Arithmetic of the Asai L-function for Hilbert modular forms by Adam Rubin Kaye A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan 2016 Doctoral Committee: Professor Kartik Prasanna, Chair Professor Stephen M. DeBacker Professor Jeffrey C. Lagarias Professor Andrew Snowden Professor Martin J. Strauss TABLE OF CONTENTS CHAPTER I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 II. Hilbert Modular Surfaces and their cohomology . . . . . . . . . . . . . . . . 10 2.1 Classical Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Adelic description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Compactification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Cohomology of Hilbert Modular Surfaces, Hecke Operators, and Hilbert Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 III. Beilinson’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.1 Class Number Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 Beilinson’s Conjecture for Surfaces. . . . . . . . . . . . . . . . . . . . . . . . 19 3.4 Beilinson’s Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.5 Ramakrishnan’s results for Hilbert modular surfaces . . . . . . . . . . . . . . 24 IV. p-adic Beilinson Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.1 p-adic Rankin-Selberg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.2 p-adic Beilinson for a product of modular curves . . . . . . . . . . . . . . . . 27 4.3 p-adic Asai L-function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 V. Algebraicity of the Asai L-function . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.1 Summary of Shimura’s proof . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.2 Automorphic forms on a quaternion algebra . . . . . . . . . . . . . . . . . . 33 5.3 The Asai L-function on a quaternion algebra . . . . . . . . . . . . . . . . . . 36 5.4 Automorphic Forms on an Orthogonal Group . . . . . . . . . . . . . . . . . 37 5.5 Integral representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.6 CM Points on the orthogonal Shimura variety . . . . . . . . . . . . . . . . . 41 5.7 Evaluating E at CM Points . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 k 5.8 Relating D(s) and G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 k 5.9 Evaluating G at CM Points . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 k 5.10 Proof of proposition V.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.11 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 VI. Rationality of the Asai L-function I . . . . . . . . . . . . . . . . . . . . . . . . 51 ii 6.1 Rational version of Shimura’s Period . . . . . . . . . . . . . . . . . . . . . . 51 6.2 Rankin-Selberg Convolution and the Rational Period . . . . . . . . . . . . . 54 6.3 Rationality of G at CM points . . . . . . . . . . . . . . . . . . . . . . . . . 55 k 6.4 Rationality of E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 k VII. Integrality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 VIII. Rationality of the Asai L-function II . . . . . . . . . . . . . . . . . . . . . . . . 73 8.1 Asai’s Integral Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 73 8.2 Rationality with Harris’ Period. . . . . . . . . . . . . . . . . . . . . . . . . . 76 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 iii CHAPTER I Introduction In this thesis, we will study the arithmetic of the Asai L-function for Hilbert modular forms over a real quadratic field. This is motivated by a p-adic analog of the Beilinson conjecture for Hilbert modular surfaces. This p-adic analog, which is expected to involve p-adic deformations in the weight direction (and is thus not covered by the general p-adic Beilinson conjecture of Perrin-Riou [30]), does not seem to be stated anywhere in the literature. The relevant p-adic L-function should interpolate critical values of the Asai L-function for Hilbert modular forms of non- parallel weight. Thus we are led to study the arithmetic of the special values of such L-functions, building on previous work of Shimura. Our main results include two theorems on rationality of such L-values that refine previous work of Shimura. We also make some partial progress towards integrality results. Let F be a real quadratic field and let O be the ring of integers in F. For F simplicity, we will also assume throughout that F has narrow class number equal to 1. Let H denote the complex upper half plane and f : H2 → C a holomorphic Hilbert modular form of weight (k ,k ). We assume that f is a normalized Hecke 1 2 eigenform. It turns out there are two natural L-functions that one can associate with f. One 1 2 such, the standard L-function is defined by (cid:88) L(f,s) = C(a)N(a)−s, a where the sum is over integral ideals a of O and the terms C(a) denote the Fourier F coefficientsoff. ThisL-functionistheonethatappearsintheBirchandSwinnerton- Dyer conjecture for the form f. AnotherL-functionthatonemayassociatewithf istheso-calledAsai L-function, defined by ∞ (cid:88) LAs(f,s) = ζ(2s−k −k +2) C(n)n−s, 1 2 n=1 where ζ(s) is the Riemann-Zeta function. Note that the sum is over the positive integers rather than integral ideals of O . Asai [1] used Rankin’s method to show F that this L-function admits an analytic continuation and a functional equation. The Asai L-function for forms f of weight (2,2) is related to the Hasse-Weil zeta function of Hilbert modular surfaces. Specifically, the L-function L(H2(X),s) for a Hilbert modular surface is a product of Asai L-functions corresponding to such forms, and consequently it is the Asai L-function which arises in the statement of Beilinson’s conjectures for H2 of Hilbert modular surfaces. We will now recall this connection. Let X be a smooth projective surface over Q. Then for each prime l, H2(X × ´et Q Q¯,Q ) is a Gal(Q¯/Q)-module and as l varies these form a compatible system of l finite-dimensional l-adic representations of Gal(Q¯/Q). Let L(2)(X,s) denote the L-function associated with this compatible system. The (conjectural) functional equation satisfied by this L-function relates the values L(2)(X,s) with L(2)(X,3−s). Then Beilinson’s conjecture (stated in [4]) for the motive H2(X) concerns the special values L(2)(X,s) for integers s ≥ 2 or equivalently the leading term in the Taylor expansion at integers s ≤ 1. These special values are expected to be related to 3 regulators of elements in certain higher Chow groups attached to X. Precisely, the higher Chow group (or motivic cohomology) in question that is conjecturally associated with L(2)(X,j) is the group H3 (X,Q(j)) = CHj(X,2j −3)⊗Q. M Beilinson then constructs a regulator map (1.0.1) H3 (X,Q(j)) → H3(X,R(j)) M D where H∗(X,R(·)) denotes real Deligne cohomology. The target of the map (1.0.1) D admits a natural Q-structure. The conjecture then says that the map (1.0.1) is an isomorphism and the difference between the determinant of this Q-structure and the one induced from (the image of) H3 (X,Q(j)) is given by the L-value L(2)(X,j), up M to certain elementary factors. In fact, one needs to be a bit more careful for two reasons. Firstly, one must work not with H3 (X,Q(j)) but rather with the subspace M of integral elements denoted H3 (X ,Q(j)). Secondly, for the point j = 2 there M Z is a correction factor involving the Neron-Severi group, that is related to the Tate conjecture for X. We will ignore such subtleties for the moment. We now specialize to the case of a Hilbert modular surface X. Since such a surface is not compact, one should consider the intersection cohomology IH2; for the purposes of the introduction we ignore the issue of noncompactness relegating the more precise description to Chapter II. For such an X, the representation H2(X× ´et Q Q¯,Q )decomposesaccordingtotheactionoftheHeckealgebraattachedtothegroup l G = Res GL . LetAdenotetheadelesoverQ. Theautomorphicrepresentations F F/Q 2 π of G (A) that contribute to H2(X × Q¯,Q ) can be classified and the interesting F ´et Q l representations are those where the infinite components are discrete series of weight 2, namely those that correspond to Hilbert modular forms f of parallel weight two. 4 Fixing such a π (and f), let W (π) denote the π-isotypic component of H2(X × (cid:96) ´et Q Q¯,Q ). We write W(π) for the compatible system of Galois representations W (π). (cid:96) (cid:96) The key point that connects Beilinson’s conjecture to the Asai L-function is the following result of Brylinski-Labesse. Theorem I.1 (Brylinski-Labesse, [9]). There exists a finite set of primes Q such that L (W(π),s) = LAs(f,s) Q Q where L denotes the L-function with Euler factors at primes dividing Q removed. Q As a result, Beilinson’s conjecture reduces more or less to a corresponding con- jecture relating the π-isotypic component of H3 (X,Q(j)) to special values of the M Asai L-function LAs(f,s). Let us now briefly recall previous work on Beilinson’s conjecture (and p-adic versions). conjecture: (A) The original paper of Beilinson [4] treats the case of the Rankin-Selberg L- function L(f ×g,2) where f and g are classical modular forms of weight two. This may be viewed as a special case of the setting described above by taking F to be the quadratic Q-algebra Q×Q instead of a real quadratic field. (B) An unpublished preprint of Ramakrishnan [35] discusses the case of a Hilbert modular form of weight (2,2) at s = 2. (C) Thepaper[25]ofGuidoKingstreatsthecaseofHilbertmodularformsofweight (2,2) at s ≥ 3. More recently, there has been interest in formulating and proving p-adic versions of Beilinson’s conjecture, in which the regulator is replaced by a p-adic regulator and theL-functionbyap-adicL-function. Inthesettingof(A)above, thiswasstudiedin 5 the paper [6] of Bertolini, Darmon and Rotger. The p-adic L-function in question is that obtained by interpolating (the algebraic parts of) critical values of the Rankin- Selberg L-functions L(f ,g,s) where F is a Hida family of forms containing f and κ κ is an arithmetic point of F corresponding to a classical modular form f of weight κ ≥ 3. The point is that the original Rankin-Selberg L-function L(f × g,s) has no critical values since the weights of f and g are equal but the L-function L(f ×g,s) κ is critical at the points 2 ≤ s ≤ k − 1 where k is the weight of f . We note that κ to define the algebraic part of L(f ×g,j), one needs to divide it by an appropriate κ period. In this case, the period may be taken to be the Petersson inner product (cid:104)f ,f (cid:105) of the form f , up to certain elementary factors. κ κ κ We consider instead the case (B) with a Hilbert modular form f of parallel weight two. Again, the Asai L-function L(fAs,s) has no critical points. The idea then is to vary such an f in a Hida family F of forms whose weights are (k,2). When we specialize F to a form f of weight (k,2) with k > 2, the Asai L-function L(fAs,s) κ κ is critical in the range 2 ≤ s ≤ k − 1. Thus one may hope to construct a p-adic L-function by interpolating the algebraic parts of these L-values. However, one runs into a problem since the transcendental period that one must divide by is no longer so easy to define. Indeed the form f is not a “product” of forms as in the case κ F = Q×Q and the period must somehow distinguish one of the two infinite places of the field F. It turns out that there are two way to do this and thus two periods that can be defined. We remark that the exact relation between these two periods is not obvious and is an open question that deserves further study. Motivated by the above discussion, for the rest of the introduction, we use the symbol f to denote a Hilbert modular form (for F) of weight (k,2), and discuss in turnthetwoperiodsthatonecanattachtosuchanf thatcapturethetranscendental 6 part of critical values of the Asai L-function of f. The first period is defined by using theJacquet-Langlandscorrespondence. TheideahereisduetoShimura(in[?]). One assumes that there exists a quaternion algebra B over F which is split at one infinite place and ramified at the other infinite place. In other words, we need to assume that the representation π is discrete series for at least one finite place. Let τ ,τ be 1 2 the infinite places of F and B be a quaternion algebra that is split at τ and ramified 1 at τ . Let πB denote the Jacquet-Langlands transfer of π to B×. Also let fB be a 2 nonzero vector in πB of weight (k,0) at infinity. The form fB may be viewed as a section of an automorphic vector bundle V on the Shimura curve X attached to B B B×. Now using the fact that X and V admit models over Q¯ and even over F, the B B form fB can be normalized up to a non-zero element in the field FQ(f) which is the compositum of F with the field Q(f) generated by the Hecke eigenvalues of f. In fact, one can also integrally normalize fB at least at good primes using suitable integral models of X and V , as in [24] §1. The first period is then defined to be B B the Petersson inner product of fB with respect to a suitable measure. Q(f,B) := (cid:104)fB,fB(cid:105). The following theorem is due to Shimura. Theorem I.2 (Shimura [43], [44]). LAs(f,k −1) LAs(f,k −1) = lies in Q¯. πk(cid:104)fB,fB(cid:105) πkQ(f,B) Our first main result refines this to Theorem I.3. (See Chapter VI , Cor. VI.18) Suppose that Q(f) ⊆ F and that every prime ideal p of F dividing the discriminant of B is split over Q. Then LAs(f,k −1) lies in F , πkQ(f,B) ab 7 where F denotes the maximal abelian extension of F. ab The proof of this result is obtained by carefully analyzing Shimura’s proof of algebraicity, which is rather difficult and involves several steps. The brief idea is as follows. One views B as a quadratic space over F equipped with the quadratic form given by the reduced norm. Then fB × fB : H2 → C can be viewed as an automorphic form for the identity component G of the group GO(B), since G is B B isomorphic to (B××B×)/F×. Shimura defines another automorphic form E˜ on G k B such that (cid:104)fB ×fB,E˜ (cid:105) ∼ LAs(fB,k −1)(cid:104)fB,fB(cid:105). k Here ∼ denotes equality up to certain well-understood factors. He then shows that E˜ is defined over Q¯ which implies (cid:104)fB ×fB,E˜ (cid:105) ∼ C(cid:104)fB,fB(cid:105)2 with C ∈ Q¯. The k k ˜ idea then is to refine Shimura’s proof by showing that E is in fact defined over a k smaller field. The Hermitian symmetric domain for G does not have any cusps, so the ratio- B ˜ nality properties of E cannot be studied through q-expansions. Instead the idea k ˜ is to evaluate E at certain CM points and show that the resulting values, suitably k normalized, are rational. Let K be a totally imaginary degree 2 extension of F and w ∈ H2 a CM point corresponding to K. One needs to study the value E˜ (w ) K k K divided by an appropriate CM period attached to K. Now K itself can be viewed as a quadratic space over Q of dimension 4. The Shimura variety GO(K) has a distinguished CM point z associated with K and one can define an automorphic K form G on GO(K) with the property that k ˜ E (w ) ∼ G (z ). k K k K The problem then reduces to showing that the form G on GO(K) is rational. To k