretation of quod nunc Rubrum ad mare patescit was first suggested by the great humanist Justus Lipsius, on the basis of a passage from Eutropius' Breviarum (see below, n. 86), and was defended by strong supporters 15

  Rubrum mare as what we know today as the Red Sea. Epigraphical evidence that turned up after the publication of Syme's Tacitus 20 has proved that the Roman province of Arabia, annexed in 106, actually extended along the coast of the Red Sea on the northwest side of the Saudi Arabian peninsula. Syme and virtually everyone else had formerly discounted the possibility of a province of such extent."21 Bowersock's judgment is confirmed by further evidence and arguments which will appear in the third section of the present article 22 . The historical problem having been solved, it is easier to grasp the meaning of Tacitus' observation: when Germanicus visited Egypt, the remotest border of the imperium Romanum in this part of the inhabited world was situated at the Egyptian city of Syene. In 106 AD, with the annexation of the Nabataean kingdom by Trajan and the creation of the province of Arabia, the Empire was expanded 23 , its new boundary being formed by the mare Rubrum : exim ventum Elephantinen ac Syenen, claustra olim Romani imperii quod nunc Rubrum ad mare patescit. This implies that, in Tacitus' view, the frontier of the Roman Empire had for the first time reached this body of water. Tacitus affirm that the annexation of Arabia led to the expansion of the Roman Empire? As far as we know, the southernmost -inland -limit of the Arabia provincia was located as Hegra / 20 G. W. Bowersock, The Greek-Nabataean bilingual inscription at Ruwwafa, Saudi Arabia, in J. Bingen -G. Cambier -G. Nachtergael (eds.), Le monde grec. Pensée, littérature, histoire, documents. Hommages à Claire Préaux, Bruxelles 1975, 513-522, 518-519. Several years before, Beaujeu (n. 14) 222-225, adopted this solution as the best one. Also see . H. Volkmann, Zur Datierung der Annalen des Tacitus, Gymnasium 60, 1953, 236-239, 238-239; Sage (n. 13) 957), who argues that the first part of the Annals must have been composed before Armenia became a province (114 AD), and logically rejects the mare Rubrum (= sinus Persicus) option; Goodyear (n. 4) 392-393; R. Martin, Tacitus, Berkeley-Los Angeles 1981, 247, n.16. 21 G. W. Bowersock, Tacitus and the Province of Asia, in: T. J. Luce & A. J. Woodman (eds), Tacitus and the Tacitean Tradition,

		P Pu ub bl li is sh he ed d i in n K Kl li io o 9 93 3/ /1 1 ( (2 20 01 15 5) ) 1 13 35 5--1 15 56 6
	interpreting The reader at this stage is left with two geographical problems. First, considering that Egypt -with
	its ports lying on the Red Sea shore -was a part of the Roman Empire since 30 BC (Aegyptum imperio
	populi Romani adjeci	24 ), the mare Rubrum was already the boundary of Rome in 19 AD	25 . Secondly,
	how could
		4

  and Beaujeu. The southernmost place where the Roman army was garrisoned is Hegra / Mada'in Saleh, in the hinterland (Sartre [n. 19] 621; Bowersock [n. 21]. Latitude of Mada'in Saleh: 26° 47 N ; latitude of Aswan: 24° 05 N (n.b. : the approximate latitude of the Euphrates' mouth [Characene] is 30°N).

27

Wellesley (n. 23) 146-147.

  See, eg., Plin. nat. 6.168 : Berenice oppidum, matris Philadelphi nomine, ad quod iter a Copto diximus.

	35 Plin. nat. 5, 60 (trans. H. Rackham) ; Aristeid. Aiguptios 115 (transl. C. A. Behr). Also see Str. (n. 36); Lucan. 10,313-315; Dion
	Chrys. 32,36; Jos. b.Jud. 4.608 [4.10.5]; Stat. Theb. 1.263-265.

  N.b. : we read in Pliny's encyclopaedia the name Berenice urbs Trogodytarum (Plin. nat. 2, 183).39 Strab. 17,1,2 (τὰ δὲ κατωτέρω ἑκατέρωθεν Μερόης παρὰ μὲν τὸν Νεῖλον πρὸς τὴν Ἐρυθρὰν Μεγάβαροι καὶ Βλέμμυες, Αἰθιόπων ὑπακούοντες, Αἰγυπτίοις δ´ ὅμοροι· παρὰ θάλατταν δὲ Τρωγλοδύται -transl. H. C. Hamilton).40 Strab. 17,1,53 (εἰσὶ δ´ οὗτοι νομάδες καὶ οὐ πολλοὶ οὐδὲ μάχιμοι, δοκοῦντες δὲ τοῖς πάλαι διὰ τὸ λῃστρικῶς ἀφυλάκτοις ἐπιτίθεσθαι πολλάκις -transl. H. C. Hamilton).41 The relationship between nomad peoples and the Roman Empire is a complex problem (see, eg., P. Trousset, La frontière romaine et ses contradictions, in: La frontière (Travaux de la Maison de l'Orient 21), Lyon 1993, 25-33). they used to do previously (palai). On the other hand, with regard to the definition of Rome's frontier, they did not really belong to the imperium, for the countries "where the inhabitants only lead a miserable and nomadic life" are worthless 42

	P Pu ub bl li is sh he ed d i in n K Kl li io o 9 93 3/ /1 1 ( (2 20 01 15 5) ) 1 13 35 5--1 15 56 6
	roads, which
	Hamilton)
	37 See Wellesley (n. 23) 137-146 ; S. E. Sidebotham, Berenike and the Ancient Spice Route, Berkeley-Los Angeles-London 2011,
	85-86.
	7

38

  ὅτι Ῥωμαίοις ἀπό τε δύσεων καὶ τοῦ πρὸς ἑσπέραν ὠκεανοῦ ἐπὶ τὸ Καύκασον ὄρος καὶ ποταμὸν Εὐφράτην καὶ ἐς Αἰθίοπας τοὺς ἄνω Αἰγύπτου καὶ <δι'> Ἀράβων ἐπὶ τὸν ἑῷον ὠκεανὸν ἡ ἀρχὴ διεξέρχεται, καὶ ὅρος ἐστὶν αὐτοῖς ὁ ὠκεανὸς ἀνερχομένου τε καὶ δυομένου τοῦ θεοῦ). Syme (n. 2) 768, believed that Tacitus referred to the mare Rubrum in a very general way and did not really think of the (Persicus) sinus : "Now Rubrum mare as all agree has the general meaning of 'the Indian Ocean'. The Persian Gulf and the Red Sea are its two great inlets as such, each of which can bear the name. But Tacitus is not thinking of either inlet as such. Rather the eastern ocean, the one end of the world, like Gades the other (omnibus in terris quae sunt a Gadibus usque Auroram et Gangen [Juvenal 10, 1-2]). To elucidate Tacitus, the language of poets, of orators, and of historians is relevant, the theme of Alexander and the theme of world conquest." No doubt, as we will see below, Alexander is present in this ideological geography. On the other hand, should we admit that "Tacitus is not thinking of either inlet", we would have to deduce that he did not refer to any historical event (either the creation of the province of Arabia, or the brief annexation of Mesene), which seems to me rather dubious. As Beaujeu (n. 14) 215, states, "il est évident que cette notation (…) doit se référer à un fait précis, survenu entre l'année 19 p.C. et les premières années du règne d'Hadrien." 83 Plut. Ant. 36,1-2 (transl. B. Perrin). 84 Plut. Pomp. 38,2-3 (transl. B. Perrin). 85 Appian. Proem. 34 ). Appianus continues as follows : "(…) But the greatest sea-power of the Medes and Persians included only the Gulf of Pamphylia and the single island of Cyprus or perhaps other small islets belonging to Ionia in the Mediterranean. They controlled the Persian Gulf also, but how much of that is open sea (Μήδοις δὲ καὶ Πέρσαις ἥ τε πλείστη θάλασσα [ἢ] ὁ Παμφύλιος κόλπος ἦν καὶ μία νῆσος ἡ Κύπρος ἢ <εἰ> τί που ἄλλο σμικρὸν τῆς Ἰωνίας ἐν θαλάσσῃ• τοῦ τε Περσικοῦ κόλπου (καὶ γὰρ τοῦδε ἐκρατοῦν) πόσον τι καὶ τὸ τοῦτου πέλαγός ἐστιν; -transl. H. White).

	P Pu ub bl li is sh he ed d i in n K Kl li io o 9 93 3/ /1 1 ( (2 20 01 15 5) ) 1 13 35 5--1 15 56 6
	85 "
	80 Goodyear (n.4) 393. Also see Sage (n. 13) 958: "This possibility gains greater weight when we take into account the general
	weakness of the author."
	81 Compare with the imperial space of Alexander (see Strab. 11,5,5; 15,1,8, who accuses Alexander's companions of geographical
	forgeries and flattery).
	12

82

  It is well known that Antiochos I and Ptolemy II launched expeditions respectively on the Caspian Sea and the Red Sea in order to equal Alexander. It is also agreed that Pompey followed in Alexander's footsteps 91 . Thus, when Trajan annexed Arabia extending to the mare Rubrum, and consequently made the Eastern ocean92 the boundary of its Empire, the memory of Alexander the Great was probably in his mind Despite a phenomenal expenditure of perverse ingenuity in numerous attempts to explain away that passage, which in turn fixes the scope of the more general reference to expansion of empire in IV.4.3, the Rubrum Mare here is necessarily the Rubrum Mare of XIV. 2 5. 2 (i.e. the Arab-Persian Gulf), and Tacitus patently refers to an epochal enlargement of the empire, not a trivial rectification of frontiers. And since nunc means "now," Tacitus wrote that passage after hearing the first news of Trajan's conquest of Mesopotamia in 115 and before that territory was abandoned by Hadrian soon after the death of Trajan in August, 117".89 Volkmann (n. 21) 238. Cf. also Symes (n. 2) 768 ; Borzsák (n. 15) 468-469 ; Beaujeu (n. 14) 215 etc. 90 Arrian. an. 6,19,3-5. 91 See Nicolet (n. 3) 36-42; 45-47 ; G. Parker, The Making of Roman India, Cambridge 2008, p. 207-227. 92 About Rome and the northern edge of the world, see Chr. Krebs : "Borealism: Caesar, Seneca, Tacitus, and the Roman discourse about the Germanic North,"in: E. S. Gruen, Cultural Identity and the Peoples of the Ancient Mediterranean, Los Angeles, 2010, 202-221 ; id., Magni viri: Alexander, Pompey, and Caesar in Catullus 11, Philologus 152, 2008, 223-229. 93 Volkmann (n. 21) 238: "Trajan lebte besonders stark in der Tradition Alexanders d. Gr.". He quotes the famous passage byCassius Dio (Hist. Rom. 68, 28-29): " Then he (sc. Trajan in Mesene) came to the ocean (sc. the bottom of the Persicus sinus) itself, and when he had learned its nature and had seen a ship sailing to India, he said :' I should certainly have crossed over to the Indians, too, if I were still young (…) For he began to think about the Indians and was curious about their affairs, and he counted Alexander a lucky man. Yet he would declare that he himself had advanced farther than Alexander, and would so write to the senate, although he was unable to preserve even the territory that he had subdued" -transl. H. B. Foster). Also Eutr. Breviarum ab urbe condita 8,3,1 : Messenios vicit ac tenuit. Usque ad Indiae fines et mare Rubrum accessit). On the presence of Alexander the Great in Annals 2, 59-61, see B. Kelly, Tacitus, Germanicus and the kings of Egypt, CQ 60.1, 2010, 221-237.

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	13

90

.

93

.

86

Contra, Beaujeu (n.

14)

223, who thinks that Tacitus' spatial representations emanate from a cartographic perception of the Red Sea area. 87 N.b. : see how Aelius Aristide, who wrote his speech Eis Rhômên in ca 144 AD, describes the expansion of the Roman Empire: "Now, indeed, the boundaries of your empire are of extent not to be despised, not such that their interior can be measured. (…) Nothing escapes you, neither city, nor nation, nor harbor, nor land, unless you have condemned something as useless (πλὴν εἴ τινων ἄρα ἀχρηστίαν κατέγνωτε). The Erythra thalassa (= mare Rubrum), the cataracts of the Nile (= Syene) and lake Maeotis, which former men spoke of as the ends of the earth, are for this city "the fence of courtyard". As to the ocean, which certain writers neither believed existed at all not that it flowed around the earth (…)you have discovered this ocean so thoroughly etc." (Eis Rhômên 28, p. 205 Jebb -transl. C. A. Behr).

88

On patescit, see Beaujeu (n. 14) 214; Goodyear (n. 4) 391-392. From this point of view, the annexation of Arabia was not a minor alteration of the Roman frontier, contrary to Oliver's opinion (n. 16) 297 : "

Cf. Strab. 17,3,24. "Of Africa (Libyê), the whole sea-coast on the Mediterranean is in their (sc. Rome) power; the rest of that

The

Let us now turn to the second part of the sentence, which contains the allusion to the creation of the provincia Arabia in 106 AD : (Romani imperii) quod nunc Rubrum ad mare patescit.

What must be first examined is the relationship between Arabia and the mare Rubrum. The question is why, unlike Egypt, the Nabataean kingdom was not disconnected from the Red Sea, from the Greco-Roman perspective. Of course, "we may remember that, as Beaujeu persuasively argues, the Red Sea counted for Tacitus as entirely within Arabia, not as adjacent to Egypt." 52 But more important than the whole of Arabia is the specific case of Nabataea. Long before the reign of Trajan, the Greeks were already aware that the Nabataeans, who had once been a largely nomadic people, had become a completely sedentary nation 53 : Nabataea was a kingdom, that is to say a state with its king, administration, army, and territory. As regards the last feature, at some point the Nabataean territory comprised the north-eastern end of the Red Sea, viz. Aila (today Aqaba) which played a part in the Nabataean trade activity, as Bowersock states :"The Nabataean trade routes up from 'Aqaba and westward to Gaza are clearly indicated by forts, cisterns, and abundant pottery 54 ". In particular, Nabataean piracy emerged in the Gulf of Aila in the 2 nd (?) century BC, when the Nabataeans "realized that the new traffic by sea (sc. via Myos Hormos and Berenikê) would mean a gradual decline in the overland commerce which had been the basis of their prosperity and their sedentarization.

55 " At any rate, from the Greco-Roman point of view, the mare Rubrum was clearly a part of the Nabataean kingdom and therefore one of its borders. For instance Diodorus, quoting Agatharchides, states: "After one has sailed past this country the Laeanites Gulf (= the Gulf of Aila) comes next, about which are many inhabited villages of Arabs who are known as Nabataeans. This tribe occupies a large part of the coast and not a little of the country which stretches inland etc.

56

" About two centuries later, Pliny points out: "Beyond the Pelusiac Mouth is Arabia, which extends to the mare Rubrum 57 ". As for sea ports, beside Aila / Ailana (the modern Aqaba) 58 , Leukê Komê, the location of which is still unknown, was under Nabataean control 59 . To put it in other words, in contrast with the west coast of the Red Sea, which was the domain of the Trogodytes, the Gulf of Aqaba belonged to Nabataea; in contrast with Berenikê and Myos Hormos which could be defined as "Egyptian enclaves" in Trogodyte territory, Aila (and Leuke Komê) were integrated into the Arabian territory.

Thus the creation of the province of Arabia "automatically" extended the frontier of the empire as far as the mare Rubrum. Here it is worth quoting a remarkably interesting passage of Cassius Dio (ca 150-235 AD), for it recalls Ann.2.61 in a striking way: "… while Syria and Phoenicia had become tranquil, Pompey turned against Aretas (= the Nabataean king Aretas III -Pompey's undertaking took place in 64 BC). The latter was king of the Arabians, now subjects of the Romans, as far as the Erythra thalassa ( . On the contrary, some political messages are likely to have been embedded in it. Let us again consider the two sides of the Red Sea: on the African coast lived the destitute Troglodytes. How could Rome take pride in including such tribes in its empire? On the opposite side lay the rich and prestigious kingdom of Arabia, which in the decades before 106 AD had witnessed "an unprecedented prosperity and growth." 62 In this case, the name mare Rubrum was worth mentioning, since it provided a considerable glory to both Rome and Trajan : the country "extending to the mare Rubrum and to the Arabia known by the surname of Happy and famous for its perfumes and its wealth" 63 , which Alexander the Great wanted -or planned 64 "there are no known coins advertising Aretas as king in 3, 2, and 1 BC" thinks that the Nabatean kingdom was annexed for this brief interval. This assumption, says Bowersock, receives support from Strabo: "The geographer Strabo stated unambiguously, in a passage that has been curiously neglected, that in his day (nun de) the Nabataeans, like the Syrians, were subjects of the Romans." This text, however, is quite ambiguous. It reads : "The Nabataeans and Sabaeans, situated above Syria, are the first people who occupy Arabia Felix. They were frequently in the habit of overrunning this country before the Romans became masters of it, but at present both they and the Syrians are subject to the Romans (Πρῶτοι δ´ ὑπὲρ τῆς Συρίας Ναβαταῖοι καὶ Σαβαῖοι τὴν εὐδαίμονα Ἀραβίαν νέμονται, καὶ πολλάκις κατέτρεχον αὐτῆς πρὶν ἢ Ῥωμαίων γενέσθαι· νῦν δὲ κἀκεῖνοι Ῥωμαίοις εἰσὶν ὑπήκοοι καὶ Σύροι -transl. H. C. Hamilton)". If this interpretation is accepted, the Nabataeans may have been "subject" to Rome in the same way as the Sabaeans (see Res Gest. div. Aug. 26). 66 . Incidentally, I wonder whether another political message was linked with the name mare Rubrum. The Parthian kings were the great rivals of Rome at that time. A common opinion was that the inhabited world was dominated by the Roman and Parthian empires. Tacitus,for instance, says: "There was also to be read what tributes were imposed (sc. by the ancient Pharaohs) on these nations (…) a revenue as magnificent as is now exacted by the might of Parthia or the power of Rome. 74 " In fact the mare Rubrum (= Arab-Persian Gulf) was a boundary of the Parthian empire : "From there, they (sc. Alexander and his army) set off for the country of the Parthians, then an unknown people. Now they dominate all nations settled beyond the Euphrates and Tigris who are bordered by the mare Rubrum" 75 . We even come across the idea that the mare Rubrum was their own sea : such is the plausible meaning of an expression that we read in Lucanus' Pharsalia : "… their sea has a different colour from ours … " (= the mare Rubrum belongs to the Parthians) 76 ". However, once the imperium Romanum, in turn, reached the mare Rubrum, the Parthian empire was no longer the only one to "possess" this sea. That was also the glory of Rome and Trajan 77 Now let us go further. Tacitus stresses the fact that the annexation of Arabia enabled the Romanum imperium to expand. However, as I stated in the beginning, topographical maps contradict this assertion 78 : Syene is further south than the Nabataean Red Sea. Some scholars have argued that the Roman knowledge of this area was inaccurate and distorted: ""As Beaujeu argues, the Romans saw Egypt and especially the Delta as belonging to Asia and not Africa. They were mistaken as well on its orientation which they took to be east-west and underestimated its size. Thus it would be possible for Tacitus to think of the annexation of 106 as an extension of Roman control in the west Asiatic area towards the southeast on the same basic line of orientation as the Red Sea coast of Egypt." 79 Tacitus' ignorance has also been pointed out: : : "Granted T.'s geographical incompetence and the considerable haziness about the region of Egypt and Arabia evident in Pliny and elsewhere, also the possibility T. did not recall when the Romans reached the west of the Red Sea, and the certainty he knew of the annexation of Arabia Petraea, then it seems likely a conflation of ideas could have prompted him to remark at this point, when Germanicus comes to the limit of his Egyptian travel, that in contrast to the position in AD 19, Roman control in the area extended by his own time to the Red


Nicolas Captier¹²³⁶
email: nicolas.captier@curie.fr

Jane Merlevede¹²³

Askhat Molkenov⁴

Ainur Seisenova⁴

Altynbek Zhubanchaliyev⁴

Petr V Nazarov⁵

Emmanuel Barillot¹²³

Ulykbek Kairov⁴

Andrei Zinovyev¹²³
email: andrei.zinovyev@curie.fr

BIODICA: a computational environment for Independent Component Analysis

Keywords:

We developed BIODICA, an integrated computational environment for application of Independent Component Analysis (ICA) to bulk and single-cell molecular profiles, interpretation of the results in terms of biological functions and correlation with metadata. The computational core is the novel Python package stabilized-ica which provides interface to several ICA algorithms, a stabilization procedure, meta-analysis and component interpretation tools. BIODICA is equipped with a user-friendly graphical user interface, allowing nonexperienced users to perform the ICA-based omics data analysis. The results are provided in interactive ways, thus facilitating communication with biology experts.

Introduction

The recent progress of high throughput omics technologies has made molecular data more accessible and has fostered the development of many computational analyses to exploit the rich information they offer. Such analyses require efficient tools to handle the high dimensionality of these data and reveal the underlying biological processes. Independent Component Analysis (ICA) is a statistical and computational method which aims to represent observed signals as linear mixtures of independent latent factors. ICA has been successfully applied to omics data with the hypothesis that observed molecular profiles result from linear combinations of unobserved biological and technical processes [START_REF] Liebermeister | Linear modes of gene expression determined by independent component analysis[END_REF]. In particular, it has been shown to extract interpretable and reproducible components and has stood out from other popular methods like Principal Component Analysis (PCA) or Non-negative Matrix Factorization (NMF) (Sompairac et al., 2019).

Here we present BIODICA, a complete computational environment for a user-friendly application of ICA to omics data. It encompasses a set of tools to extract and interpret reproducible independent components, using methods that already proved to be successful in multiple studies [START_REF] Biton | Independent component analysis uncovers the landscape of the bladder tumor transcriptome and reveals insights into luminal and basal subtypes[END_REF][START_REF] Aynaud | Transcriptional programs define intratumoral heterogeneity of ewing sarcoma at single-cell resolution[END_REF].

Methods

Stabilization procedure for extracting reproducible components

The computational core of BIODICA is the Python package stabilized-ica. It implements a stabilization procedure which addresses the variability of the solutions of ICA algorithms when run multiple times [START_REF] Himberg | Icasso: software for investigating the reliability of ica estimates by clustering and visualization[END_REF]. When applied to transcriptomics data, not only did this procedure provide a quantification of the significance of the independent components but it also extracted more reproducible ones than standard ICA [START_REF] Cantini | Assessing reproducibility of matrix factorization methods in independent transcriptomes[END_REF]. Besides, it allowed the development of an approach for selecting the optimal number of independent components to extract from omics data [START_REF] Kairov | Determining the optimal number of independent components for reproducible transcriptomic data analysis[END_REF], which is also available in BIODICA.

Biological annotation of extracted components

BIODICA provides a unique toolbox to help the biological interpretation of the extracted components, combining different annotation and visualization methods which already proved their usefulness (Teschendorff et al., 2007;Kondratova et al. 2019) 1 . Several knowledge-based annotation methods are proposed, such as functional enrichment analysis using ToppFun (Chen et al., 2009) 1 , Gene Set Enrichment Analysis (Subramanian et al., 2005) 1 , or network-based enrichment analysis using known graphs of protein-protein interactions (Kairov et al., 2012) 1 . BIODICA also integrates an insightful visualization tool to project the independent components on comprehensive maps of molecular interactions using NaviCell (Bonnet et al., 2015) 1 .

Studying inter-data sets reproducibility of extracted components

BIODICA provides a tool, based on application of Mutual Nearest Neighbors (MNN), to match the components extracted from several independent omics data sets. Studying the reproducibility of independent components across multiple data sets may help distinguishing biological signals that are specific to a particular disease/data type or technical biases that are specific to particular conditions [START_REF] Biton | Independent component analysis uncovers the landscape of the bladder tumor transcriptome and reveals insights into luminal and basal subtypes[END_REF][START_REF] Cantini | Assessing reproducibility of matrix factorization methods in independent transcriptomes[END_REF].

Implementation

BIODICA comes with a user-friendly Graphical User Interface called BIODICA Navigator, providing non-experienced users a no-code access to all the BIODICA functionalities. It facilitates the communication with biology experts, producing sortable and interactive HTMLbased reports. The interface has been designed and validated in several studies, including a study of Ewing sarcoma at single-cell level [START_REF] Aynaud | Transcriptional programs define intratumoral heterogeneity of ewing sarcoma at single-cell resolution[END_REF].

For more advanced users, the stabilized-ica package can be used as a standalone tool to integrate some of the functionalities of BIODICA into Python analysis pipelines.

Future developments

For now, ICA-based blind deconvolution has been mainly applied to transcriptomics data. However, other omics technologies are now often incorporated into biological research and could also benefit from the ICA analysis proposed by BIODICA. A few studies recently used ICA to deal with DNA methylation data (e.g., [START_REF] Meunier | Dna methylation signatures reveal the diversity of processes remodeling hepatocellular carcinoma methylomes[END_REF]. We plan to foster the application of such methodologies to other omics data by adding new tools to BIODICA to interpret and exploit the independent components, taking into account the specificities of each technology. We also plan to integrate multi-omics analysis tools in BIODICA, building on the recent efforts that have been made for the integration of multiple omics layers [START_REF] Teschendorff | Tensorial blind source separation for improved analysis of multi-omic data[END_REF].

Fig

  Fig. 1. BIODICA workflow

These references are available at https://sysbio-curie.github.io/biodica-environment/docs/references/