Alexander D Rahm
email: alexander.rahm@weizmann.ac.il

Théorie des groupes/Group Theory Algèbre homologique/Homological Algebra Homology and K-theory of the Bianchi groups

We reveal a correspondence between the homological torsion of the Bianchi groups and new geometric invariants, which are effectively computable thanks to their action on hyperbolic space. We use it to explicitly compute their integral group homology and equivariant K-homology. By the Baum/Connes conjecture, which holds for the Bianchi groups, we obtain the K-theory of their reduced C * -algebras in terms of isomorphic images of the computed K-homology. We further find an application to Chen/Ruan orbifold cohomology.

Homologie et K-théorie des groupes de Bianchi Résumé. Nous mettons en évidence une correspondance entre la torsion homologique des groupes de Bianchi et de nouveaux invariants géométriques, calculables grâce à leur action sur l'espace hyperbolique. Nous l'utilisons pour calculer explicitement leur homologie de groupe à coefficients entiers et leur K-homologie équivariante. En conséquence de la conjecture de Baum/Connes, qui est vérifiée pour ces groupes, nous obtenons la K-théorie de leurs C*-algèbres réduites en termes d'images isomorphes de la K-homologie calculée. Nous trouvons d'ailleurs une application à la cohomologie d'orbi-espace de Chen/Ruan.

Version franc ¸aise abrégée

Nous étudions la géométrie d'une certaine classe de groupes arithmétiques (les groupes de Bianchi), à travers une action propre sur un espace contractile. Nous accédons à leur homologie de groupe et leur K-homologie équivariante. En plus de détail, considérons un corps de nombres quadratique imaginaire Q( √ -m), où m est un entier positif ne contenant pas de carré. Soit O -m son anneau d'entiers. Les groupes de Bianchi sont les groupes SL 2 (O -m ). Les groupes de Bianchi peuvent être considérés cruciaux pour l'étude d'une classe plus large de groupes, les groupes Kleiniens, qui ont déjà été étudiés par Henri Poincaré [9]. En fait, chaque groupe Kleinien arithmétique non-cocompact est commensurable avec un groupe de Bianchi [START_REF] Maclachlan | The arithmetic of hyperbolic 3-manifolds[END_REF]. Un éventail d'informations sur les groupes de Bianchi peut être trouvé dans les monographies [START_REF] Elstrodt | Groups acting on hyperbolic space[END_REF][START_REF] Fine | Algebraic theory of the Bianchi groups[END_REF][START_REF] Maclachlan | The arithmetic of hyperbolic 3-manifolds[END_REF]. Ces groupes agissent d'une manière naturelle sur l'espace hyperbolique à trois dimensions, qui est isomorphe à l'espace symétrique qui leur est associé. Le noyau de cette action est le centre {±1} des groupes, ce qui rend utile l'étude du quotient par le centre, PSL Théorème 1.1 (cf. theorem 2.2). La partie ℓ-primaire de la deuxième page de la suite spectrale équivariante convergeante vers l'homologie des groupes de Bianchi dépend, hors de sa ligne inférieure, seulement du type d'homéomorphisme du sous-complexe de ℓ-torsion.

Dans tous les cas d'anneaux principaux non-Euclidiens, les sous-complexes de 2-torsion, respectivement de 3-torsion, sont homéomorphes, ce qui explique les résultats de la proposition 2.1. Derrière le théorème ci-dessus, il y a la correspondance suivante entre les sousgroupes cycliques non-triviaux des stabilisateurs des sommets, et les lignes géodésiques autour desquelles ils effectuent une rotation. Il convient d'appeler ces lignes des axes de rotation.

Théorème 1.2 (cf. theorem 2.3). Soit v un sommet quelconque dans l'espace hyperbolique. L'action de son stabilisateur sur l'ensemble des axes de rotation passant par v, induite par l'action du groupe de Bianchi, est équivalente à l'action par conjugaison de ce stabilisateur sur ses sous-groupes cycliques non-triviaux.

Une étude cas par cas [START_REF] Co | homologies and K-theory of Bianchi groups using computational geometric models[END_REF] pour tous les six types de sous-groupes finis dans les groupes Bianchi nous permet de déduire du théorème 1.2, le lemme 2.4 utilisé pour obtenir le théorème 1.1. Des exemples pour le théorème 1.1 sont donnés pour la 3-torsion homologique de trente-six groupes de Bianchi sur le tableau 1. K -théorie. Avec l'information sur l'action des groupes de Bianchi, que nous obtenons en nous servant des énoncés et méthodes décrits ci-dessus, nous pouvons calculer l'homologie de Bredon de groupes de Bianchi, et en déduire leur K-homologie équivariante. Des résultats sont présentés dans le théorème 2.5. En conséquence de la conjecture de Baum/Connes, qui est vérifiée pour les groupes de Bianchi [START_REF] Julg | Operator K-theory for the group[END_REF], nous obtenons la K-théorie des C * -algèbres réduites des groupes de Bianchi comme images isomorphes.

Introduction

We study the geometry of a certain class of arithmetic groups (the Bianchi groups) by means of a proper action on a contractible space. This helps to determine their group homology and their equivariant K-homology. In more detail, we denote by Q( √ -m), with m a square-free positive integer, an imaginary quadratic number field, and by O -m its ring of integers. The Bianchi groups are the groups SL 2 (O -m ). The Bianchi groups may be considered as a key to the study of a larger class of groups, the Kleinian groups, which dates back to work of Henri Poincaré [9]. In fact, each non-cocompact arithmetic Kleinian group is commensurable with some Bianchi group [START_REF] Maclachlan | The arithmetic of hyperbolic 3-manifolds[END_REF]. A wealth of information on the Bianchi groups can be found in the monographs [START_REF] Elstrodt | Groups acting on hyperbolic space[END_REF][START_REF] Fine | Algebraic theory of the Bianchi groups[END_REF][START_REF] Maclachlan | The arithmetic of hyperbolic 3-manifolds[END_REF]. These groups act in a natural way on hyperbolic three-space, which is isomorphic to the symmetric space SL 2 (C)/SU 2 associated to them. The kernel of this action is the centre {±1} of the groups. Thus it is useful to study the quotient of a Bianchi group by its centre, namely PSL 2 (O -m ), which we also call a Bianchi group. In 1892, Luigi Bianchi [START_REF] Bianchi | Sui gruppi di sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginarî[END_REF] computed fundamental domains for this action when m = 1, 2, 3, 5, 6, 7, 10, 11, 13, 15 and 19. Such a fundamental domain has the shape of a hyperbolic polyhedron (up to a missing vertex at certain cusps, which represent the ideal classes of O -m ), so we will call it the Bianchi fundamental polyhedron.

The computation of the Bianchi fundamental polyhedron has been implemented for all Bianchi groups [START_REF] Rahm | Open source program (GNU general public license)[END_REF]. The images under SL 2 (O -m ) of the facets of this polyhedron equip hyperbolic three-space with a cell structure. In order to view clearly the local geometry, we pass to the refined cell complex, which we obtain by subdiving this cell structure until the cell stabilisers fix the cells pointwise. We will see how to exploit this cell complex in different ways, in order to see different aspects of the geometry of these groups.

Group homology. An essential invariant of groups is their homology (defined for instance in [START_REF] Brown | Cohomology of groups[END_REF]). We can compute it for the Bianchi groups using the refined cell complex and the equivariant Leray/Serre spectral sequence which starts from the homology of the stabilisers of representatives of the cells, and converges to the group homology of the Bianchi groups. We will now state the results for simple integer coefficients in the cases m = 19, 43, 67 and 163, which are the non-Euclidean principal ideal domain cases. In contrast to these, the Euclidean principal ideal domain cases are already known from [START_REF] Schwermer | The integral homology of SL2 and PSL2 of Euclidean imaginary quadratic integers[END_REF].

For some results in class number 2, see [START_REF] Rahm | The integral homology of PSL2 of imaginary quadratic integers with non-trivial class group[END_REF]. The virtual cohomological dimension of the Bianchi groups is 2. In degrees strictly above 2, we express their homology in terms of the following Poincaré series at the primes ℓ = 2 and ℓ = 3:

P ℓ m (t) := ∞ q = 3 dim F ℓ H q PSL 2 O -m ; Z/ℓZ t q .
These two primes are the only numbers which occur as orders of non-trivial finite elements of PSL 2 (O -m ). So it has been shown [START_REF] Co | homologies and K-theory of Bianchi groups using computational geometric models[END_REF] that the integral homology of these groups is, in all the mentioned degrees, a direct sum of copies of Z/2Z and Z/3Z.

Proposition 2.1. The integral homology of PSL 2 (O -m ), for m ∈ {19, 43, 67, 163}, is of

isomorphism type H q (PSL 2 (O -m ); Z) ∼ = Z β 1 -1 ⊕ Z/4Z ⊕ Z/2Z ⊕ Z/3Z, q = 2, Z β 1 , q = 1,
where m 19 43 67 163

β 1 1 2 3 7
gives the Betti number β 1 , and is in all higher degrees a direct sum of copies of Z/2Z and Z/3Z, with the number of copies specified by the Poincaré series

P 2 m (t) = -t 3 (t 3 -2t 2 +2t-3) (t-1) 2 (t 2 +t+1) and P 3 m (t) = -t 3 (t 2 -t+2) (t-1)(t 2 +1) .
We remark that in these four cases, the torsion in the integral homology of PSL 2 (O -m ) is of the same isomorphism type. To understand this, we consider, for a prime ℓ, the subcomplex of the orbit space consisting of the cells with elements of order ℓ in their stabiliser. We call it the ℓ-torsion subcomplex. The following statement on how its homeomorphism type determines the equivariant spectral sequence is proven by the reduction of the torsion subcomplex carried out in [START_REF] Co | homologies and K-theory of Bianchi groups using computational geometric models[END_REF]. This technique uses lemma 2.4 to determine the possible type of stabiliser of a vertex v with exactly two adjacent edges which have ℓ-torsion in their stabilisers. Then these two edges, together with v, are replaced by a single edge; and theorem 2.3 as well as some homological information about the finite groups in question are used to check that the induced morphisms on homology produce the same terms on the second page of the equivariant spectral sequence as before the replacement.

Theorem 2.2. The ℓ-primary part of the second page of the equivariant spectral sequence converging to the integral homology of PSL 2 (O -m ) depends outside the bottom row only on the homeomorphism type of the ℓ-torsion subcomplex.

Examples for this theorem are given for the prime ℓ = 3 and thirty-six Bianchi groups in figure 1 (for ℓ = 2, see [START_REF] Co | homologies and K-theory of Bianchi groups using computational geometric models[END_REF]). In all the non-Euclidean principal ideal domain cases, the 2-torsion, and respectively 3-torsion subcomplexes are homeomorphic, which explains the results in proposition 2.1. Underlying theorem 2.2, there is the following correspondence between the non-trivial cyclic subgroups of the vertex stabilisers and the geodesic lines around which they effect a rotation, and which we shall call rotation axes. 39 [START_REF] Co | homologies and K-theory of Bianchi groups using computational geometric models[END_REF]). For any vertex v in hyperbolic space, the action of its stabiliser on the set of rotation axes passing through v, induced by the action of the Bianchi group, is equivalent to the conjugation action of this stabiliser on its non-trivial cyclic subgroups.

-2t 3 t-1 + -t 3 (t 2 -t+2) (t-1)(t 2 +1) Theorem 2.3 ( [
A case by case study [START_REF] Co | homologies and K-theory of Bianchi groups using computational geometric models[END_REF] for all the six types of finite sub-groups in the Bianchi groups allows us to deduce from theorem 2.3 the following lemma, which is useful in order to obtain theorem 2.2. Lemma 2.4. Let v be a vertex in the refined cell complex. Then the number n of orbits of edges in the refined cell complex adjacent to v, with stabiliser in PSL 2 (O -m ) isomorphic to Z/ℓZ, is given as follows for ℓ = 2 and ℓ = 3.

Isomorphism type of the vertex stabiliser

{1} Z/2Z Z/3Z Z/2Z × Z/2Z S 3 A 4 n for ℓ = 2 0 2 0 3 2 1 n for ℓ = 3 0 0 2 0 1 2.
Here we have written S 3 for the symmetric group on three letters and A 4 for the alternating group on four letters. Note that we obtain the same table for SL 2 (O -m ) after replacing the edge and vertex stabiliser types by their pre-images under the projection SL 2 (O -m ) → PSL 2 (O -m ), which are respectively: Z/2Z, Z/4Z, Z/6Z, the 8-elements quaternion group, the 12-elements binary dihedral group and the binary tetrahedral group. K -theory. With the above information about the action of the Bianchi groups, we can further compute the Bredon homology of the Bianchi groups, from which we can deduce their equivariant K-homology. The results of the computations [START_REF] Co | homologies and K-theory of Bianchi groups using computational geometric models[END_REF] are the following.

K Γ 0 (EΓ) Z 6 Z 5 ⊕ Z/2Z Z 5 ⊕ Z/2Z Z 3 Z 4 ⊕ Z/2Z Z β1-1 ⊕ Z 3 ⊕ Z/2Z K Γ 1 (EΓ) Z Z 3 0 Z 3 Z 3 Z ⊕ Z β1 .
The remainder of the equivariant K-homology of Γ is given by Bott 2-periodicity. By the Baum/Connes conjecture, which holds for the Bianchi groups [START_REF] Julg | Operator K-theory for the group[END_REF], we obtain the K-theory of the reduced C * -algebras of the Bianchi groups as isomorphic images.

Complex orbifolds. The information we have concerning the action of the Bianchi groups on real hyperbolic three-space provides explicit orbifold structures. These orbifolds serve as models in Cosmology [START_REF] Aurich | Numerical computation of Maass waveforms and an application to cosmology[END_REF]. We can complexify these orbifolds. In [START_REF] Co | homologies and K-theory of Bianchi groups using computational geometric models[END_REF], the product structure on their Chen/Ruan orbifold cohomology has been determined for all Bianchi groups; and an algorithm has been given to compute the underlying vector space structure. This is a step towards checking Ruan's cohomological crepant resolution conjecture [START_REF] Chen | A new cohomology theory of orbifold[END_REF], which is of importance in Mathematical Physics and is completely open in 3 complex dimensions outside the global quotient case, hence these orbifolds are interesting test cases.

Figure

  Figure 1. Results for the 3-torsion homology, expressed in P ℓ m (t)

Theorem 2 . 5 .

 25 Let β 1 be the Betti number specified in proposition 2.1. For O -m principal, the equivariant K-homology of Γ := PSL 2 (O -m ) is isomorphic to

  2 (O -m ), que nous appellerons également un groupe de Bianchi. En 1892, Luigi Bianchi[START_REF] Bianchi | Sui gruppi di sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginarî[END_REF] a calculé des domaines fondamentaux pour cette action pour quelques uns de ces groupes. Un tel domaine fondamental est de la forme d'un polyèdre hyperbolique (à quelques sommets manquants près), et nous l'appellerons le polyèdre fondamental de Bianchi. Le calcul du polyèdre fondamental de Bianchi a été implémenté sur ordinateur pour tous les groupes de Bianchi[START_REF] Rahm | Open source program (GNU general public license)[END_REF]. Les images sous SL 2 (O -m ) des faces de ce polyèdre munissent l'espace hyperbolique d'une structure cellulaire. Pour mieux observer la géométrie locale, nous passons au complexe cellulaire raffiné, que nous obtenons en subdivisant cette structure cellulaire jusqu'à ce que les stabilisateurs dans SL

	d'un ensemble représentatif de
	cellules, et qui converge vers l'homologie du groupe de Bianchi. Nous précisons dans la
	proposition 2.1, l'homologie entière de PSL 2 (O -m ) dans les cas m = 19, 43, 67 et 163, qui
	constituent tous les cas d'anneaux principaux non-Euclidiens. Les cas d'anneaux Eucli-
	diens sont déjà connus de [13]. Des résultats récents pour des cas de groupe de classe
	d'idéaux non-trivial se trouvent dans [12]. Nous remarquons que dans les quatre cas
	d'anneaux principaux non-Euclidiens, la torsion dans l'homologie de PSL 2 (O -m ), est du
	même type d'isomorphisme. Pour comprendre ceci, nous considérons, pour un nombre pre-
	mier ℓ, le sous-complexe de l'espace d'orbites, composé des cellules ayant des stabilisateurs
	qui contiennent des éléments d'ordre ℓ. Nous l'appellerons le sous-complexe de ℓ-torsion.
	L'énoncé suivant traite la manière dans laquelle son type d'homéomorphisme détermine
	la suite spectrale équivariante. Il est démontré par la réduction des sous-complexes de
	torsion effectuée dans [11]. Cette technique utilise le lemme 2.4 pour cerner le type de sta-
	bilisateur d'un sommet v qui est adjacent à exactement deux arêtes dont les stabilisateurs
	admettent de la

2 (O -m ) fixent les cellules point par point. Nous allons exploiter ce complexe cellulaire de différentes manières, afin de cerner des aspects différents de la géométrie de ces groupes.

Homologie de groupes. Un invariant essentiel des groupes est leur homologie (définie par exemple dans

[START_REF] Brown | Cohomology of groups[END_REF]

). Nous pouvons la calculer pour les groupes de Bianchi en nous servant du complexe cellulaire raffiné. À ce fin, nous utilisons la suite spectrale équivariante de Leray/Serre qui part de l'homologie des stabilisateurs ℓ-torsion. Ensuite, ces deux arêtes et v sont remplacés par une seule arête. Le théorème 1.2 et quelques informations homologiques sur les groupes finis en question sont utilisés pour vérifier que les morphismes induits en homologie produisent les mêmes termes sur la deuxième page de la suite spectrale équivariante qu'avant le remplacement.

  1. Results for the 3-torsion homology, expressed in P ℓ m (t)

	m specifying the Bianchi group	3-torsion subcomplex, homeomorphism type	P 3 m (t)
	2, 5, 6, 10, 11, 15, 22, 29, 34, 35, 46, 51, 58, 87, 95, 115, 123, 155,		-2t 3 t-1
	159, 187, 191, 235, 267
	7, 19, 43, 67, 139, 151, 163		-t 3 (t 2 -t+2) (t-1)(t 2 +1)
	13, 37, 91, 403, 427	2	-t 3 (t 2 -t+2) (t-1)(t 2 +1)

Acknowledgements. The author thanks Stephen Gelbart for support and encouragement.

He thanks Philippe Elbaz-Vincent, Louis Funar and Anthony Joseph for a careful lecture of the manuscript.

And he would like to thank again all the people acknowledged in [11].


Mathieu Tillier

In this article, I examine the appearance of Muslim women before the judge during the Abbasid period (132-334/750-945), both in theory and practice. The cases involving women found in law books suggest that they came freely to the court, especially for familial or marital purposes, and that the judges employed some women as court auxiliaries. However, a comparison of judicial manuals and the biographical literature shows that a woman's appearance before the judge could create a social disturbance and that not all women were allowed to appear in court. I argue that the social distinction between those who could leave their houses -and thus come before the judge -and those who could not correlated with the social hierarchy.

Based on Islamic court registers (sijillāt), several specialists in Ottoman history have analyzed the social and gender distribution of the litigants who appeared before the qāḍī, the Islamic judge. Some studies show that women represented a non-negligible percentage of the litigants -often more than ten percent. 1 Unfortunately, because of the lack of documents for the first centuries of Islam, researchers cannot reach precise results. 2 Judicial practices can be studied only through literary sources, like biographical dictionaries and law compendia.

I wish to thank David Powers, Christopher Melchert and Karen Bauer for their comments on a previous draft of this article. 1 One of the first quantitative studies of women in Ottoman courts was undertaken by R. C. Jennings, "Women in Early 17 th Century Ottoman Judicial Records -The Sharia Court of Anatolian Kayseri," JESHO, 18 (1975), 53-114. See also Y. J. Seng, "Invisible Women: Residents of Early Sixteenth-Century Istanbul," in G. R. G.

Hambly, Women in the Medieval Islamic World. Power, Patronage, and Piety (London, MacMillan, 1998), 241-68. For a review of studies on women under the Ottomans, see I. Agmon, "Women's History and Ottoman Sharia Court Records," Hawwa, 2 (2004), 175 ff. 2 The main exception is represented by the documents of the Geniza, which allowed S. D. Goitein to analyze the phenomenon of Jewish women appearing in the rabbinical court of Fusṭāṭ, mostly during the 11 th and 12 th centuries. S. D. Goitein, A Mediterranean Society. The Jewish Communities of the Arab World as Portrayed in the Documents of the Cairo Geniza (Berkeley-Los Angeles-London: University of California Press, 1967-93), 3:332-6.

Nevertheless, the frequent appearance of women in the available sources suggests that the situation was not very different in early Islam.

Muslim women reportedly were confined to private space, while public space was reserved for men. The qāḍī's tribunal, usually held in the chief mosque of the city, served as public space par excellence: anyone could attend it, enter a complaint, or simply hear others' complaints. It is necessary to investigate the sources in an effort to reconstruct the actual position of women at the qāḍī's court. An exploration of the legal literature reveals what kinds of cases involving women were expected to be brought before the qāḍī. I rely especially on the Iraqi al-Khaṣṣāf (d. 261/874), a Ḥanafī jurist who, in the middle of the ninth century, wrote an important adab al-qāḍī manual, and on its tenth-century commentary by al-Jaṣṣāṣ (d. 370/980). In fact, these two works can hardly be separated: al-Khaṣṣāf's original text is included in al-Jaṣṣāṣ's commentary. I will argue that, despite the absence of legal restrictions on the appearance of women before a judge, the prevailing social norm discouraged some women from appearing in court, and that gender issues must be viewed within their specific social context.

Legal cases involving Muslim women

Women as litigants

Women appear frequently in law books about family matters. Ḥanafī jurists discuss several types of cases in which women might be involved in a lawsuit.

The first category deals with personal status, that is, cases relating to marriage and divorce. A man could allege, for example, that a woman was his wife. 3 As al-Khaṣṣāf explains, a woman whose husband went on a long journey and did not return may have remarried; then the first husband might come back and claim that he is her legal husband. 4 Other cases come to mind: David Powers mentions the case of a woman in the fifteenthcentury Maghrib who fled from her husband and remarried in a neighboring village. 5 A woman also could allege that a man was her husband. 6 Women were more often involved in 3 divorce lawsuits. A woman who wanted to be separated from her husband might claim that he had repudiated her three times, 7 or she might claim that her marriage was null and void. 8 A husband could ask the judge to nullify his marriage if his wife suffered from a crippling defect like madness or severe illness (e.g. baraṣ, leprosy). 9

The second category relates to material claims on a husband. Several kinds of cases can be singled out. (1) A woman might complain about the dower (mahr): although the claimant was usually the wife's father, the judge could summon the wife to ask her about her father's claim. 10 (2) She could complain about the nafaqa or maintenance allowance provided by the husband for his wife's daily needs. Women filed complaints against husbands who did not provide them with enough money or equipment, 11 or who had to go on a journey of several months, and they asked the qāḍī to appoint a guarantor (kafīl) who would be responsible for paying their nafaqa. 12 Alternatively, a woman could ask her brother or another relative to pay her nafaqa. If her husband disappeared, she could go to the qāḍī and ask him to appoint an agent (wakīl) to manage the husband's property and pay her the nafaqa. (3) A woman could ask for material commodity, like new dresses, 13 or ask for the employment of a cook. 14 (4) Finally, some divorced women appeared before the judge to receive permission to take their children away from their father to another city. 15 Another category is sexual intercourse. A wife might accuse her husband of engaging in illicit sexual relations with another woman (zinā). 16 The jurists commonly dealt with the case in which one of the spouses could not or did not fulfill his or her sexual obligations. For example, a woman complained that her husband had sworn not to touch her and respected this oath for four mouths (īlā). In this instance, the qāḍī could impose a divorce between them. 17 Sexual incapacity was also the subject of lawsuits: divorce was automatic if the wife was 4 physically incapable of sexual intercourse. 18 More frequently, however, jurists deal with the incapacity of men: according to Ḥanafī law, the qāḍī should pronounce the divorce immediately if a husband appears to be castrated. 19 If he was sexually impotent, the qāḍī instituted a long procedure (sometimes lasting more than a year) to determine whether or not he would regain his sexual potency. If the wife proved that her husband was impotent, she was given the choice of remaining married or divorcing him. 20 Al-Jaṣṣāṣ explains that sexual intercourse is a wife's right. 21 Although in theory Islamic law restricts a woman's capacity to ask for divorce, the cases relating to sexuality suggest that some women used judicial strategies to obtain it. This hypothesis is confirmed by al-Subkī, who relates how a woman obtained a divorce after complaining that her husband's penis was too large (shakat ilay-hi imra'a kubr ālat zawji-hā). [START_REF] Al-Khaṣṣāf | Ṭabaqāt[END_REF] An ill-treated woman might file a complaint against her husband. In Ḥanafī law, a woman cannot obtain a divorce if her husband harms her. [START_REF] Marìn | Disciplining Wives: A Historical Reading of Qur'ân 4:34[END_REF] However, according to al-Jaṣṣāṣ, the qāḍī must prevent her husband from abusing her. 24 He can order a man to let his wife live in the house of respectable neighbors (ṣāliḥīn) who will inform the qāḍī about the husband's behavior with his wife. If it appears that he is in fact maltreating her, the qāḍī will reprimand him or sentence him to physical punishment. 25 According to Maribel Fierro, such a procedure, found also in Maghribī Mālikī law, may have originated in local Berber customs. [START_REF] Fierro | Ill-Treated Women Seeking Divorce: the Qur'ānic two Arbiters and Judicial Practice among the Malikis in al-Andalus and North Africa[END_REF] The mention of this procedure in early Ḥanafī law, however, suggests that this practice originated in the east or, at least, was widespread in the Islamic empire from an early period.

Finally, women were involved in lawsuits about inheritance, 27 kinship (for example if a woman claims that a foundling is her brother and asks the judge to entrust her with the child), 28 or criminal matters (murder or illegal sexual intercourse). 29 Al-Khaṣṣāf relates the story of a young woman who, upon reaching puberty, asked the qāḍī for permission to choose which of her divorced parents she would live with. 30 On the whole, most of the cases involving women in al-Khaṣṣāf's Adab al-qāḍī are related to family law and personal status. Al-Khaṣṣāf refers to women when their gender is essential to the substance of the case. However, he does not cite women involved in lawsuits where their gender is immaterial, as commercial litigations. Al-Khaṣṣāf's Adab al-qāḍī, thus, reflects only women's involvement in typically female cases, and leaves in the dark their possible participation in other types of lawsuits.

Women as auxiliaries in the qāḍī's court

The qāḍī's court was not reserved exclusively for men. Many familial matters compelled the judge to use female auxiliaries. He needed female judicial experts to investigate women's virginity or pregnancy, their sexual defects, and matters relating to hidden parts of the female body (like abortion, birth and suckling); this type of expertise was commonly recommended in lawsuits about female slaves, e.g., when a buyer accuses the seller of fraud. 31 Such experts would determine if a young girl was physically capable of engaging in sexual relations before she got married, 32 or would examine young wives who reportedly were incapable of engaging in sexual relations. 33 According to al-Khaṣṣāf and al-Jaṣṣāṣ, several women might act as experts for the same case.

The status of a female expert's word and its value before the judge is striking. Usually, male Muslim experts are recognized as witnesses, and their word is considered as a testimony 27 (šahāda) or information (khabar). 34 According to al-Jaṣṣāṣ, however, a woman's word cannot cancel a contract, and her statement must be treated as a claim (daʿwā). For this reason, a litigant who disagrees with her may swear that she is wrong. The judge considers him innocent and the woman's word is rejected. 35 According to some Ḥanafī jurists, however, virginity, pregnancy and menstruation can be established only by female experts, and their word must always be recognized as valid in these matters. 36 Al-Sarakhsī (d. 483/1090) tends to recognize women's expertise as authentic testimony. He defines a female's report concerning the intimate parts of another female as information (khabar), similar in nature to a report going back to a single authority (al-āḥād); but it is also a testimony (shahāda), as she must be a free Muslim and use the formulary "I testify" (ashhadu) in the judicial proceeding.

Moreover, her word is binding, as the qāḍī bases his judgment on it. 37 In addition, the qāḍī employed women as auxiliaries when a defendant (male or female) refused to appear in court and hid at home. He had to send two trustworthy men with several women and servants (khadam): the male auxiliaries would encircle his house and prevent anyone from leaving, while the women and the servants would enter the house, drive away its female inhabitants and look for the defendant. If they found him, they compelled him to appear at the qāḍī's court. 38 Thus, women were needed to enter the private space of the house, which had to remain hidden from the eyes of any man outside the inner circle of the family. 39 Women also investigated female witnesses in an effort to determine whether they were trustworthy and honorable. According to al-Khaṣṣāf, only women who are used to mixing with people in the public sphere can conduct this type of investigation. 40 34 J.-P. Van Staëvel, "Savoir voir et le faire savoir : l'expertise judiciaire en matière de construction, d'après un auteur tunisois du 8 e /XIV e siècle," Annales Islamologiques, 35 (2001), 654. 35 Al-Jaṣṣāṣ, in al-Khaṣṣāf, Adab al-qāḍī, 558. 36 Ibid., 560. Cf. M. Fadel, "Two Women, One Man: Knowledge, Power and Gender in Medieval Sunni Legal Thought," IJMES, 29 (1997), 195. 37 R. Shaham, "Women as Expert Witnesses," 48-49. 38 Al-Khaṣṣāf, Adab al-qāḍī, 252. See F.J. Ziadeh, "Compelling Defendant's Appearance at Court in Islamic Law," Islamic Law and Society, 3 (1996), 310. 39 On the private status of the house, which could not be entered legally without permission, see E. Alshech, ""Do Not Enter Houses Other Than Your Own": the Evolution of the Notion of a Private Domestic Sphere in Early Sunnī Islamic Thought," Islamic Law and Society, 11 (2004), 295ff. This procedure suggests that al-Khaṣṣāf still held the interventionist view of the Ḥanafī Abū Yūsuf (d. 182/798) who, unlike the jurists of other madhhabs, allowed entering the house without the permission of its inhabitants. Ibid., 298. 40 Ibid., 314, 422; see also al-Jaṣṣāṣ, in al-Khaṣṣāf, Adab al-qāḍī, 314.

Finally, a woman might serve as trustworthy agent (amīn) to look after another woman, usually a slave, who claimed to be free, or when two litigants claimed to be her legal owner.

The qāḍī entrusted the slave girl to a female agent in order to prevent her from having sexual intercourse with one of the two litigants until the resolution of the case. 41 This procedure might also involve a free woman who claimed that she had been repudiated three times, in order to take her away from her husband before the qāḍī formally pronounced the divorce. 42 These cases show that the judiciary needed female auxiliaries who might intervene in all situations in which there was a violation of the ḥaram -the domestic, private or female space -forbidden to any man outside the inner circle of the family. In theory, Ḥanafīs even recognized a woman's right to be a qāḍī. 43 Indeed, in the first Ḥanafī adab al-qāḍī books, women were to a large extent integrated into the court. They appeared in this literature both as plaintiffs and as judicial auxiliaries.

Looking for the archetypal female litigant

The fact that fiqh books devote considerable attention to women appearing in court leads us to wonder whether religious models contributed to the shaping of legal theory. Few if any explicit injunctions calling on women to appear in the qāḍī's court are found in the fiqh literature. Although the jurists do not mention any explicit model, implicit models may 41 Al-Khaṣṣāf, Adab al-qāḍī, 377. 42 Al-Jaṣṣāṣ, in al-Khaṣṣāf, Adab al-qāḍī, 377. Cf. M. Fierro, "Ill-Treated Women," 338. nevertheless have shaped mentalities. We will concentrate here on two examples taken from the Qur'ān and the Sīra.

First, the Qur'ān exhorts the Muslim community to resort to justice when a married couple is confronted with discord: "If you fear dissension between a married couple, send forth an arbiter from his family and an arbiter from her family. If they desire reconciliation, God will bring them together. God is All-Knowing, All-Discerning." (Q. 4:35) 44 According to Maribel Fierro, this verse "was never understood to refer to every case of marital discord, given the husband's right to repudiate his wife that is clearly established in other Qur'ānic verses." 45 The exhortation was understood in the context of Q. 4:34, which allows the husband to take action against his wife if she disobeys him: "if after having been admonished, banished and beaten, the rebellious wife still does not obey the husband, then the two arbiters should be appointed. Presumably, it is the husband who asks for arbitration because, for whatever reason, he has no desire to divorce his rebellious wife." 46 However, jurists also understood this rule as "a mechanism for a wife to seek a judicial divorce on the ground of harm." 47 Q. 4:35 was revealed in an Arabian context where there were no state institutions such as the office of qāḍī. The Qur'ān mentions two ḥakams or arbiters chosen by the litigants, who determine whether they should stay married or divorce. 48 This understanding of Q. 4:35 is reported by al-Ṭabarī (d. 310/923). At the end of the 3 rd /9 th century, however, the sulṭān or his delegate was probably the main supervisor of the procedure: the litigants had to complain before him, and he had to choose the two ḥakams if he lacked a clear understanding of the situation of the spouses. Here, the ḥakams act as agents (wakīls) who negotiate an agreement between the litigants. 49 If the spouses refuse to invest them with the power of decision, they intervene as experts or witnesses before the sulṭān to determine which of the spouses is at fault. 50 As Fierro has shown, when this procedure was enforced in practice, it was usually the 44 Translation by Tarif Khalidi, The Qur'an (London: Penguin Classics), 2008, 67. 45 M. Fierro, "Ill-Treated Women," 325. 46 Ibid., 325. The commentators explain that the "rebellion" of the wife may be caused by the conduct of her husband, or by too small a nafaqa. See references infra. 47 Ibid., 326. 48 See Sufyān al-Thawrī (d. 161/778?), Tafsīr al-Thawrī (Beirut: Dār al-kutub al-ʿilmiyya, 1403 H.), 94-95. 49 This corresponds to the Ḥanafī doctrine, which considers the ḥakamān to be mere agents of the spouses. Therefore, their decision is not binding. According to the Mālikīs, however, the arbiters are entitled to impose a divorce. Fierro, "Ill-Treated Women," 329, 331. qāḍī who appointed the two ḥakams: 51 this suggests that the litigants had already brought suit before the qāḍī. Whatever the modes of application of this procedure may have been in time and space, the Qur'ān itself was regarded as encouraging both men and women to seek justice.

The commentators explain that other verses are related to a judicial context. According to al-Ṭabarī, Q. 58:1ff. were revealed when an Anṣārī woman (named Khawla or Khuwayla) 52 came to the Prophet and asked him about her marital situation after her husband told her that "her back was like his mother's back" (a case of ẓihār). In one narrative, al-Ṭabarī compares the situation to that of a trial in which the Prophet was supposed to act as a judge and to deliver a judgment (fa-yaqḍī fiyya wa-fīka amra-hu). 53 In addition to this Anṣārī woman, the example of some early Muslim women may have encouraged women to appear in court. It is well known that Arab women held an important political and religious position in early Islam, especially when they belonged to noble families. 54 Of the many women who played an important role in early Islamic politics, three of them may have influenced women's position in the legal sphere.

(1) Khadīja, Muḥammad's first wife, is remembered as an independent woman who managed her business affairs by herself. Although her model does not seem to have influenced women's legal status in Islam, her independence may have encouraged women to undertake judicial actions before the qāḍī, especially in case of business litigations.

(2) ʿA/'isha, the youngest of Muḥammad's wives, is an ambiguous character: she appears simultaneously as the archetypal child-wife "that God offered to the pious mature man" 55 and as the woman who opposed ʿUthmān and revolted against ʿAlī, the fourth caliph. During the first years of her marriage to the Prophet, she was accused of illegal sexual intercourse (the so-called "lie case", ifk). 56 Strictly speaking, this was not a judicial case, as there was no 51 M. Fierro, "Ill-Treated Women," 330. 52 On the disagreements about her nasab, see al-Ṭabarī, Tafsīr, 28:1. Cf. Muqātil b. Sulaymān, Tafsīr, 3:329. 53 Al-Ṭabarī, Tafsīr, 28:5. Earlier tafsīrs, however, do not treat the mujādila as a plaintiff in a trial, but rather as a woman seeking the opinion of the Prophet. See Muqātil b. Sulaymān, Tafsīr, 3:329. arbiter or judge. Nevertheless, this episode serves as an example -at least for Sunnis -of a high-ranking woman involved in a legal matter during the Prophet Muḥammad's life. 57 (3) Muḥammad's daughter Fāṭima was involved in a legal case: after her father's death, she publicly contested the first caliph's refusal to award her a share of her father's estate. [START_REF] Lammens | Fāṭima et les filles de Mahomet[END_REF] These three women contributed to the model of the ideal Muslim female. [START_REF] Spellberg | [END_REF] Reports about these exemplary women may have encouraged them to appear in the qāḍī's court despite the limited public role enjoyed by women in Middle Eastern societies. 60

Looking for the practice

Biographical sources and their data

Although law books put forth rules for cases in which women might appear in the qāḍī's court, legal doctrine does not necessarily coincide with practice. The jurists talk about women only when dealing with female-related issues like dower, nafaqa or divorce. Women were deeply involved in economic life, working in crafts (mainly as weavers and spinsters), trade, and services to other women (hairdressers, midwives, nurses). [START_REF] Shatzmiller | Labour in the Medieval Islamic World[END_REF] Although they were supposed to work at home or in other women's houses, [START_REF] Shatzmiller | [END_REF] far from the public eye, they probably became involved in disputes with sellers, clients and partners. 63 As disputes of this nature are not gender-related, the jurists do not explicitly mention women in connection with these "asexual" cases. Therefore, we cannot know whether they were expected to appear in the court or not. If we want to understand whether women did in fact appear before the qāḍī, we have to look for other sources. Biographical dictionaries contain narratives that may reflect social practice, especially those dedicated to qāḍīs such as the books of Wakīʿ (d. 306/918), al-Kindī (d. 350/961) and Ibn Ḥajar al-ʿAsqalānī (d. 852/1449). 64 These dictionaries, which contain many references to lawsuits, have usually been regarded as reliable historical sources with regard to judicial practices. 65 The following table presents results of several recent studies devoted to women. 81 Fatwā literature, usually regarded as representative of actual practice, also corroborates the important position of women at the qāḍī's court. 82 The scarcity of references to non-matrimonial cases is striking. Only four accounts deal with inheritance, contracts or criminal law (one case of qadhf or slanderous accusation of fornication). 83 The biographical dictionaries dedicated to qāḍīs are generally interested in a much larger panel of legal cases. Wakīʿ and al-Kindī show a strong concern for commercial matters. Altogether, theirs books contain about 215 reports dealing with commercial claims or debts. Therefore, one would expect to find a higher number of narratives referring to the involvement of women in commercial litigation. Two complementary hypotheses may be formulated.

(1) Biographies of qāḍīs do not portray actual judicial practice, but rather illustrate theoretical points developed in the legal literature. This would explain why most of the cases involving women quoted in the biographical literature fit the typically female cases found in the law books. Because the narratives were intended to exemplify the theoretical thinking of the fuqahā', there was no need to mention women appearing before the judge in less gender-specific cases. If this hypothesis is confirmed, it would mean that the biographical literature complements fiqh books and is part of a wider theoretical literature. The mention of women's cases in biographical literature does not provide a representative picture of how often they frequented the court.

(2) Despite their strong participation in economic life, women appealed to the qāḍī less frequently than men. It thus remains to examine the extent to which women had access to the qāḍī's court.

Female plaintiffs and social hierarchy

Although it is now acknowledged that, in the pre-modern period, women were present in public space, 84 the cases brought before the judge could give rise to social disturbance. Many cases involving women were related to family or sexual conflicts, private matters that were made public before the court which usually was held in the chief mosque. 85 Anyone could attend the qāḍī's court and hear the litigants. According to the Ḥanafīs, women appearing as litigants or witnesses had to unveil to establish their identities. 86 This put the woman in an awkward position before her relatives and the community. A trial that took place in Baṣra in the Umayyad period revealed that a woman had been seduced by a young man, who was present in court; her husband was so ashamed that he had no choice but to repudiate her. 87 In 286/899 in Rayy, a woman appeared before the qāḍī Mūsā b. Isḥāq al-Khaṭmī, together with her guardian (walī), who claimed that her husband had not paid the dower (mahr). When the qāḍī asked the witnesses to examine the woman without her veil, her husband immediately acknowledged his debt to protect her (and probably himself) from public shame. The wife was so grateful to her husband that she renounced the mahr and offered it to him. 88 The fact that judicial procedure broke the social norm could have negative effects on the qāḍī's duty to "give everyone his due" . 89 Al-Khaṣṣāf dedicates several pages to the specific procedures relating to female plaintiffs.

First, the qāḍī must hear them on a particular day. Whereas men had to write a petition (ruqʿa) with their names and their complaints (the qāḍī called them according to the order in which he collected the petitions), women did not have to do so, to protect their reputation (satr). Al-Khaṣṣāf advises the judge not to call a woman by her name when summoning her. 90 The Ḥanafī jurist clearly understood that unveiling a woman in court contravened the social norm. For the sake of justice, not only did the judge and the witnesses have to recognize a female plaintiff, but also she had to unveil before the court secretary who was writing the minutes (maḥḍar) of the lawsuit. Al-Khaṣṣāf advises the qāḍī to look at her and to describe her face to the scribe himself. The qāḍī had to see her in any case: the less she was seen by auxiliaries, the better for her honor. 91 According to al-Jaṣṣāṣ, the judge has to protect the 85 See M. Tillier, "Un espace judiciaire entre public et privé," Annales Islamologiques, 38 (2004), 511.

86 See Wakīʿ, Akhbār al-quḍāt, 2:44-45. The Mālikī madhhab permitted women to appear with their faces veiled, on the condition that two witnesses could confirm their identity. See Shatzmiller, Labour in the Medieval Islamic World, 363.

87 Wakīʿ, Akhbār al-quḍāt, 1:357-58. reputation of young and attractive women. If they are old and ugly, however, it is not scandalous that everyone see their faces. 92 Because the tension between exhibition in court and the social ideal of privacy could not be solved in a satisfying way, it is unlikely that all women did go to the qāḍī's court. Khālid b. Ṭalīq, a Baṣran judge from the second half of the 2 nd /8 th century, refused any representative (jarī) in court unless the male or female litigant was ill. When witnesses testified that a female litigant appointed a representative because of illness, the qāḍī asked them to prove that she was ill by bringing a sample of her urine to court. 93 Such reports suggest that women who did not want to appear in court appointed representatives.

Muslim jurists distinguish between women who can leave their houses and those who cannot. The Ḥanafī and Iraqi scholar al-Simnānī (d. 444/1052), for instance, distinguishes between the "barza" woman, who is allowed to appear in public, and the "mukhaddara", who is kept in seclusion. 94 Such a distinction has important consequences for legal procedure. The Egyptian Shāfiʿī jurist al-Muzanī (d. 264/877-78) allows a woman to appoint a representative (wakīl), whether she is one of "those who go out or those who do not." 95 Ḥanafī jurists did not regard the appointment of a wakīl as a satisfactory solution. According to al-Khaṣṣāf, when a woman involved in a lawsuit is one of "those who do not go out", the qāḍī (who cannot leave the court) must send her a trustworthy agent (amīn) who replaces him and delivers his judgment. The amīn could hear her acknowledgement and make her swear in place of the judge. 96 The same distinction is made in connection with a physical examination to determine if a girl is marriageable: the qāḍī can make the determination himself if the girl is from "those who go out", but he must send female experts to her house if that is not the case. 97 The swear at the mosque, while the second go there only for serious cases, at night, so as not to be seen; in a minor case, however, they may swear at home before a qāḍī's agent. 98 The biographical literature confirms that some women did not dare come to the qāḍī's court: at the end of the 2 nd /8 th century, the qāḍī of Kūfa Ḥafṣ b. Ghiyāth was walking in an alley (zuqāq) with his scribe when a young and pretty woman implored him to marry her to someone, since her brothers ill-treated her. 99 Apparently this woman could approach the judge only in the private or semi-private space of her alley. Similarly, in Ibn Ḥajar's book on Egyptian judges, we read that the qāḍī Ibn Zabr (d. 377/987) was approached by a man whose wife wanted to file a complaint against him. The husband would not let her appear in court, because he had sworn to divorce her if she left her house. The qāḍī followed him to a blind alley, where the wife explained her case from the roof and the qāḍī pronounced his judgment without getting off his mule. 100

distinction
Who were these women who "did not go out" and could not appear in court? Several studies on the Near East in late Antiquity have shown that women's behavior correlated with their social status. According to Joëlle Beaucamp, many women went to the court in Byzantine Egypt. 101 However, not every woman could do so easily. Beaucamp observes that between the 4 th and the 7 th centuries, one in three women turned to a representative when involved in a lawsuit. 102 She concludes that their behavior probably corresponded to a social norm: a woman was inclined to ask a man (usually her husband) to represent her before the judge. This social norm had a counterpart in Byzantine law: in the Code of Justinian, a constitution assigned to Constantine and dated 315 CE forbids women to mix with men at meetings and in lawsuits, lest they act indecently and shamelessly. The Byzantine court was a masculine public space in which a virtuous woman should not be present; so long as she had a father or a husband who could deal with her business, she had to avoid the court. 103 practices. Respectable women received only women as guests in their private apartments and never showed their faces outside. Lower-class women behaved differently: "sex segregation can only function in societies in which there are working women who are not veiled or secluded, because they serve as intermediaries between secluded women and the public, and make the seclusion of some women feasible." 104 Do such class-related practices explain the reluctance of some Muslim women to appear in the qāḍī's court?

The legal literature suggests that a woman's social position correlates with whether or not she appears in court. Despite the theoretical equality of the litigants, 105 Islamic law recognizes the existence of a social hierarchy and recommends that the qāḍī take it into consideration in his judgments. When a wife asks for maintenance (nafaqa), for example, he must take her economic and social status into account. If she is from a modest background, the qāḍī must award her a small maintenance even if her husband is rich and a spendthrift. If she is a rich woman, the judge must award her a large maintenance. 106 In the case of seclusion (during a woman's ʿidda, 107 or if her husband prohibits her from leaving the house without his consent), 108 al-Sarakhsī proposes that the ruling should be adapted to the woman's social status. Unlike free women, who must stay at home during the entirety of the ʿidda period, female slaves -even one who has been manumitted (mudabbara), is an umm walad 109 or enjoys a contract of enfranchisement (mukātaba) -generally can leave their houses during their ʿidda, especially if they have to work for their master or to purchase their freedom.

Conclusion

In contrast to Byzantine law, Ḥanafī doctrine do not discourage women from coming to the court: Muslim women have the right to come to the mosque, to complain, to testify and to act as judicial agents. The example of the Prophet's wives and daughters may have contributed to this entitlement. It is more complicated to understand the actual practice of women. At first sight, evidence for this practice is found in biographical reports about women.

However, the close correspondence of these reports to the typical "feminine" cases treated in law books calls into question their connection with actual practice. Because of their important contribution to economic life, the scarcity of references to commercial litigations involving women seems suspect. Women who were involved in commercial matters may have brought their disputes before the qāḍī only infrequently because of the socio-economic structure of labor (many women worked for their husbands or their masters and therefore were not autonomous "partners"), 117 but this does not explain the whole phenomenon.

The legal and biographical literature both suggest that practice was more varied. First, the necessity to unveil before the judge, in a public space, was perceived as shameful and source of dishonor to women. Second, Muslim jurists had to distinguish between women who could come easily and complain before the qāḍī and "respectable" women who did not leave their houses. This distinction appears in the writings of jurists belonging to different madhhabs and provinces (Ṭabaristān, Iraq, Egypt, Maghrib), and it suggests that such a distinction spread throughout the Islamic empire. In the Abbasid period, the women who did not leave their houses were probably high-ranking women. 118 According to a prevalent social practice in the late antique Near East, high-ranking women must avoid public exposure. It is likely that their conduct served as a model that spread to other women. From the 3 rd /9 th century onwards, the qāḍī's court became a space that was attended by men and low-status women. A nonwomen at the rabbinical court of Fusṭāṭ that "with the exception of teenage girls […], women were as active in court as men." Goitein, A Mediterranean Society, 3:323, 336. negligible number of cases involving women may have been brought before the more informal institutions of mediation and arbitration, those of wise and learned men or women locally recognized, in a quarter or a town, for their skills in solving conflicts. 119 This does not mean that women (even high-ranking women) could not use the judicial system. Rather the system had to adapt, encouraging women to use a representative and allowing them to take the oath before a judge's agent. The issue of women's appearance in court highlights the adaptability of Muslim jurists and the ongoing and continuous interaction between legal theory and actual social practices.

3

  Al-Khaṣṣāf, Adab al-qāḍī, ed. Farḥāt Ziyāda (Cairo: The American University in Cairo Press, 1978), 138, 378, 419, 531. 4 Ibid., 379. 5 D. S. Powers, "Four Cases Relating to Women and Divorce in al-Andalus and the Maghrib, 1100-1500," in M. K. Masud et al., Dispensing Justice in Islam. Qadis and their Judgments (Leiden-Boston: Brill, 2006), 402.

43

  Al-Jaṣṣāṣ, in al-Khaṣṣāf, Adab al-qāḍī, 354. Whereas Ḥanafīs limited women's adjudication to non-ḥudūd cases, al-Ṭabarī allowed them to adjudicate any dispute. The Shāfiʿīs and Ḥanbalīs held that only men can serve as judges. Al-Māwardī, al-Aḥkām al-sulṭāniyya (Constitutiones Politicae), ed. M. Enger (Bonn: Adolphus Marcus, 1853), 107-8; id., Adab al-qāḍī, ed. Muḥyī Hilāl al-Sarḥān (Baghdad: Maṭbaʿat al-irshād, 1971), 1:625-8; Ibn Qudāma, al-Mughnī, ed. ʿAbd Allāh b. ʿAbd al-Muḥsin al-Turkī and ʿAbd al-Fattāḥ al-Ḥulw (Cairo: Hajr, 1986-90), 14:12-13 (I owe this reference to Christopher Melchert). Byzantine law also excluded women from serving as judges. J. Beaucamp, "Les femmes et l'espace public à Byzance : le cas des tribunaux," Dumbarton Oaks Papers, 52 (1998), 132. Despite the theoretical recognition of women serving as judges in Ḥanafī law, E. Tyan found only one female judge in Islamic pre-modern history: Thaml/Thumal was appointed at the maẓālim court in 306/918-19, under al-Muqtadir. E. Tyan, Histoire de l'organisation judiciaire en pays d'Islam (Leiden: E.J. Brill, 1960), 162; see also Ibn al-Jawzī, al-Muntaẓam fī tawārīkh al-mulūk wa'l-umam, ed. Suhayl Zakkār (Beirut: Dār al-fikr, 1995), 8:12; N. M. El-Cheikh, "The Qahramâna in the Abbasid Court: Position and Functions," Studia Islamica, 97 (2003), 52-53.

  50 Al-Ṭabarī, Tafsīr (Beirut: Dār al-fikr, 1404 H.), 5:70-71, 75-76. Cf. Muqātil b. Sulaymān, Tafsīr Muqātil b. Sulaymān, ed. A. Farīd (Beirut: Dār al-kutub al-ʿilmiyya), 2003, 1:228.

54

  See for example N. Abbott, "Women and the State in Early Islam," Journal of Near Eastern Studies, 1 (1942), 106-126. 55 Th. Bianquis, La famille arabe médiévale, 86. 56 See Ibn Hishām, al-Sīra l-nabawiyya, Beirut: Dār al-kutub al-ʿilmiyya, 2001, 674 ff; EI 2 , s.v. ʿA/'isha bint Abī Bakr (W. M. Watt); D. A. Spellberg, Politics, Gender, and the Islamic Past. The Legacy of ʿA'isha bint Abi Bakr (New York: Columbia University Press, 1994), 61 ff.

88

  Al-Khaṭīb, Ta'rīkh Madīnat al-salām, 15:53. On the shamefulness of unveiling at the qāḍī's court, see also al-Iṣfahānī, al-Aghānī (Beirut: Dār Iḥyā' al-turāth al-ʿarabī, 1987), 12:68. 89 See al-Jaṣṣāṣ, in al-Khaṣṣāf, Adab al-qāḍī, 96. 90 Al-Khaṣṣāf, Adab al-qāḍī, 54. 91 Ibid., 99.

117

  See the hypothesis of S. Delmare-Lafont about Assyrian working women in "Quelques femmes d'affaires au Proche-Orient ancien," in Mémoires de la Société pour l'Histoire du Droit et des institutions des anciens pays bourguignons, comtois et romands, Dijon, 2008, 28-9. 118 M. van Berkel has shown that high-ranking women of al-Muqtadir's harem could play a political role only in so far as they relied on male allies occupying key positions within the state. M. van Berkel, "The Young Caliph and his Wicked Advisors: Women and Power Politics under Caliph Al-Muqtadir (r. 295-320/908-932)," Al-Masāq, 19 (2007), 10, 13, 14.

  20

119

  abbasside (Damascus: IFPO, 2009), 301 ff.

  Al-Khaṣṣāf, Adab al-qāḍī, 398, 404.

	28 Ibid., 532.
	29 Ibid., 349.
	30 Ibid., 690.
	31 Ibid., 178, 558, 561, 562.
	32 Ibid., 629.
	33 Ibid., 630, 633. See R. Shaham, "Women as Expert Witnesses in Pre-Modern Islamic Courts," in R. Shaham
	(ed.), Law, Custom, and Statute in the Muslim World. Studies in Honor of Aharon Layish (Leiden-Boston: Brill,
	2007), 49-63. Cf. D. Serrano, "Rape in Maliki Legal Doctrine and Practice (8 th -15 th Centuries C.E.)," Hawwa, 5
	(2007), 178-9.

  the main types of cases involving women found in biographies of 239 qāḍīs who held office in Iraq (193 qāḍīs, mainly in Baghdad, Baṣra, Kūfa and Wāsiṭ) and Egypt (46 qāḍīs, in Fusṭāṭ) during the

	Maintenance (nafaqa) 70	3
	Child care 71	2
	Adultery/liʿān 72	3
	Problems with sexual intercourse 73	3
	Undefined marital dispute 74	1
	Inheritance 75	2
	Contract/property 76	2
	Legal penalties (ḥadd) 77	1
	Unknown 78	7
	Total number of female litigants appearing before a qāḍī	38
	Representative acting for a woman 79	2
	Female witness 80	2
	Abbasid period (132-334/750-945).
	CASE TYPE	NUMBER
	Marriage (woman requesting or refusing marriage) 66	3
	Alleged/irregular marriage 67	2
	Divorce 68	5
	Dower (mahr) 69	4
	Dā'irat al-maʿārif al-ʿuthmāniyya, n.d.), 1:148 (also Ibn al-Jawzī, al-Muntaẓam, 6:230); al-Khaṭīb, Ta'rīkh 78 Al-Kindī, Akhbār quḍāt Miṣr, 374; Ibn Ḥajar, Rafʿ al-iṣr, 185/154; Wakīʿ, Akhbār al-quḍāt, 2:87, 108, 116; al-
	Madīnat al-salām, ed. Bashshār ʿAwwād Maʿrūf (Beirut: Dār al-gharb al-islāmī, 2001), 9:74. Ṣaymarī, Akhbār Abī Ḥanīfa wa-aṣḥābihi (Beirut: ʿA/lam al-kutub, 1985), 153; al-Khaṭīb, Ta'rīkh Madīnat al-
	67 Wakīʿ, Akhbār al-quḍāt, 2:44-45; 3:137. salām, 8:555.

al-Marāghī (Cairo: Maṭbaʿat al-saʿāda, 1947-1950), 1:261; ʿAbd al-Razzāq al-Ṣanʿānī, Muṣannaf ʿAbd al-Razzāq, ed. Ḥabīb al-Raḥmān al-Aʿẓamī (Beirut: al-Maktab al-islāmī, 1403 H.), 8:273-74; al-Shāfiʿī, al-Umm (Beirut: Dār al-maʿrifa, 1393 H.), 7:34. (I owe this reference to Christopher Melchert.) For Fāṭimid examples, see Goitein, A Mediterranean Society, 3:314. 64 Wakīʿ, Akhbār al-quḍāt, op. cit.; al-Kindī, Akhbār quḍāt Miṣr, in Kitāb al-wulāt wa-kitāb al-quḍāt, ed. R. Guest (Leiden: E.J. Brill, 1912) ; Ibn Ḥajar, Rafʿ al-iṣr ʿan quḍāt Miṣr, ed. ʿAlī Muḥammad ʿUmar (Cairo: Maktabat al-Khānjī, 1998). 65 See e.g. Tyan, Histoire de l'organisation judiciaire; W. B. Hallaq, The Origins and Evolution of Islamic Law (Cambridge: Cambridge University Press, 2005). Although these biographical dictionaries are widely used by historians of legal practices, little reflection has been devoted to their value. I am currently preparing a study of this issue. 66 Wakīʿ, Akhbār al-quḍāt, 3:188; Ibn Abī l-Wafā', al-Jawāhir al-muḍiyya fī ṭabaqāt al-ḥanafiyya (Ḥaydarābād: 68 Ibn Ḥajar, Rafʿ al-iṣr, 430/100 (the first number refers to Ibn Ḥajar's Arabic text; the second number refers to my partial French translation of this book: M. Tillier, Vies des cadis de Miṣr (Cairo: IFAO, 2002)); Wakīʿ, Akhbār al-quḍāt, 2:63, 68; 3:290 (also Ibn Abī l-Wafā', al-Jawāhir, 1:209); al-Tanūkhī, Nishwār al-muḥāḍara wa-akhbār al-mudhākara, ed. ʿAbbūd al-Shāljī (s.l., 1971-73), 3:227.

The data fits very well the typology analyzed earlier. Of forty-two accounts of trials involving women, twenty-six deal with marital disputes (almost two-thirds of the total). The actual rate is probably higher, for six accounts do not explain the nature of the case brought before the qāḍī. At first sight, one might conclude that the legal typology reflected in the adab al-qāḍī literature was close to judicial practice, and that both the legal and the biographical sources portray women as being deeply involved in judicial practice. This would confirm the

69

Al-Kindī, Akhbār quḍāt Miṣr, 375, 381; al-Khaṭīb, Ta'rīkh Madīnat al-salām, 15:53; al-Tanūkhī, Nishwār, 2:28. 70 Ibn Ḥajar, Rafʿ al-iṣr, 178/111; 351/142; Wakīʿ, Akhbār al-quḍāt, 2:79. 71 Ibn Ḥajar, Rafʿ al-iṣr, 330/181; Wakīʿ, Akhbār al-quḍāt, 3:135-6. 72 Ibn Ḥajar, Rafʿ al-iṣr, 119/43, 330/181; al-Khaṭīb, Ta'rīkh Madīnat al-salām, 8:554. 73 Ibn Ḥajar, Rafʿ al-iṣr, 328/178, 330/181; al-Subkī, Ṭabaqāt, 3:452. 74 Wakīʿ, Akhbār al-quḍāt, 3:138. 75 Wakʿ, Akhbār al-quḍāt, 2:173; Ibn Ḥajar, Rafʿ al-iṣr, 121/46. 76 Ibn Ḥajar, Rafʿ al-iṣr, 275/95; Wakīʿ, Akhbār al-quḍāt, 3:170 (see also Ibn al-Jawzī, al-Muntaẓam, 5:417-9). 77 Ibn Khallikān, Wafayāt al-aʿyān (Beirut: Dār Ṣādir, 1994), 4:180. 79 Wakīʿ, Akhbār al-quḍāt, 2:128; al-Kindī, Akhbār quḍāt Miṣr, 375. 80 Wakīʿ, Akhbār al-quḍāt, 3:134; Ibn Khallikān, Wafayāt al-aʿyān, 1:277.

  also appears in Maghribī Mālikī law: concerning women's oaths, Saḥnūn distinguishes between those who can go out and those who cannot. The first category must92 Al-Jaṣṣāṣ, in al-Khaṣṣāf, Adab al-qāḍī, 100. Cf. Q. 24:60.

	93 Wakīʿ, Akhbār al-quḍāt, 2:128.
	94 Al-Simnānī, Rawḍat al-quḍāt wa-ṭarīq al-najāt, ed. Salāḥ al-Dīn al-Nāhī (Beyrouth-Amman: Mu'assasat al-
	risāla -Dār al-furqān, 1984), 1:177.
	95 Al-Muzanī, Mukhtaṣar al-Muzanī (Beirut: Dār al-maʿrifa, n.d.), 110.
	96 Al-Khaṣṣāf, Adab al-qāḍī, 241, 627. See also al-Jaṣṣāṣ, in al-Khaṣṣāf, Adab al-qāḍī, 244. Cf. Al-Simnānī,
	Rawḍat al-quḍāt, 1:177, 188.

  In 6 thcentury Najrān, E. A. Doumato notes, the veiling and seclusion of women were class-related

	Dār al-kutub al-ʿilmiyya, 2001), 1:159-60. On the testimony of women and its relation to public space in
	classical fiqh, see M. Fadel, "Two Women, One Man," 192-95. On the influence of such procedures on Jewish
	law, see G. Libson, Jewish and Islamic Law. A Comparative Study of Custom During the Geonic Period
	(Cambrige: Harvard University Press, 2003), 107-9.

98

Saḥnūn, al-Mudawwana l-kubrā (Beirut: Dār ṣādir, n.d.), 5:136. Cf. Ibn Farḥūn, Tabṣirat al-ḥukkām (Beirut:

  110 A widow may go out during the day in case of necessity, on the condition that she sleeps in her house. Abū Yūsuf, Kitāb al-āthār, ed.

	Abū l-Wafā' al-Afghānī (Beirut: Dār al-kutub al-ʿilmiyya, n.d.), 142; al-Sarakhsī, al-Mabsūṭ (Beirut: Dār al-
	maʿrifa, n.d.), 6:32.
	108 Al-Shaybānī, al-Aṣl, ed. Abū l-Wafā' al-Afghānī (Karachi: Idārat al-Qur'ān wa'l-ʿulūm al-islāmiyya, n.d.),
	3:272 ff ; al-Muzanī, Mukhtaṣar, 295 ; al-Sarakhsī, al-Mabsūṭ, 8:173 ff.
	109 As for the umm walad, Saḥnūn recognizes her status as much closer to that of a free woman, and states that
	"some of them go out and others do not." Saḥnūn, al-Mudawwana l-kubrā, 5:136.
	110 According to al-Sarakhsī, the husband can forbid her to go out only if he pronounces a revocable divorce (al-

104 E. A. Doumato, "Hearing Other Voices," 184. 105 M. Tillier, "La société au miroir du tribunal," Annales Islamologiques, 42 (2008), 157-8. 106 Al-Khaṣṣāf, Adab al-qāḍī, 647. 107 Al-Shaybānī, al-Jāmiʿ al-ṣaghīr (Beirut: ʿA/lam al-kutub, 1404 H.), 230 ; al-Shāfiʿī, al-Umm, 5:226. Ḥanafī jurists usually prohibit a divorced woman from leaving her house until the end of her ʿidda. ṭalāq al-rajʿī), in order to save his honor and to prevent her from giving birth to the child of another man. Al-Sarakhsī, al-Mabsūṭ, 6:33.

Al-Khaṣṣāf, Adab al-qāḍī, 531.

Al-Khaṣṣāf, Adab al-qāḍī, 629.

Some jurists expressly state that the appearance of women before the qāḍī depends on their social status. In the 4 th /10 th century, the Shāfiʿī Ibn al-Qāṣṣ (d. 335/946), who originated from Ṭabaristān and lived in Iraq and Syria, mentions in the context of oaths sworn by women "noble women who do not go out" (nisā' min ahl al-sharaf lā takhruj): like al-Khaṣṣāf and the Mālikī jurists, he holds that the qāḍī can make women swear at home, and must send them two amīns who take their oath. He also emphasizes the problem of women unveiling before the qāḍī, stating that a female witness may remain veiled when testifying, provided that two honorable witnesses (ʿadl) testify to her identity. 111 The fact that some high-ranking women were reluctant to appear in the qāḍī's court is confirmed in biographical accounts. Ca.

155/772 in Baṣra, a low-ranking Nabaṭī (i.e. Aramean Iraqi autochthon) complained against a noble woman, the daughter of a former governor of the city. The qāḍī Sawwār b. ʿAbd Allāh summoned the woman but she refused to come, and the governor blamed the qāḍī for asking such a high-ranking woman to appear in court. 112 When the Egyptian qāḍī Ghawth b.

Sulaymān was ordered to adjudicate a matrimonial conflict between the caliph al-Manṣūr and his wife, Umm Mūsā -a noble woman from the tribe of Ḥimyar -he asked Umm Mūsā to appoint a wakīl to represent her before him: 113 this suggests that the caliph's wife was not supposed to appear before a judge. In the same period, however, some women of high socioeconomic status complained directly to the qāḍī. These include a landlady who sued the Kūfan governor before Sharīk b. ʿAbd Allāh, 114 and a noble woman from the Banū ʿIjl who asked the qāḍī Ḥafṣ b. Ghiyāth to marry her to someone. 115 These counter-examples may be markers of a progressive transition toward stricter seclusion within the upper classes of large Abbasid cities. 116 The increasing weight of this social norm at the end of the 2 nd /8 th century probably led to the exclusion of some women from the court.