Cohomology of Arithmetic Groups, L-Functions and Automorphic forms

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Keywords: Eisenstein series ; automorphic cohomology ; automorphic L-functions ; arithmetic groups. Export Citation. User Account Log in Register Help. Search Close Advanced Search Help. My Content 1 Recently viewed 1 Eisenstein series, coh Show Summary Details. More options …. Forum Mathematicum. Managing Editor: Bruinier, Jan Hendrik. See all formats and pricing. Online ISSN See all formats and pricing Online. Prices are subject to change without notice.

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Frontmatter Erratum to The distribution of the logarithm in an orthogonal and a symplectic family of L-functions [Forum Math. Volume 30 Issue 6 Nov , pp. We can circumvent this by a combination of global and local means. The global tool is simply the following observation.

Observation Let II be as in Theorem 2. Then II is quasi-automorphic as in those theorems. The only thing to observe is that if. So, by either Theorem 2. Now, if we begin with n automorphic on H A , we will take T to be the set of finite places where n, is ramified. For applying Theorem 2. We will now take 77 to be highly ramified at all places v E T. So at v E T our twisting representations are all locally of the form unramified principal series highly ramified character.

We now need to know the following two local facts about the local theory of L-functions for H. Once again, for these applications it is crucial that the local theory of L-functions is sufficiently developed to establish these results on the local y-factors.

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Both of these facts are known for GL,, the multiplicativity being found in [33] and the stability in To utilize these local results, ;hat one now does is the following. At the places where a, is ramified, choose II, to be arbitrary, except that it should have the same central character as a,.

This is both to guarantee that the central character of It is the same as that of r and hence automorphic and to guarantee that the stable forms of the y-factors for r, and II, agree. Choose our character q so that at the places v E T we have that the L- and y-factors for both rv8 q,, and n, q, are in their stable form and agree.

We then twist by 'T 8 q for this fied character q. So at the places v E T we have. SO L II x r , s is nice and we may proceed as before. We have, in essence, twisted all information about a and I I a t those v E T away. In essence, the converse theorem fills in a correct set of data a t those places in T to make the resulting global representation automorphic.

Theorems 2. It has had many applications which we would now like to catalogue for completeness sake.

In their original paper [31], Jacquet , Piatetski-Shapiro, and Shalika used the known holomorphy of the Artin L-function for three dimensional monomial Galois representations combined with the converse theorem to establish the strong Artin conjecture for these Galois representations, that is, that they are associated to automorphic representations of GL3. Gelbart and Jacquet used this converse theorem to establish the symmetric square lifting from GL2 to GL3 [14]. Jacquet, Piatetski-Shapiro and Shalika used this converse theorem to establish the existence of non-normal cubic base change for GL2 [32].

These three applications of the converse theorem were then used by Langlands [43] and Tunnel1 [63] in their proofs of the strong Artin conjecture for tetrahedral and octahedral Galois representations, which in turn were used by Wiles [67] Patterson and Piatetski-Shapiro generalized this converse theorem to the three fold cover of GL3 and there used it to establish the existence of the cubic theta representation [47], which they then turned around and used to establish integral representation for the symmetric square L-function for GL3 [48].

More recently, Dinakar Ramakrishnan has used Theorems 2. The basic properties of this L-function are known through the work of Garrett [13], Piatetski-Shapiro and M l i s [52], Shahidi [58], and Ikeda [25], [26], [27], [28] through a combination of integral representation and Eisenstein series techniques. Rarnakrishnan himself had to complete the theory of the triple product L-function. Once he had, he was able to apply Theorem 2. After he had established the tensor product lifting, he went on to apply it. We should note that Ramakrishnan did not handle the ramified places via highly ramified twists, as we outlined above.

Instead he used an ingenious method of simultaneous base changes and descents to obtain the ramified local lifting from GL2 x GL2 to GL4. If v is Archimedean, we take II, as the local Langlands lift of a, as in [3, If v is non-Archimedean and rr, is unrarnified, we take II, as the local Langlands lift of a, as defined via Satake parameters [3,40]. If v is finite and rr, is ramified, we take II. To show that II is a weak Langlands lifting of? The Rankin-Selberg theory of integral representations for these L-functions has been worked out by several authors, among them Gelbart and Piatetski-Shapiro [15], Ginzburg [17], and Soudry [61, We know that for these L-functions most of the requisite properties for the lifting are known.

The basics of the local theory can be found in 17, 61, The multiplicativity of gamma is due to Soudry [61, The stability of gamma was established for this purpose in [8]. As for the global theory, the meromorphic continuation of the L-function is established in [15], [17]. The global functional equation, a t least in the case where the infinite component rr, is tempered, has been worked out in conjunction with Soudry. The remaining technical difficulty is to show that L a x T, s is entire and bounded in strips for T E Ts n - 1 8 q.

This L-function has been studied by Jacquet and Shalika [37] from the point of view of Rankin-Selberg integrals and by Shahidi by the method of Eisenstein series. We know that the JacquetShalika version is entire for T E Ts n - 1 8 q , but we know that it is the Shahidi version that normalizes the Eisenstein series and so controls the poles of L a x T, s. Gelbart and Shahidi have also shown that, away from any poles, the version of the exterior square L-function coming from the theory of Eisenstein series is bounded in vertical strips [16]. So, we would essentially be done if we could show that these two avatars of the exterior square L-function were the same.

This is what we are currently pursuing.. We should point out that Ginzburg, Rallis, and Soudry now have integral representations for L-functions for Spz, x GL, for generic cusp forms [19], analogous to the ones we have used above for the odd orthogonal group. What should be true about the amount of twisting you need to control in order to determine whether II is automorphic?

There are currently no conjectural extensions of Theorem 2. However conjectural extensions of Theorems 2. The most widely believed conjecture, often credited to Jacquet, is the following. Conjecture Let us briefly explain the heuristics behind this conjecture. The idea is that the converse theorem should require no more than what would be true. Now, if II were automorphic but not cuspidal, then still L II x r , S should have meromorphic continuation, be bounded in vertical strips away from its poles, and satisfy the functional equation.

The above conjecture states that, all other things being nice, this is the only obstruction to II being cuspidal automorphic. There should also be a version with limited ramification as in Theorem 2. The most ambitious conjecture we know of was stated in [4] and is as follows.

For n 4 we can no longer expect to be able to take II' to be II. All of these cannot belong to the space of cusp forms on GL4 A , since the space of cusp forms contains only a countable set of irreducible representations. For example, Kim and Shahidi have have shown that for non-dihedral cuspidal representations n of GL2 A the symmetric cube L-function is entire along with its twists by characters [38]. F'rom Conjecture This would produce a weak symmetric cube lifting from GL2 to GL4. If these conjectures are to be attacked along the lines of this report, the first step is carried out in Section 4 above.

What new is needed is a way to push the arguments of Section 6 beyond the case of abelian Y,. The most immediate extension of these converse theorems would be to allow the L-functions to have poles. One would then try t o invert these interpretations along with the integral representation. We hope to pursue this in the near future. This would be the analogue of Li's results for GL2 [43, If one could establish a converse theorem for GL, allowing an arbitrary finite number of poles, along the lines of the results of Weissauer and Raghunathan [53], these would have great applications.

Finiteness of poles for a wide class of L-functions is known from the work of Shahidi , but to be able to specify more precisely the location of the poles, one usually needs a deeper understanding of the integral representations see Rallis [54] for example. A first step would be simply the translation of the results of Weissauer and Raghunathan into the representation theoretic framework.

An interesting extension of these results would be converse theorems not just for GL, but for classical groups. The earliest converse theorem for classical groups that we are aware of is due to Mad3 [46]. He proved a converse theorem for classical modular forms on hyperbolic n-space? He inverts the Mellin transform of holomorphic Siegel modular forms on the Siegel upper half space 3, but does not achieve a full converse theorem. For Sp4 a converse theorem in this classical context was obtained by Imai [29], extending Koecher7sinversion in this case, and requires twisting by M a d forms and Eisenstein series for G4.

It seems that, within the same context, a similar result will hold for Sn,. Duke and Imamoglu have used Imai7s converse theorem to analyze the Saito-Kurokawa lifting [12]. It would be interesting to know if there is a representation theoretic version of these converse theorems, since they do not rely on having an Euler product for the L-function, and if they can then be extended both to other forms on these groups as well as other groups. Another interesting extension of these results would be to extend the converse theorem of Patterson and Piatetski-Shapiro for the three-fold cover of GL3 [47] to other covering groups, either of GL, or classical groups.

References [I] H. Bass, K-theory and stable algebra, Publ. IHES 22 , Bass, J. Milnor, and J-P. IHES 33 , Gelbart and F. Shahidi, Boundedness in finite vertical strips for certain L-functions, preprint.

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Ginzburg, 1. Piatetski-Shapiro, and S. Rallis, L-functions for the orthogonal group, Memoirs Amer. Ginzburg, S. Rallis, and D. Soudry, L-functions for the symplectic group, Bull. France , IHES 79 Gurevic, Determining L-series from their functional equations Math. USSR Sbornik 14 , Hamburger, h e r die Riemannsche Funktionalgleichung der 5funktion, Math.

Harris and R. Taylor, On the geometry and cohomology of some simple Shimura varieties, preprint, Annalen , Henniart, Une preuve simple des conjectures de Langlands pour GL n sur un corps p-adique, preprint, Ikeda, On the functional equations of the triple L-functions, J. Kyoto Univ. Pure Math. Ikeda, On the location of poles of the triple L-functions, Compositio Math. Conrey and D. Farmer, An extension of Hecke's Internat. Duke and 0. Imamoglu, A converse theorem and the SaitoKurokawa lift, Internat.

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Garrett, Decomposition of Eisenstein series; Rankin triple products Ann. Gelbart and H. Gelbart, 1. Ikeda, On the Gamma factors of the triple L-functions, 11, J. Imai, Generalization of Hecke's correspondence to Siege1 modular forms, Amer. Jacquet and R. Jacquet, 1. Piatetski-Shapiro, and J. M a d , h e r eine neue Art von nichtanalytischen automorphen Funktionen une die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Jacquet; 1.

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Shalika Relevement cubique non normal, C. Paris Ser. Shalika Rankin-Selberg convolutions, Amer. Hamburg 16 , Jacquet and J. Shalika On Euler products and the classification of automorphic representations, I, Amer. Patterson and 1. Piatetski-Shapiro, A cubic analogue of cuspidal theta representations, J. Pures and Appl.

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Shalika, A lemma on highly ramified r-factors, Math. Gelfand, ed. Kim and F. Koecher, a e r Dirichlet-Reihen mit Funktionalgleichung, J. Studies 96 Princeton University Press, Piatetski-Shapiro and S. Rallis Rankin triple L-functions, Compositio Math. Raghunathan, A converse theorem for Dirichlet series with poles, C. Paris, S6r. Math , Razar, Modular forms for r o N and Dirichlet series, Trans.

Riemann, h e r die Anzahl der Primzahlen unter einer gegebenen Grosse, Monats. Berliner Akad. Shahidi, On the Ramanujan conjecture and the finiteness of poles of certain L-functions, Ann. Shalika, The multiplicity one theorem for GL,, Ann. Tunnell, Artin's conjecture for representations of octahedml type, Bull. Notes Math. The purpose of this note is threefold: i to describe this conjecture in the simplest non-trivial situation: the case when F is a real quadratic field; ii to mention some recent numerical work of Goto [12] and Hiraoka in support of the conjecture; this work nicely compliments the computations of Doi, Ishii, Naganuma, Ohta, Yamauchi and others, done over the last twenty years cf.

Section 2. The conjectures in [7] of Doi, Hida and Ishii go back to ideas of Doi and Hida recorded in the unpublished manuscript [6]. Some of the material in this note appears, at least implicitly, in [7]. We wish to thank Professor Hida for useful discussions on the contents of this paper. J c IF will denote a subset of IF. There are three spaces of cusp forms that will play a role in this paper. Let k 2 2 denote a fixed even integer. We recall this now.

Finally, let. For more details the reader may refer to [17]. Let Z denote the center of G, and let 2, denotk the center of G,. Here, the Fourier coefficients c g, f only depend on the fractional ideal generated by the finite part gf of the idele g. Moreover, one may check that m tt c m, f vanishes outside the set of integral ideals. Each f E SkjIF OF may be realized as a tuple of functions fi on Z satisfying the usual transformation property with respect to certain congruence subgroups ri defined below.

Then Y K is the set of complex points of a a quasi-projective variety defined over 0. These algebras are constructed in the usual way as sub-algebras of the algebra of Z-linear endomorphisms of the corresponding space of cusp forms generated by all the Hecke operators. There is a one to one correspondence between simultaneous eigenforms f E S of the Hecke operators normalized so that the 'first' Fourier the set of Z-algebra homomorphisms X coefficient is I , and Spec T of the corresponding Hecke algebra T into Finally, the Fourier expansion of f induces the usual Fourier expansion of the f,.

The subfields K t of generated by the images of such homomorphisms, that is the field generated by the Fourier coefficients of f are called Hecke fields. Since T is of finite type over Q, Kf is a number field. Moreover, it is well known that Kf is either totally real or a CM field. This would contradict the non-abelianess of the Galois representation when. We leave the precise argument to the reader. In any case, we have f f,, from which it follows that Kf is a CM field. As we have seen, eigenforms in S- do not have 'complex multiplication' if by the term complex multiplication one understands that j has the same eigenvalues outside the level, as the twist of f by its nebentypus.

Such a phenomena might be called 'generalized complex multiplication' or better still, 'genus multiplication', since it is connected to genus theory. Let us give an example which was pointed out to us by Hida. Briefly, this stat? Using Galois representations, and their associated Artin L-functions, we give here a heuristic reason as to why the above analytic properties should hold. Then, using standard properties of Artin L-functions, we have. The spaces Sf, S- are intimately connected to the space S via base change.

This heuristic argument was carried out by Doi and Naganuma in [8] and [24], in a purely analytic way with no reference to Galois representations. In any case, from now on we will assume the process of base change as a fact. Let us denote the two base change maps f e f by. In any case, we have f f,, from which it follows that Kf is a C M field. As we have seen, eigenforms in S- do not have 'complex multiplication' if by the term complex multiplication one understands that f has the same eigenvalues outside the level, as the twist of f by its nebentypus.

We shall come back to the phenomena of 'genus multiplication' later. This heuristic argument was carried out by Doi and Naganuma in [8] and [24],in a purely analytic way with no reference to Galois representations. In any case, from now on we will assume the proces of base change as a fact. These maps are defined on normalized eigenforms f E S f , and then extended linearly to all of S f.

By the perfectness of the pairing 3. At unramified primes we have:. Indeed, a comparison of Euler products in 4. Before we proceed further we would like to investigate which elliptic cusp forms can base change to the space S. Let us start with some observations. Similarly, twists of forms in S- by XD,or by genus characters, also base change to S, but in this case the twisting operation preserves the spaces. We have P. On the other hand, L s, f xD does not have any Euler factors at the primes plD.

Since both L s, f? Equivalently, one might have to replace the cusp form f xDby the unique normalized newform f ' which has the same Hecke eigenvalues as f xDoutside D. In this case, f ' is just the newform f,, defined above 3. Thus, the following Euler factors need to be added to L s, f xD at the primes pl D:. Proof Let pf denote the Xadic representation attached to f? The identity 4. A comparison of the determinant on both sides of 4. On the other hand, since pi! This yields a contradiction since the right hand side of 4. Thus we may assume that ai 2, for all i.

Khare has pointed out that an alternative argument may be given using the local Langlands' correspondence. Indeed if pilJN, for some i, then the local representation at pi of the automorphic representation corresponding to f would be Steinberg. S u p pose now that in addition. The left hand side of 4.

This shows that the exact level is one. Thus f respectively f' is the twist of a level one form, by the genus character XD, respectively xD2. Then we have that DIN. On the other hand, Lp s,f' has degree at most 1 in p-'. This is because f' must again have p in its level, since p divides the conductor of its nebentypus. Thus we get the usual contradiction, since the right hand side of 4.

Then, again an argument involving Euler factors yields a contradiction. Indeed, the same theorem of Atkin-Li shows that pi 1 N', so Lpi s, f ' has degree at most 1in pi-'. Presumably, an alternative argument using the local Langlands' correspondence could be given here as well, but we have not worked it out. The above discussion shows that we may further assume that N', plp2 - -.

Now say that ql ,q2,. We have. For simplicity, we now make two assumptions for the rest of this article. Thus 4. Under 4. The formulas 4. Consequently o fixes the minimal primes in I- corresponding to base-change eigenforms, and permutes the minimal primes corresponding to non-basechange forms amongst themselves. We now assume that the algebra I- is 'F-proper' cf. Let h denote a fixed element of this orbit. Since o must preserve the corresponding minimal prime ideal ker Xh of 7 we see that there must exists an automorphism T of Kh such that. Remark 4.

The former inclusion is usually expected to e an equality. But the latter inclusion is never an equality since Kg is a totally real field, whereas Kg is a CM field. This phenomena will be reflected in some of the numerical examples of Section 8; see also Remark For the readers convenience we recall the definition of the imprimitive adjoint L-function attached to f. For each prime p, define a, and Pp via. Let K: denote the subfield of Kh fixed by T. Thus we have the following decompositions of finite semisimple commutative Qalgebras :.

When f E S- we omit the factors corresponding to the primes p with p D. Thus the value L l, Ad f is a critical value in the sense of Deligne and Shimura. Similarly, we define the twisted adjoint L-function by. Here [ ] denotes a representative of a Galois orbit, and all the decompositions are induced by the algebra homomorphisms of 3. This has been checked numerically, at least for weights k 5 cf. We can now finally state the main conjecture. Define the sets: Again, the Hecke algebra 7 acts on both sides, and 6 is equivariant with respect to this action.

Let M and [f, f ] denote the eigenspaces with respect to these actions. Then as before, for a p. Conjecture 6. The following conjecture will be crucial for the analysis of congruences in terms of adjoint L-values. It relates the Eichler-Shimura periods of a OF. In the following sections we will sketch how one might attempt to prove the main conjecture. Briefly the idea is this: The first step is to show that a prime p lies in N if and only if there is a congruence mod p between a base-change form in S and a non-basechange form in S.

To see this, one first relates untwisted adjoint L-values over Q respectively F to congruence primes. This has been worked out in the elliptic modular case by Hida in a series of papers , [15] and It is indeed possible that this image may not be the full ring of integers of Kh, in which case one should really consider the relative discriminant of these 'smaller' orders.

We ignore the complications arising from such a possibility in the sequel. A natural identity between all the adjoint L functions involved, along with the period relations in Conjecture 6. The second step identifies the congruence primes above with the primes in D see Proposition Using the simplicity of the non-basechange part of the Hecke algebra recall the assumption made in 4. The converse is more difficult, but would follow from a weak version of Serre's conjecture on the modularity of mod p representations.

We emphasize again that the plan of proof outlined above is due essentially to Hida, and has been learned from him through his papers, or through conversations with him. Before elaborating on the details of the 'proof' of the main conjecture we first would like to give a sample of some numerical examples in support of it.

These computations are but a small sample of those done by Doi and his many collaborators Ishii, Goto, Hiraoka, and others, over the last twenty years. If f E Sf, set. The computations in the - case, and the Hecke fields of the F-proper part of the Hecke algebra can be found in the table in Section 2. We refer the reader to that table and to the references in [7] for other numerical examples in the - case.

The method of computation in this case relies on a formula of Zagier expressing the twisted adjoint L-values of f E S- in terms of the Petersson inner product f, 4 for an explicit cusp form 6 E S see Theorem 4 and equation 90 of [32]. These authors used instead an identity of Hida see Theorem 1. In this section we recall how untwisted adjoint L-values are related to congruence primes.

We treat the elliptic modular00case first. Fix a prime p. For the definition of the set Sinvariant, and for more general results, we refer the reader to [lo]. Establishing a converse to Theorem 9. However a proof should now be accessible cf. The following beautiful theorem of Hida completely characterizes congruence primes for f as the primes dividing a special value of the adjoint L-function of f : Theorem 9. T h e n p is a congruence prime for f if and only i f. A partial result in the Hilbert modular situation has been worked out in [lo]. There we establish one direction, namely that almost all prime that divide the corresponding adjoint L-value are congruence primes.

Moreover, we show that the primes that are possibly omitted are essentially those that 'divide' the fundamental unit of F. More precisely, we have:. In this section we outline a method for establishing Conjecture 7.

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The arguments presented here have not been worked out in detail, and we therefore offer our apologies to the reader for the occasional sketchiness of the presentation. We hope that this section will serve, if nothing more, as a guide for future work. Lemma Suppose that there is a congruence. Theorem 9. Let L -be. Then by comparing determinants on the two sides of either of the possibilities I , we get a congruence between the trivial character and mod p. This is impossible since p 2. Proposition Assume in addition that p is not a congruence prime for any f E s f. Then p E N if and only if there is a congruence A.

Remark It is expected to hold most of the time. It is however conceivable that a prime p may divide both the terms on the right hand side of In the sequel we have ignored the complications arising from this second possibility. Also, it can be shown cf. Lemma 3. Thus p must divide one of the two terms on the right hand side of By Theorem 9. Since we have assumed that p is not a congruence prime for f , we must in fact have that. That is, p E N. This shows one direction. The above argument is essentially reversible.

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Suppose that p divides the twisted adjoint L-value for some f E Sf. Then divides the left hand side of the the identity By Lemma If f , g E S-, then the relations The upshot of all this is that h' is a non-base-change form, and so is a Galois twist of h by the standing assumption 4. By replacing f with h mod XJ' for some a Galois twist, we have a congruence of the form p' p, and this proves the other direction. Now, by Theorem 9. However, the relations 4. Then any such prime, being lost on lifting, would not occur in the numerator of the left hand side of the relation A numerical check cf.

Then if r denotes the automorphism of Kh extended to K making the diagram of display 4. To show the other inclusion, suppose that p E V. Then as above, we see that. We now establish the connection of the primes in N with the primes in V. Recall the well known: Coqjecture Then is modular. Then by Proposition Thus the apparent obstruction NFIQ ek-' - I , which arose in [lo] as a measure of the primes of torsion of certain boundary cohomology groups, is likely to be more a short coming of the method of proof used there, rather than a genuine obstruction. Interestingly, Urban has some results towards the proposition that circumvents using these relations.

His idea is that the primes dividing the twisted adjoint L-values are related to the primes dividing the Klingen-Eisenstein ideal for and so, to the primes dividing an appropriate twisted Selmer group.