On the density of S-adic integers near some projective G-varieties
aa r X i v : . [ m a t h . N T ] A p r ON THE DENSITY OF S -ADIC INTEGERS NEAR SOMEPROJECTIVE G -VARIETIES YOUSSEF LAZAR
Abstract.
We provide some general conditions which ensure that a systemof inequalities involving homogeneous polynomials with coefficients in a S -adic field has nontrivial S -integral solutions. The proofs are based onthe strong approximation property for Zariski-dense subgroups and adelicgeometry of numbers. We give two examples of applications for systemsinvolving quadratic and linear forms. Introduction
Given a finite set of places S of Q which contains the archimedean one, weconsider a finite family of homogeneous polynomials ( f i,p ) p ∈ S (1 i r ) whereeach f i,p has coefficients in the completion of Q relative to the place p ∈ S .We are interested in the following problem, given any real ε > , can we findan nonzero vector x ∈ Z nS such that(1) < | f i,p ( x ) | p ≤ ε for every p ∈ S and i = 1 , . . . , r ? Despite its apparent simplicity, this question is extremely difficult to solve ingeneral and as far as we know, only few cases have been settled. Our point ofdeparture is the case of a single isotropic quadratic form f = Q for which asolution was found only quite recently by Borel and Prasad [5] for S = S ∞ andcompleted in the general case as soon Ratner gave a complete solution to theRaghunathan conjecture [2] in full generality. Their result is an S -arithmeticgeneralization of Margulis’ proof of the Oppenheim conjecture [20]. For thereader interested in such dynamical methods and the applications of Ratner’stheory to number theoretical problems we refer to (e.g. [15].).The main tool we are going to use in order to treat the question (1) withthe highest level of generality is the strong approximation property for alge-braic groups . In other words, we will be merely focusing on the arithmeticalproperties of groups actions rather than their ergodic behaviour. It is not verysurprizing that strong approximation could solve the same density problems asRatner’s orbit closure theorem does since both results take place in groups gen-erated by one-dimensional unipotent elements. The bridge between these twonotions is the Kneser-Tits conjecture which asserts that any simply-connectedgroup which is simple and isotropic over a local field, is generated by its onedimensional unipotent elements which was solved by Platonov for such groups(see e.g. §7.2, [26]). Our main purpose is to provide a substitute in the casewhen Ratner’s theorem fails to hold. One illustration of the great advantage of using strong approximation rather than Ratner’s theory is that we get ridof the heavy task of classifying intermediate Lie groups, indeed in most casecan such classification is unfeasible unless in the rare cases when we are able toreduce to lower dimension. Fortunately this is the case of the original proof ofthe Oppenheim conjecture which has been proved for n = 3 and for Gorodnik’sresult for pairs ( Q, L ) which reduces to the dimension four but for pairs theclassification of intermediate is much more involved (see [9]).In the same circle of ideas, recently Ghosh, Gorodnik and Nevo developped in([10], [11], [12], [13]) the metric theory of diophantine approximation on ho-mogeneous varieties of semisimple groups in the S -arithmetic setting. Amongmany other deeps results, they proved analogs of Khintchine’s and Jarnik’stheorems for S -adic homogeneous spaces using both ergodic theory and strongapproximation for algebraic groups combined with deep concepts coming fromthe theory of automorphic forms and representation theory. Their methodalso provides a quantitative version of the strong approximation theorem inhomogeneous spaces of semisimple groups.Finally one should mention that for higher degrees, i.e. when the numberof variables and the degrees of the f i ’s are greater than the number of r ofpolynomials, the circle method of Hardy and Littlewood still remains the mostpowerful method for proving (1) in great generality. In fact, it is providingalso sharp quantitative estimate of the numbers of solutions with boundedheights of the number of points lying exactly on a variety following the programpromoted by Y. Manin in the seventies.1.1. Background and notations.
Let us denote by Σ Q the set of all places in Q , these are given by the set of all prime numbers and the archimedean placecorresponding to ∞ . Let S be a finite set of places in Σ Q p which containsthe archimedean one, and let us denote by S f the subset of all finite (prime)places in S , thus we have S = S f ∪ {∞} . For each prime p , we can definethe p -adic absolute value is denoted by | . | p over Q and we denote by Q p thecorresponding completion of Q . The product of the Q p ( p ∈ S ) is denotedby Q S . The set of p -adic integers is denoted by Z p and is defined to be theset of x ∈ Q such that | x | p ≤ . The ring of S -integers of Q is the set Z S which elements are integral outside S i.e. such that x ∈ Z p for p / ∈ S . Foreach p ∈ S f , Q p is a locally compact (additive) group, hence it is equippedwith a Haar measure characterized by the formula µ p ( a Ω p ) = | a | p µ p (Ω p ) for all a ∈ Q p and Ω p is a measurable subset of Q p of finite measure. We normalize itby prescribing the value of the measure µ p over the basis of open sets in Q p bytaking µ p ( a + p n Z p ) = p − n , in particular, µ p ( Z p ) = 1 . The set of adeles A of Q is the subset of the direct product Q p Q over all the places of Q consistingof those x = ( x p ) such that x ∈ Z p for almost all places. The set of adeles A is a locally compact ring with respect to the adele topology given by thebase of open sets of the form Q p ∈ S Q p × Q p / ∈ S Z p where S ⊂ Σ is finite with S ⊃ S ∞ . For any finite subset S ⊂ Σ with S ⊃ S ∞ , the ring of S -integraladeles is defined by: N THE DENSITY OF S -ADIC INTEGERS NEAR SOME PROJECTIVE G -VARIETIES 3 A S = Q p ∈ S Q p × Q p / ∈ S Z p , thus we can see that A = [ S ⊃ S ∞ A S .By definition Z S = A S ∩ Q , in addition, it can be proved that Z S is a latticein Q S i.e. a discrete subgroup of finite covolume. We can also realize Z S as acocompact lattice in A S . One of most fundamental result in arithmetic is thefollowing fact which says that for any nonempty set of places S , the image of Q under the diagonal embedding is dense in A S . This property is called thestrong approximation for the field Q . For a brief review about this propertyin the framework of algebraic groups, we invite the reader to read the recentaccount about this question in ([27]), for more details we advice one the mostcomplete reference about this topic ([26]). Quadratic forms over local fields . A quadratic form in n variables over a localfield k (i.e of characteristic zero) is given by a symmetric bilinear form B over k such that Q ( x ) = B ( x, x ) for any x ∈ k n . We say that Q is nondegenerate in k n if the rank of the matrix associated to B has maximal rank. A quadraticform Q s ( x ) is isotropic over Q s if there exists a nonzero vector x such that Q s ( x ) = 0 . Over S -adic products, a quadratic Q = ( Q s ) s ∈ S over Q S is saidto be nondegenerate (resp. isotropic) if and only if Q s is nondegenerate (resp.isotropic) over Q s for each s ∈ S . The special orthogonal group of a quadraticform ( Q s ) s ∈ S is the product of S of orthogonal groups SO( Q s ) , the latter is aLie group which is semisimple as soon as Q s is nondegenerate. The orthogonalgroup SO( Q s ) is isotropic over Q s (i.e. it has no nontrivial characters over Q s )if and only if Q s is isotropic. It is well-known that over local fields, SO( Q s ) isisotropic if and if it is noncompact. If H = Q s ∈ S H s is a product of s -adic Liegroups and S ⊆ S be a finite subset of places, then H is said to be isotropicover S if for every s ∈ S , H s is isotropic over Q s . A quadratic form ( Q s ) s ∈ S issaid to be (globally) rational if there exists a form Q with rational coefficientssuch that Q = λQ for some nonzero λ ∈ Q S , and irrational otherwise. Acrucial fact is that a quadratic form Q = ( Q s ) s ∈ S can be irrational (globally)with Q s being rational at some place s ∈ S . Any quadratic space ( V, Q ) over k , can be decomposed in virtue of the Witt’s decomposition theorem as follows V = V an ⊕ rad( Q ) ⊕ P ⊕ . . . ⊕ P r where the restriction Q to V an is anisotropic over k , rad( Q ) is the radical of Q it is equal to zero if Q is nondegenerate and P i (1 i r ) are hyperbolicplanes such that the restriction of Q to each P i is isotropic. In particular, P ⊕ . . . ⊕ P r is the maximal isotropic subspace, r is called the isotropy indexand r is called the Wiit index denoted i ( Q ) , remark that Q is isotropic over k if and only if i ( Q ) ≥ . Algebraic actions on projective varieties . If we consider an algebraic varietydefined over Q by a prime ideal I = h f , . . . , f r i for which closed points ofgiven by X = { x ∈ Q n | f i ( x ) = 0 , ≤ i ≤ r } . YOUSSEF LAZAR
The polynomials defining the prime ideal I are supposed to be homogeneous in n variables with rational coefficients. For each ≤ i ≤ r , if we denote d i = deg f i thus we must have that for any λ ∈ Q , f i ( λx ) = λ d i f i ( x ) , inparticular the zero locus X = V ( I ) can be seen an algebraic projective variety .Neverthless for practical reasons we need our varieties to be embedded invectors spaces thefore we will be stuck with the affine point view.Let G be the special linear algebraic group SL n | Q defined over Q and let usconsider its left action on the Q -vector space V = Q [ x , . . . , x n ] which is givenfor each g ∈ G and f ∈ V by g.f ( x ) = f ( g − x ) for all x ∈ Q n .We would like to define a subgroup of G which leaves globally invariant thevariety X = V ( I ) but also every element of the ideal of definition I . For thispurpose, we introduce the subgroup H of G defined over Q given by H = { h ∈ G | h.f i = f i , ≤ i ≤ r } . From its very definition H is an algebraic group which acts linearly on thevariety X . The automorphism group of X under the action of G ( PSL n in theprojective setting) is defined as Aut G ( X ) = { x ∈ X | g.x = x } . It is immediate to see that we have the following inclusions
Aut G ( X ) ⊆ H ⊆ G. In particular, one can note that X can have a large group H while having asmall or maybe even finite automorphsim group. For this reason, we prefer toconsider the action of H rather than Aut G ( X ) which in the ideal case wouldbe a semisimple noncompact Lie group.Let us consider a variety X/ Q embedded diagonally in a finite product ofcompletions relative to a finite set of places S in Q containing the archimedeanone. To do so, we first consider the natural diagonal embedding of Q in Q S whose image is the direct product of completions Q p for each p in S . Usingthis embedding we can consider the family of polynomials ( f i,p ) p ∈ S over Q S and for each p ∈ S we define X p to be the zero set of the ( f i,p ) (1 ≤ i ≤ r ) ,this defines an affine variety over Q p . Therefore X S is the direct product of thecompletions of X p ( p ∈ S ) in Q s ∈ S Q np and we can define in the same way bytaking the S -product H S = Q p ∈ S H p . Clearly the action of H on X induces anequivariant action of H S on X S with respect to the diagonal embedding. Atsome point, we will have to consider the set of Q p -points of X p which is givenby X p ( Q p ) = X p ∩ Q np , it is equipped with the p -adic topology induced by thebase field Q p . For each p ∈ S the set X p ( Q p ) of Q p -points of X p can endowedwith a structure of analytic variety over Q p . Using some analogy with analyticcomplex geometry, given any real ε > we define the ε - tubular neighborhood of X S for the S -adic topology by X εS = { x ∈ Q nS : | f j,s ( x ) | p ≤ ε for every ≤ j ≤ r and every p ∈ S } . N THE DENSITY OF S -ADIC INTEGERS NEAR SOME PROJECTIVE G -VARIETIES 5 The elements of Q nS which are in X εS are called ε -near vectors to X S . The factthat we have chosen the f i ’s to be homogeneous implies that for any ε > , theintersection of X εS with the lattice Z nS contains at least the null vector. Thusfor any ε > , showing that X εS ∩ Z nS = { } amounts to find a nonzero x ∈ Z nS such that for every p ∈ S and ≤ i ≤ r we have, | f i,p ( x ) | p ≤ ǫ. If we consider the polynomial map φ : Q nS −→ Q rS associated to X S , given by φ ( x ) = ( f ( x ) , . . . , f r ( x )) . then X εS ∩ Z nS = { } if and only if the origin in Q rS is an accumulation pointin φ ( Z nS ) for the S -adic topology. A more ambitious question regarding ourinitial problem in (1) is to ask the density of φ ( Z nS ) in Q rS , note that for realplaces (1) , i.e. non-discreteness at the origin, is equivalent to density. Notations. • If A is a subset of Q , we denote by A ( p ) its p -adic closure in Q p and for anyset of places S in Σ Q , A ( S ) = Q p ∈ S A ( p ) . • If A and B are two sets, we denote A + B by A + B = { a + b | a ∈ A and b ∈ B } . More generally if A , . . . , A l are sets, their Minkowski sum is denoted A + . . . + A l = ( M ) X ≤ i ≤ l A i = { a + . . . + a n | a i ∈ A i for 1 ≤ i ≤ l } . In particular for every integer n > , the n -times sum A + . . . + A is denotedby n ∗ A . • For each s ∈ S , we denote by Sym ( Q ns ) the set of bilinear symmetric formswith coefficients in Q s with n variables. We identify this set with the set of allquadratic forms in n variables with coefficients in Q s since we are in charac-teristic zero.1.2. Main results.
The main result gives sufficient conditions in order to en-sure that the system (1) has nontrivial solutions. These conditions are realizedif we can find a rational triple ( H , X , f ) where H acts on X = V ( f ) andwhere H satisfies some assumptions prior to the application of the strongapproximation property. As an application, we provide an answer to (1) forsystems involving one quadratic form and one/several linear form(s) whichare well-understood in the real case, i.e. S = {∞} . The three dimensionalcase was treated by S.G. Dani and G.A. Margulis in [8] where solutions to(1) was provided for pairs ( f , f ) = ( Q, L ) . In higher dimensions a similarresult has been proved by A. Gorodnik [9]. The case of values of quadraticforms restricted to affine subspaces defined linear forms has been treated by YOUSSEF LAZAR
S.G. Dani [7]. Very recently by the dual case concerning the values of linearforms on a quadric hypersurfaces, O. Sargent was able to prove an Oppenheimtype density result in the real case [29]. All these results relies on deep resultsfrom the ergodic theory of unipotent flows on homogeneous spaces. The mostpowerful tool to prove such density results is Ratner’s orbit closure theorem.The Oppenheim conjecture is a direct consequence of the Raghunathan con-jecture, but this conjecture was not yet proved at the time when Margulisand then Borel-Prasad published their results. The Raghunathan conjecturewas proved by M.Ratner in full generality and also for p -adic Lie groups, thelatter has been proved by different methods by Margulis and Tomanov [21].This was fortunate, at least for our work, that Borel and Prasad did not haveRatner’s on hand. This has forced them to coin a very interesting method in(§4, [5]) which proves (1) for a quadratic form with coefficients in Q S in n ≥ variables. Under the assumption that Q is globally irrational unless at someplace v ∈ S where Q v is rational, Borel and Prasad proved that (1) holds.This local rationality condition is absolutely paramount in order to apply the strong approximation property to the orthogonal group of Q . The theorem 1.1is generalization of their method in the most general setting so that we canensures solutions of (1) for homogeneous polynomials of arbitrary degrees. Togive some credit to the point of view made by Borel and Prasad, we apply ourmain theorem to varieties of the form V ( Q, L ) and V ( Q, L , . . . , L r ) where L , l , . . . , L r are linear forms. Indeed for both varieties we are able to givesufficient conditions so that (1) holds. We expect that it holds for more gen-eral examples of G -varieties. Theorem 1.1.
Let S be a finite set of places in Q containing the archimedeanone. For each s ∈ S , we are given a projective algebraic variety X s over Q s defined by a homogeneous prime ideal I s of Q s [ x , . . . , x n ] and let H s bethe algebraic Q s -subgroup of SL n ( Q s ) leaving invariant every generator of I s .Assume that the following subset of places S ⊂ S f is nonempty, S = { p ∈ S f | H p is rational over Q } . If there exists a connected algebraic subgroup H of G rational over Q and ahypersurface X = V ( f ) defined over Q such that (1) H is a semisimple absolutely almost simple algebraic Q -group. (2) For every prime p ∈ S , X p = X and H p = H . (3) Every Q -simple factor of H is isotropic over S .Then for any ǫ > , there exists a nonzero S -integral vector lying ǫ -near X S i.e. X εS ∩ Z nS = { } . As an application we can use the theorem to prove existence of near integralvectors to the variety X S = { Q = L = 0 } which can be see as the hypersurfaceconsisting of one nondegenerate quadric Q = 0 cutted out by the hyperplane ofequation { L = 0 } . In the following result we extend the validity of a previous N THE DENSITY OF S -ADIC INTEGERS NEAR SOME PROJECTIVE G -VARIETIES 7 result of the author(Corollary 2.2, [17]) proved under the condition that anynontrivial linear combinaison α s Q s + β s L s should be irrational at all places s ∈ S . Indeed we are able to prove that the same result holds when if we onlyassume that αQ + βL is (globally) irrational over Q S leaving the possibilitythat it is rational at some nonarchimedean place. Corollary 1.2.
Assume S is a finite set of places in Q containing the archimedeanone and let Q = ( Q s ) s ∈ S be a quadratic form and L = ( L s ) s ∈ S be a linear formon Q nS with n ≥ and L s = 0 for all s ∈ S . Suppose that the pair ( Q, L ) satisfies the following conditions, (1) Q is nondegenerate. (2) Q | L =0 is nondegenerate and isotropic. (3) For any choice α, β in Q S with ( α, β ) = (0 , , the form αQ + βL is ir-rational with at least one place v ∈ S f where α v Q + β v L is proportionalto a rational form.Then for any ε > , there exists a nonzero x ∈ Z nS such that | Q s ( x ) | s < ε and | L s ( x ) | s < ε for each s ∈ S . In the same vein, we are also able to treat the case when we consider severallinear forms instead of one. This result is a variation of a recent due to O.Sargent [29] in the S -arithmetic setting which is proved using Theorem 1.1. Corollary 1.3.
Assume S is a finite set of places in Q containing the archimedeanone and suppose that Q is an isotropic nondegenerate quadratic form in n ≥ variables over Q S with rational coefficients. Let M = ( L , . . . , L r ) be a linearmap M : Q nS → Q rS where ( L i,s ) s ∈ S (1 i r ) are linear forms of rank r over Q nS in n ≥ variables which satisfies the following conditions: (1) n > max s ∈ S (dim ker M s ) + 2 . (2) rank ( Q s | Ker M s ) = r and Q | Ker M s is isotropic and , for every s ∈ S . . (3) For each choice of α , . . . , α r in Q S with ( α , . . . , α r ) = (0 , . . . , , thelinear form α L + . . . + α r L r is irrational unless at one place v ∈ S f where it is proportional to a rational form.Then for any ε > , there exists a nonzero x ∈ Z nS such that Q s ( x ) = 0 and | L i,s ( x ) | s < ε for each s ∈ S and i r . Remarks. (1)
The proof of Theorem 1.1 is based on a strenghtening of thestrong approximation theorem which apply to Zariski dense subgroups of re-ductive groups proved by Matthews, Vaserstein and Weisfeiller ([18], [31]) andlater by Nori [25]. In the meanwhile, Venkataramana (Proposition (5 . , [30])proved that there exists Zariski dense subgroup of integral points of SL n ( Z ) which contains no unipotent elements and thus such subgroup might be eligiblefor strong approximation even in it is far from being unipotent or generatedby unipotents elements. For more general details about strong approximation YOUSSEF LAZAR in algebraic groups and more particularly this version, we advice the readerthe recent survey of A.S. Rapinchuk [27]. (2)
The method used in Theorem 1.1 is an adaptation of the work of Borel-Prasad in the case when S contains at least one place where the form is rational(see §4, [5]). (3) A nice feature of the proof of Corollaries 1.2 and 1.3 is that we do nothave to reduce to lower dimension. Indeed, the reduction process was a pre-condition for proceeding to the classification of intermediate subgroups aringfrom the application Ratner’s theorem. (4)
In the statement of Corollary 1.3 we have not tried to find optimal andthus it can be seen as a partial S -adic generalization of the main result of O.Sargent in the real case [29]. It may be possible to refine the conditions butour aim was to give an easy example of application of the main theorem.2. Proof of Theorem 1.1
We denote by S the set of places of S which are disjoint from S , in partic-ular S contains only nonarchimedean places with let us say S = { p , . . . , p s } and S = { q , . . . , q l − , ∞} . Let us define Λ to be the stabilizer of the standardlattice Z nS in H ( Q ) i.e. Λ = { h ∈ H ( Q ) | h ( Z nS ) = Z nS } . Under the diagonal embedding, Λ can be seen an S -arithmetic which is discretein H ( Q S ) . Consider the universal Q -isogeny π : ˜ H → H , here f H is asemisimple simply connected group defined over Q . Let us choose an arbitrary S -arithmetic subgroup e Λ of ˜ H ( Q ) embedded as a discrete subgroup in ˜ H ( Q S ) .Since ˜ H is absolutely almost simple over Q and every simple factor H i of f H is noncompact at every place in S , in particular P s ∈ S rank Q v H i > .Borel’s density theorem (see e.g. proposition 3.2.10 ([19]) applied to ˜ H with K = k = Q gives us that the discrete subgroup ˜Λ is Zariski-dense in ˜ H . Aspecial instance of the strong approximation for Zariski-dense subgroups (seee.g. [26], Theorem 7.14) applied in ˜ H for the set of primes S = { p , . . . , p l } implies that the closure ˜Λ is open in ˜ H ( A S ) . The latter adelic group isdescribed by the following product Y p ∈ S e H ( Q p ) × Y q ∈ S e H ( Z q ) × Y q / ∈ S ∪ S e H ( Z q ) Thus for every q ∈ S , the projection e Λ ( q ) is open in ˜ H ( Z q ) . The universal Q -isogeny π transforms ˜Λ into an arithmetic subgroup π ( ˜Λ) in H ( Q ) suchthat π ( e Λ) ( S ) is open in H ( Q S ) . The two arithmetic subgroups π ( ˜Λ) and Λ N THE DENSITY OF S -ADIC INTEGERS NEAR SOME PROJECTIVE G -VARIETIES 9 are commensurable in H , thus π ( ˜Λ) ∩ Λ is a finite index subgroup in π ( ˜Λ) (see e.g. cor. 3.2.9 [19]). For each p ∈ S , we define U p to be the projection of π ( ˜Λ) ∩ Λ onto its p -component, that is,(2) U p := ( π ( ˜Λ) ∩ Λ) p . As we have seen above, the p -adic closure U p ( p ) lies in the open subgroup π ( ˜Λ) ( p ) of H ( Q p ) for each p ∈ S and U S is contained in Λ ( S ) by definition.Now let us introduce the following subset of vectors in Q np defined for each p ∈ S , by: X p = U − p X p ( Q p ) − X ( Q p ) where U − p = { a − p | a p ∈ U p } is an open subset of invertible matrices in H ( Q p ) where U p was defined in (2).We claim that X p is a nonempty open blunt cone in Q np for p ∈ S . Indeed,since X p is not rational over Q for each p ∈ S while X is, this forces X p = X for p ∈ S . The Zariski density of X p ( Q p ) (resp. X ( Q p ) ) in X p (resp. X )implies that X p ( Q p ) = X ( Q p ) for each p ∈ S . In particular for each p ∈ S ,there exists an x ∈ X p ( Q p ) − X ( Q p ) , thus taking a p = I n we get that x ∈ X p .Now let us fix x ∈ X p and consider y ∈ Q np sufficiently close to x , so that wecan find an g ∈ SL n ( Q p ) such that y = gx where g is close to I n in SL n ( Q p ) .Since I n ∈ U p then we can assume that g is arbitrarily close to I n in the openset U p . We have f i,p ( g p x ) = 0 for some g p ∈ U p close to I n and since y = gx weget f i,p (( g p g − ) y ) = f i,p ( g p x ) = 0 with g p g − ∈ U p thus y ∈ X p . It is clear that X p does not contains the origin and that for any λ ∈ Q ∗ p and x ∈ X p , λx ∈ X p by homogeneity of the f i ’s, the claim is proved.For each p ∈ S , denote by ∆ p ( r ) the hypercube centered at with radius r in Q np , i.e. ∆ p ( r ) = { x ∈ Q nS | | x i | p ≤ r } and denote ∆ S ( r ) = Q p ∈ S ∆ p ( r ) . Our aim is to show that ∩ ε> X εS ∩ Z nS = { } ,a first step consists to show the existence of nonzero lattice point for the domain (∆ S ( δ ) × X S ) which can be seen as a sort of approximation of X εS . Lemma 2.1.
For every δ > , we have (∆ S ( δ ) × X S ) ∩ Z nS = { } . Proof of the lemma.
Let us fix a place p ∈ S whether it is archimedean ornot, and set l = | S | . Let v ,p ∈ Q np be a nonzero in vector X p and completeit to a basis { v ,p , v ,p , . . . , v p,n } of Q np . For each real a > we introduce thehypercubes V p ( a ) in Q np and W p ( a ) in Q n − p defined as V p ( a ) = { n X i =1 α i v i,p | | α i | p ≤ a } and W p ( a ) = { n X i =2 α i v i,p | | α i | p ≤ a } .Since we know that X p is open, we can find an infinitesimal hypercube v ,p ⊕ W p ( α ) for some small enough real α > so that it is contained in X p . Howeverwe have the following fact: the resulting infinitesimal hypercube v ,p ⊕ W p ( α ) remains in the cone X p if we perform a translation in the direction of v ,p awayfrom Z p . In other words, we can find an real positive α small enough so thatfor each p ∈ S and for any given arbitrary η ∈ Q p with | η | p > (i.e. η / ∈ Z p )we have simultaneously(3) ηv ,p ⊕ W p ( α ) ⊂ X p and V q ( α ) ⊂ X p . Indeed, let us set u = ηv ,p + P ni =2 α i v i,p ∈ ηv ,p ⊕ W p ( α ) for η ∈ Q ∗ p − Z p .Thus u can be written as u = η ( v ,p + n X i =2 η − α i v i,p ) . It is clear that for every ≤ i ≤ n , | η − α i | p < α , thus v ,p + n X i =2 η − α i v i,p ∈ v ,p ⊕ W p ( α ) Therefore using the cone invariance for X p , we infer that u ∈ X p , which provesthe claim (3). For this choice of α and for each reals δ, t > let us introducethe S -adelic domain C p ( δ, t ) = ∆ S ( δ/ l ) × [0 , t ] v ,p ⊕ W p ( α/ × Y q ∈ S \{ p } V q ( α/ l ) × Y s / ∈ S Z p ⊂ A nS . Let us define C p ( δ ) to be ∗ C p ( δ, . It is a compact subset of A nS and thusit meets the discrete subgroup Z nS in finitely many points at the number of k = | C p ( δ ) ∩ Z nS | . The set of S -integral vectors Z nS is a cocompact lattice in A nS i.e. A nS / Z nS is a compact space of finite volume for the measure µ inducedby vol A S on the quotient space. Let us denote by D a fundamental domain forthe quotient A nS / Z nS and by µ ( D ) its volume, in particular we have A nS = [ y ∈D y + Z nS . We can find some τ large enough so that exists k +2 disctinct points y , . . . , y k +1 in C p ( δ, τ / such that y i − y j ∈ Z nS for any i < j k + 1 . Indeed, for thisit suffices to remark that the function t vol A S ( C p ( δ, t )) in increasing, so for τ large enough we have the inequality vol A S ( C p ( δ, τ / > ( k + 1) µ ( D ) . Now applying the adelic Blichfeldt’s principle in A nS (see e.g. Lemma 4, §5.2,[22]) we obtain the required y i ’s. Let us put K = A nS − C p ( δ ) and x i = y − y i for ≤≤ k + 1 . The previous assertion tells us that x , . . . , x k +1 are nonzeroelements of Z nS , namely we have a set of k + 1 integral vectors. But there areonly k elements in C p ( δ ) ∩ Z nS , thus exactly one of them must lie in K p ∩ Z nS , Note that if p is nonarchimedean Q p is not an ordered field neither even partially, soone has to be careful with the meaning of this assertion. The real meaning is arithmeticalrather than geometrical as it can be seen just below. N THE DENSITY OF S -ADIC INTEGERS NEAR SOME PROJECTIVE G -VARIETIES11 call it x ( p ) . By definition x ( p ) is the difference of two elements in C p ( δ, τ / ,thus x ( p ) ∈ ∗ C p ( δ, τ / .Hence for each p ∈ S , we have obtained an nonzero element x ( p ) of Z nS whichlies in K p ∩ ∗ C p ( δ, τ / , the latter subset satisfying the following inclusion K p ∩ ∗ C p ( δ, τ / ⊂ ∆ S ( δ/l ) × (1 , τ ] v ,p ⊕ W p ( α ) × Y q ∈ S \{ p } V q ( α/l ) × Y s / ∈ S ∗ Z p . Finally let us set x = P p ∈ S π S ( x ( p )) where π S is the projection to the S -factor. It is already clear that x is a nonvector vector in Z nS . It remains toverify that x ∈ ∆ S ( δ ) × X S . For the S -components, we have π S ( x ) = X p ∈ S π S ( x ( p )) ∈ ( M ) X p ∈ S π S ( K p ∩ ∗ C p ( δ, τ / ⊂ l ∗ ∆ S ( δ/l ) ⊂ ∆ S ( δ ) . On S side, we isolate the diagonal component in order to obtain π S ( x ) = x ( p ) p + X q ∈ S \{ p } x ( p ) q ∈ (1 , τ ] v ,p ⊕ W p ( α ) × Y q ∈ S \{ p } ( M ) X p ′ ∈ S \{ p } V q ( α/l ) p ′ . Remembering the choice of α > made above in (3), we infer that π S ( x ) ∈ X p × Y q ∈ S \{ p } V q ( α ) ⊂ X S . Hence for any δ > , we can always find a nonzero vector x lying in (∆ S ( δ ) × X S ) ∩ Z nS and this achieves the proof of the lemma. (cid:3) We are now ready to prove the theorem, for this let us fix an ε > . The factthat the f i ’s are homogenous polynomials and thus, in particular, continouswith f i (0) = 0 , implies that the existence of δ ( ε ) > such that ∆ S ( δ ( ε )) ⊂ X εS . Using Lemma 2.1 with δ = δ ( ε ) , one obtains a nonzero S -integral vector x ∈ Z nS such that π S ( x ) ∈ X εS and π S ( x ) ∈ X S . The latter condition meansthat for each p ∈ S and corresponding x p ∈ X p , there exists some u p ∈ U p such that f i,p ( u p x p ) = 0 . Since U p is open there exists g p ∈ U p such that < | f i,p ( g p x p ) | p ≤ ε/ . With the help of the strong approximation theorem, we have seen earlier that U S is contained in Λ ( S ) , and incidentally we find ( γ p ) p ∈ S ∈ Λ S such that forevery p ∈ S and ≤ i ≤ r , one has(4) < | f i,p ( γ p x p ) | p ≤ ε. Consider the projection Λ S ։ Λ S and let (˜ γ ) p ∈ S be a lift of ( γ p ) p ∈ S , that is, ˜ γ p = γ p for every p ∈ S . We claim that y = ˜ γx is the solution of our problem.Indeed, on the one hand the inequalities in (4) imply that π S ( y ) ∈ X εS .On the other hand since π S ( x ) ∈ X εS and X εp is H -invariant with Λ p ⊆ H for every p ∈ S , we deduce that y p = (˜ γ p x p ) ∈ X εp for every p ∈ S , i.e. ∆ p ( δ )0 γ p x p x p X ε X p = X = { f = 0 } ∆ S ( δ ) ⊂ Q nS u p x p γ p x p x p X p = T j { f j,p = 0 } X εS X S ⊂ Q nS Figure 1.
The space between the blue dotted lines is the tubu-lar neighborhood X εS . π S ( y ) ∈ X εS . This allows us to conclude the existence of a nonzero vector y ∈ X εS ∩ Z nS and this finishes the proof of the theorem.3. Proof of Corollary 1.2
Let us consider a pair ( Q s , L s ) s ∈ S over Q S satisfiying all the assumptions ofthe corollary 1.2. For each s ∈ S , we set X s to be the algebraic (projective)variety given by { Q s = L s = 0 } , geometrically this can be seen as the cone Q s = 0 cutted out by the hyperplane of equation L s = 0 in Q ns . It is moresuitable here to think X s as the quadric of equation { Q s | Ls =0 = 0 } , in that waythe assumptions (1) and (2) amounts to say that this quadric { Q s | Ls =0 = 0 } isnondegenerate and contains at least one nonzero vector in Q ns for every s ∈ S .We need to introduce the following map associated to the pair ( Q, L ) over Q S ψ : P ( Q S ) → Sym ( Q nS )( α : β ) αQ + βL . This induces at each place s ∈ S , a (local) map ψ s ( α s : β s ) = α s Q s + β s L s .The assumption (3) says that the range of ψ is in the subspace Sym ir. ( Q nS ) consisting of quadratic forms in Sym ( Q nS ) which are not proportional to ra-tional form over Q S . The complement consisting of quadratic forms over Q nS (resp. Q ns ) which are proportional to a rational form is denoted Sym rat. ( Q nS ) N THE DENSITY OF S -ADIC INTEGERS NEAR SOME PROJECTIVE G -VARIETIES13 (resp. Sym rat. ( Q ns ) ). Case 1.
If we are in the case where ψ s assumes its values in Sym ir. ( Q ns ) forevery s ∈ S , there is nothing to prove. Indeed, the corollary 2.2. in [17] alreadygives the required result.Now we treat the case of interest here. Case 2.
Suppose there exists a place v ∈ S f such that the range of ψ v isin Sym rat. ( Q nv ) . This means concretely that for any not both zero coupleof constants ( α v , β v ) ∈ Q v we have that α v Q v + β v L v is proportional to aquadratic form Q with rational coefficients, thus α v Q v + β v L v = λ v Q forsome λ v ∈ Q ∗ v , in particular ψ v (1 : 0) = Q v and ψ v (0 : 1) = L v are proportionalto a rational form. Thus we can write L v = µ v L where L is a rational linearform and µ v ∈ Q ∗ v and from the definition of Q we have that α v Q v | L v =0 = λ v Q | L =0 and in particular SO ( Q v | L v =0 ) = SO ( Q | L =0 ) . Now for each s ∈ S we denote by H s the stabilizer of the pair ( Q s , L s ) to be the subgroup of SL n ( Q s ) leaving both invariant Q s and L s . Following Lemma 4.1 in [17], since Q v and L v are both proportional to rational forms then we can arrange abasis { w , . . . , w n − , u } consisting of rational vectors with the condition that { L v = 0 } = h w , . . . , w n − i and L v ( u ) = u . Hence we can find a g ∈ SL n ( Q ) such that H v = g − (cid:20) SO ( Q v | L v =0 ) 00 1 (cid:21) g. Let us define the hypersurface defined over Q by X = { Q = L = 0 } = { Q | L =0 = 0 } and set H := g − (cid:20) SO ( Q | L =0 ) 00 1 (cid:21) g. Clearly we have X v = X and H v = H where v is as chosen above. Theform Q | L =0 is nondegenerate thus H is semisimple, it is also isotropic whichimplies that H is noncompact and isotropic at v . Moreover, from the ra-tionality of Q | L =0 we infer that H is defined over Q . Therefore the triple ( X , Q | L =0 , H ) satisfies all the conditions of Theorem 1.1 unless the con-nectedness for H which is not ensured at all. To remedy to this situationwe can consider the connected component of the identity of H which we de-note H +0 . The point is that H +0 is now connected but it is not obvious thatthe other properties (isotropy and rationality) are preserved by performingthis operation. The key fact is that H +0 has finite index in H thus since H +0 is still noncompact and therefore isotropic, in addition H +0 is defined over Q since H is (see e.g. Prop. 1.2(b), [4] and also 2.3.2 [19] and remark 2just after). A last remark concerns the conservation by the connected com-ponent under the central isogeny. This point is quite crucial since at somestage in the proof of the Theorem 1.1 we need to pass to the universal cover-ing. The fact that π : f H → H is an isogeny defined over Q , we have that H +0 = π ( f H ) + = π ( f H ) (see e.g. Cor. 1.4 (b), [4]). Hence π induces an central isogeny π + : f H → H +0 where f H is simply connected. In fact, thelatter subgroup is explicitely given by f H = g − (cid:20) Spin( Q | L =0 ) 00 1 (cid:21) g. To sum up, given an arbitrary ε > , the triple ( X , Q | L =0 , H +0 ) satisfies allthe conditions of Theorem 1.1 for S = { v } thus there exists a nonzero x ∈ Z nS such that x ∈ X εS i.e. | Q s ( x ) | s < ε and | L s ( x ) | s < ε for each s ∈ S . (cid:3) Proof of Corollary 1.3
We proceed in the same way as the previous corollary. Let us considera Q quadratic form and a linear map M = ( L , . . . , L r ) satisfying all theassumptions of the Corollary 1.3. For each s ∈ S , we define the followingvariety X s = V ( Q s , L ,s , . . . , L r,s ) defined over Q ns , for pratical reasons it ismore suitable to see this variety as X s = V ( Q s | M s =0 ) . Geometrically the locusof X s is defined by a quadratic { Q s = 0 } cutted out by the intersection ofthe hyperplanes of equations { L i,s = 0 } (1 i r ) , the later intersection isjust the kernel of M s i.e. X s = V ( Q s | ker M s ) . It is assumed that rank M s = r for every s ∈ S , thus dim ker M s = n − r , that is to say, L ,s , . . . , L r,s arelinearly independent over Q s . If we denote by B s the symmetric bilinear formassociated to Q s , we can consider the following orthogonal decompositon ofthe quadratic space ( Q ns , Q s ) with respect to B : ( Q ns , Q s ) = (ker M s , Q s | ker M s ) M ((ker M s ) ⊥ , Q s | (ker M s ) ⊥ ) and let { e , . . . , e n } a basis of Q ns adapted to this decompostion above, that is, M s = h e , . . . , e n − r i and M ⊥ s = h e n − r +1 , . . . , e n i such that B ( e k , e l ) = 0 for all k n − r and n − r + 1 k n . Now for any s ∈ S , we set the followingsubgroup of SL n ( Q s ) given in the adapted basis by H s = (cid:20) SO ( Q s | ker M s ) 00 I r (cid:21) . We claim that H s leaves invariant both Q s and M s for each s ∈ S . Indeed let h ∈ H s , and A an element of SO ( Q s | ker M s ) such that h = (cid:20) A I r (cid:21) . Note that, as given, the range of A is necessarily within the subspace ker M s .Let x be a vector in Q ns which decomposes into x = x + x with x ∈ ker M s and x ∈ (ker M s ) ⊥ . Therefore L s ( hx ) = L s ( h ( x , x ) t ) = L s ( Ax ⊕ x ) = L s ( Ax ) + L s ( x ) = 0 + L s ( x ) = L s ( x ) + L s ( x ) = L s ( x ) , thus L s is H s -invariant. In the other hand, using the same decomposition for x we get Q s ( hx ) = Q ( Ax ⊕ x ) = Q s | ker M s ( Ax ) + Q s ( x ) = Q s ( x ) + Q s ( x ) = Q ( x ) . N THE DENSITY OF S -ADIC INTEGERS NEAR SOME PROJECTIVE G -VARIETIES15 In particular, the claim shows that H s acts linearly on X s = V ( Q s | ker M s ) . Nowgiven any constants α , . . . , α r in Q S not all zero, the form α ,s L ,s + . . . + α r,s L r,s is irrational for every place s ∈ S \{ v } and proportional to a rational form for s = v . Let us set f := Q v + α ,v L ,v + . . . + α ,v L r,v , it is clear that f isproportional to a rational form and a suitable choice of constants allows us toassume that f has rational coefficients. The crucial fact is that f | ker M v = Q v | ker M v is also a quadratic form with rational coefficients since ker M v is a Q -subspace. Thus if we put H = (cid:20) SO ( f | ker M v ) 00 I r (cid:21) then H = H v is a algebraic subgroup of SL n ( Q v ) which is defined over Q andwhich acts on X = X v . Moreover, for the same reasons as in the previouscorollary, H +0 is a connected semisimple algebraic subgroup which is isotropicat v since Q v | ker M v is isotropic, in particular H +0 has no compact factors. Weobtain a rational triple by taking ( X , f | ker M v , H +0 ) , since it satisfies all theconditions of Theorem 1.1 for S = { v } , we infer that for any ε > , we canfind a nonzero x ∈ Z nS such that | Q s ( x ) | s < ε and | L i,s ( x ) | s < ε for each s ∈ S and i r .To conclude, one has to remark that since Q s is rational and x ∈ Z nS = ( A S ∩ Q ) n , Q s ( x ) ∈ Q for every s ∈ S . Since Q is discrete in every completion, wededuce that if ε is small enough we can find x ∈ Z nS − { } such that Q s ( x ) = 0 and | L i,s ( x ) | s < ε for each s ∈ S and i r . (cid:3) Existence of rational triples ( X , H , f ) for general varieties In this section we adress some remarks concerning the class of varieties X which falls into the conditions of the theorem. The varieties involved in thetheorem are called complete intersection in the litterature and they have beensubject to extensive research until now and still many problems remains openconcerning those projective varieties. From our point of view, we are moreconcerned with the invariant theory of the space of homogeneous polynomials.In order to apply the main theorem one has to find a rational triple ( X , H , f ) such that X S (resp. H S ) can be splited in the form X S = V ( f ) × X S (resp. H S = H × H S ) without loss in generality we assume that S = { v } forsome v ∈ S f thus S = S \{ v } . In particular H acts rationally on the Q -hypersurface X = V ( f ) . Bounds on the degrees ( d , . . . , d r ) of the generators of the ideal of X . • A first constrain is the equality X v = X , that is, in algebraic terms V ( f ,v , . . . , f r,v ) = V ( f ) . Applying Hilbert’s Nullstellensatz in an algebraic closure of Q v yields p ( f ) = q ( f ,v , . . . , f r,v ) where √ J is the radical of an ideal J in Q v [ x , . . . , x n ] , in particular thereexists an integer ρ > and (homogeneous) polynomials P , . . . , P r over Q v such that(5) f ρ = P f ,v + . . . + P r f r,v Let us denote by N i (resp. d i ) the total homogeneous of P i (resp. f i,v ) for each i r . Thus the previous equality reads in terms of degrees as(6) ρ deg f = max i r { N i + d i } . Let us assume that the degrees are ordered as follows N > N > . . . ≥ N r and d > d > . . . > d r , then (6) reads(7) ρ deg f = N + d . When d i = 2 (1 i r ) , upper bounds for ρ can be effectively computed, thefollowing sharp estimates for ρ are due to J. Kollár (Corollary 1.7, [14].)(8) ρ ≤ (cid:26) d d . . . d r if r ≤ nd d . . . d n − d r if r > n and for each i r (9) N i + d i = deg( P i f i,v ) ≤ (cid:26) (1 + d ) d d . . . d r if r ≤ n (1 + d ) d d . . . d n − d r if r > n where d denotes deg f . To sum up, saying that X v equals the hypersurface X = V ( f ) amounts to find some polynomials ( P j ) i r such that f ρ = P f ,v + . . . + P r f r,v where ρ satisfies condition (8) and after (9) we get that for every i r ,(10) ≤ deg( P j ) ≤ (cid:26) (1 + d ) d d . . . d r − d j if r ≤ n (1 + d ) d d . . . d n − d r − d j if r > n. The equidimensional case : If we assume that d = . . . = d r = d = 2 , we obtainthe following bounds for every i r ,(11) ≤ deg( P j ) ≤ (cid:26) (1 + d ) d r − d if r ≤ n (1 + d ) d n − d if r > n. If we want that the right hand in (6), that is, f ρ to have only degree d , thennecessarily ρ = 1 and from (11) we get that the polynomials P i should beconstant polynomials α i (1 i r ) (12) f = α f ,v + . . . + α r f r,v . • Rationality conditions X v = X = V ( f ) with f rational. The relation N THE DENSITY OF S -ADIC INTEGERS NEAR SOME PROJECTIVE G -VARIETIES17 (5) shows that if the polynomial f has rational coefficients then some linearcombination (over Q v ) of the f i,v ’s must be rational. In particular, this and(12) explain why the condition (3) in both corollaries 1.2 and 1.3 is necessary. From complete intersections towards invariant group of the ideal of definition.
The main issue is that given a variety X defined over a field of characteristiczero K , say a complete intersection, to find the largest subgroup of G = SL n | K which acts trivially on the ideal of definition I X . For instance, let us be givena complete intersection X = V ( f , . . . , f r ) where f , . . . , f r are homogeneouspolynomial of degrees d ≤ . . . ≤ d r . The ideal of definition of X is given by I X = h f , . . . , f r i . The central role is played by the stabilizer H of the idealwhich is defined to be H s = \ i r { g ∈ G | g.f i = f i } . The ideal stabilizer H s obviously does act on X s for every s ∈ S , and it givesan action of H S on X S induces by the usual of G = PSL n on the vectorspace of homogeneous polynomials in Q s [ x , . . . , x n ] . In particular H s contains Aut G ( X ) the group automorphism of X under G which is the pointwise sta-blizer of X s under G .The ideal stabilizer is an algebraic subgroup of G given by the following equa-tions in the variables ( g i,j ) f k ( gx ) = f k ( x ) and det( g i,j ) − .Let us try to solve those equation with g = ( g i,j ) i,j , for this let us explicit thecoefficients of the f k and assume that they are of form f k ( x , . . . , x n ) = X | α | = d k a ( k ) α x α . . . x α n n . Therefore we have n equations which takes places in K [ x , . . . , x n ] ( d k ) [( g i,j ) i,j ] X | α | = d k a ( k ) α ( n X j =1 g j x j ) α . . . ( n X j n =1 g nj n x j n ) α n − x α . . . x α n n ! = 0 . X | α | = d k a ( k ) α ( X | β | = α (cid:18) α β (cid:19) ( g x ) β . . . ( g n x n ) β n ) . . . ( X | β | = α n (cid:18) α n β (cid:19) ( g n x ) β . . . ( g nn x n ) β n )= X | α | = d k a ( k ) α x α . . . x α n n . We do not need to go further to observe that such compuations unless we aredealing with low degrees (i.e. d = 2 , ) leads to tremendous compuations andtrying to obtain the ideal stabilizer in such a way is quite compromized. Us-ing elimination when d = 2 , one could provide the required invariant groupswhere the solutions g = ( g i,j ) are given by functions of the coefficients a ( k ) α of the f k ’s. A last step would be to decide if the invariant group is com-pact/semisimple/isotropic which could ask some additional efforts, this task iscrucial in order to apply our main theorem. Final Comments
The main problem is to determine the ideal stabilizer H of a given projectivevariety X = V ( f , . . . , f r ) . This consists in finding a subgroup H such that K [ x , . . . , x n ] H = K [ f , . . . , f r ] . This question is dual to the
Invariant theory , indeed in invariant theory we fixa group G and we try to understand the ring of invariants k [ X ] G of the variety X . This theory has reached a good level of maturity, notably with the rise ofthe geometric invariant theory (G.I.T. [24]) and more recently with the theoryof prehomogeneous spaces for which we hope that we could derive an analogof the work of A. Yukie ([32]) using our main theorem. In terms of category,the invariant theory is an attempt to understand the image of the functor F ( X ) : Grps → Rings G → k [ X ] G . The so called
Inverse Invariant theory consists the dual situation, namelythe image of the following functor given a fixed projective variety F ( X ) ∗ : Rings → Grps A → G where we define the functor F ( X ) ∗ as follows F ( X ) ∗ ( A ) = G if A = k [ X ] G .As far as we know, this theory has been only developped for finite groups andin particular for linear groups over finite fields . The latter has been studiedin detail by Neusel using tools from algebraic topology such as Steenrod op-erations. It should be very interesting to have such a theory for linear groupsover fields in null characteristic and to have a criterion which ensures that thegroup obtained is reductive or/and noncompact. If we could have such theoryit would open a large range of applications and more particularly for solvingthe diophantine inequalities in (1). N THE DENSITY OF S -ADIC INTEGERS NEAR SOME PROJECTIVE G -VARIETIES19 References [1] Bombieri E. and Gubler W.
Heights in Diophantine Geometry
Cambridge Univer-sity Press (2006).[2] A. Borel
Values of indefinite quadratic forms at integral points and flows on spacesof lattices , Bull. Amer. Soc. , , no 2, (1995) 184-204.[3] A. Borel Introduction aux groupes arithmetiques , Hermann, Paris (1969), 126 p.[4] A. Borel
Linear algebraic groups , 2nd. enl. ed. Graduate Texts in Mathematics, ,Springer-Verlag New-York Inc. (1991).[5] A. Borel and G. Prasad,
Values of isotropic quadratic forms at S -integral points, Compositio Math. (1992) 347-372.[6] A. Borel and J. Tits, Homomorphismes "abstraits" de groupes algebriques simples
Ann. of Math. (1973) 499-571.[7] S.G. Dani, Simultaneous diophantine approximation one quadratic forms and lin-ear forms , J. Mod. Dynam., Vol (2008) 129-138 (Special issue in honor to G.A.Margulis).[8] S.G. Dani and G.A. Margulis, Orbit closures of generic unipotent flows on homoge-neous spaces of
SL(3 , R ) , Math. Ann. Vol (1990) 101-128.[9] A. Gorodnik, Values of pairs involving one quadratic form and one linear form at S -integral points , Trans. Amer. Soc., Vol (2004) 200-217.[10] A. Ghosh, A. Gorodnik, A. Nevo Diophantine Approximation exponents in homoge-neous varieties , Recent trends in Ergodic Theory and dyn. systems, Conf. in honnorof S.G. Dani, Contemp. Math. Amer. Math. Soc. Vol (2015) 181-200.[11] A. Ghosh, A. Gorodnik, A. Nevo,
Diophantine Approximation and Automorphicspectrum , I.R.M.N. Vol (2013) 5002-5058.[12] A. Ghosh, A. Gorodnik, A. Nevo, Metric Diophantine Approximation on homoge-neous varities , Compositio Math. Vol (2014) 1435-1456.[13] A. Ghosh, A. Gorodnik, A. Nevo,
Optimal density for values of generic polynomialmaps , preprint, (2018) arxiv.1801.01027.[14] J. Kollar,
Sharp Effective Nullstellensatz , J. Amer. Soc., Vol , No. 4 (1988) 963-975[15] D. Kleinbock, N.Shah, A. Starkov, Dynamics of subgroup actions on homogeneousspaces of Lie groups and application to number theory , Ch. 11. in Handbook ofdynamical systems, Vol A, Edited by B. Hasselblatt and A. Katok, Elsevier Science(2002) 1-109.[16] S. Lang
Algebraic number theory, second ed.
Graduate texts in Mathematics Vol. , Springer-Verlag, New-York 1994.[17] Y. Lazar,
Values of pairs involving one quadratic form and one linear form at S -integral points , J. Number Theory, Vol (2017) 200-217.[18] C.R. Matthews, L.N. Vaserstein, B. Weisfeiler, Congruence properties of Zariski-dense subgroups I , Proc. London Math. Soc., Vol no. 3 (1984), 514-532.[19] Margulis G.A. Discrete Subgroups of Semisimple Lie Groups , Springer Verlag,(1991).[20] Margulis G.A.
Indefinite quadratic forms and unipotent flows on homogeneousspaces , Dynamical Systems and Ergodic Theory, vol 23, Banach center Publi. PWN-Polish Scientific Publ. , Warsaw, 1989, (399-409).[21] G.A Margulis and G. Tomanov,
Invariant measures for actions of unipotent flowsover local fields on homogeneous spaces,
Invent. Math. (1-3) (1994) 347-392.[22] McFeat.
Adelic geometry of numbers
Dissertationes Math. Rozprawy Mat. 88 (1971)pp. (1-49).[23] M.D. Neusel,
Inverse Invariant Theory and Steenrod operations , Mem. Amer. Soc.Vol n0. 692 (2000). [24] D. Mumford, J. Forgaty and F. Kirwan
Geometric Invariant Theory , 3rd enlargeded. Ergebnisse der Mathematik und ihrer Grenzgebiete Vol Springer-Verlag BerlinHeidelberg (1994).[25] M.V. Nori,
On subgroups of GL n ( F p ) , Invent. Math. Vol (1987) 257-275.[26] V.P. Platonov and A.S.Rapinchuk, Algebraic groups and number theory , Volume , Academic Press (1994) 614p.[27] A.S. Rapinchuk,
Strong approximation for algebraic groups , Thin groups and super-strong approximation, MSRI Publications Volume , (2013), 269-298.[28] M. Ratner, Raghunathan’s conjectures for p -adic Lie groups , I.R.M.N., No.5 (1993),141-146.[29] O. Sargent,
Density of values of linear maps on quadratic surfaces , J. Number The-ory, Vol (2014) 363-384.[30] B.N. Venkataramana,
Zariski dense subgroups of arithmetic groups , J. Algebra., Vol (1987) 325-339.[31] B. Weisfeiler,
Strong approximation for Zariski-dense subgroups of semisimplegroups , Ann. of Math., Vol (1984) 271-315.[32] A. Yukie,
Prehomogeneous spaces and Ergodic theory I , Duke Math. J. Vol no. 1(1997), 123-147. E-mail address ::