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Addition and : The of Partitions Scott Ahlgren and Ken Ono

At first glance the stuff of partitions seems like computed the values of p(n) for all n up to 200. child’s play: He found that 4=3+1=2+2=2+1+1=1+1+1+1. p(200) = 3, 972, 999, 029, 388,

Therefore, there are 5 partitions of the 4. and he did not count the partitions one-by-one: But (as happens in ) the seemingly 200 = 199 + 1 = 198 + 2 simple business of counting the ways to break a =198+1+1=197+3=...... number into parts leads quickly to some difficult and beautiful problems. Partitions play important Instead, MacMahon employed classical formal roles in such diverse of as power identities due to Euler. , Lie theory, , To develop Euler’s recurrence, we begin with the elementary fact that if |x| < 1, then mathematical physics, and the theory of special functions, but we shall concentrate here on 1 =1+x + x2 + x3 + x4 + ... . their role in number theory (for which [A] is the 1 − x standard reference). Using this, Euler noticed that when we expand the In the Beginning, There Was Euler… infinite ∞ 1 A partition of the n is any nonin- =(1+x + x2 + x3 + ...) 1 − xn creasing sequence of natural whose sum n=1 is n (by convention, we agree that p(0) = 1). The × (1 + x2 + x4 + ...) number of partitions of n is denoted by p(n). Eighty × (1 + x3 + x6 + ...) ..., years ago Percy Alexander MacMahon, a major in the British Royal Artillery and a master , the of xn is equal to p(n) (think of the first factor as counting the number of 1’s in a par- Scott Ahlgren is assistant professor of mathematics at tition, the second as counting the number of 2’s, the University of Illinois, Urbana-Champaign. His e-mail and so on). In other words, we have the generat- address is [email protected]. ing Ken Ono is professor of mathematics at the University of ∞ ∞ n 1 Wisconsin-Madison. His e-mail address is ono@math. p(n)x = 1 − xn wisc.edu. n=0 n=1 Both authors thank the National Science Foundation for =1+x +2x2 +3x3 +5x4 + ... . its support. The second author thanks the Alfred P. Sloan Foundation, the David and Lucile Packard Foundation, and Moreover, Euler observed that the reciprocal of the Number Theory Foundation for their support. this infinite product satisfies a beautiful identity

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(also known as Euler’s Pentagonal Number Theo- Rademacher’s series converges is remarkable; for rem): example, the first eight terms give the approxi- ∞ ∞ mation − n − k (3k2+k)/2 (1 x )= ( 1) x p(200) ≈ 3, 972, 999, 029, 388.004 n=1 k=−∞ =1− x − x2 + x5 + x7 − x12 − ... . (compare with the exact value computed by MacMahon). These two identities show that To implement the method requires a   ∞ detailed study of the analytic behavior of the  p(n)xn generating function for p(n). Recall that we have n=0 ∞ n 1 × 1 − x − x2 + x5 + x7 − x12 − ... =1, F(x):= p(n)x = . (1 − x)(1 − x2)(1 − x3) ... n=0 which in turn implies, for positive n, that This is an analytic function on the domain |x| < 1. − − p(n)=p(n 1) + p(n 2) A natural starting point is Cauchy’s Theorem, − − − − − p(n 5) p(n 7) + p(n 12) + ... . which gives 1 F(x) This recurrence enabled MacMahon to perform his p(n)= n+1 dx, massive . 2πi x

Hardy-Ramanujan-Rademacher where C is any simple closed counterclockwise contour around the origin. One would hope to Asymptotic Formula for p(n) adjust the contour in relation to the singularities p(n) It is natural to ask about the size of . The an- of F(x) in order to obtain as much information as swer to this question is given by a remarkable as- possible about the . But consider for a mo- ymptotic formula, discovered by G. H. Hardy and ment these singularities; they occur at every root Ramanujan in 1917 and perfected by Hans of unity, forming an impenetrable barrier on the Rademacher two decades later. This formula is so unit circle. In our favor, however, it can be shown accurate that it can actually be used to compute that the size of F(x) near a primitive q-th root of individual values of p(n); Hardy called it “one of unity diminishes rapidly as q increases; moreover the rare formulae which are both asymptotic and the behavior of F(x) near each root of unity can be exact.” It stands out further in importance since described with precision. Indeed, with an appro- it marks the birth of the circle method, which has priate choice of C, the contribution to the integral grown into one of the most powerful tools in an- from all of the primitive q-th roots of unity can be alytic number theory. calculated quite precisely. The main contribution Here we introduce Rademacher’s result. He de- is the function T (n); a detailed analysis of the fined explicit functions T (n) such that for all n we q q errors involved yields the complete formula. have ∞ The circle method has been of extraordinary im- p(n)= Tq(n). portance over the last eighty years. It has played q=1 a fundamental role in additive number theory (in Waring type problems, for instance), analysis, and The functions T (n) are too complicated to write q even the computation of black hole entropies. down here, but we mention that T1(n) alone yields the asymptotic formula Ramanujan’s Congruences √ 1 After a moment’s reflection on the combinatorial p(n) ∼ √ eπ 2n/3. 4n 3 definition of the partition function, we have no particular reason to believe that it possesses any (In their original work, Hardy and Ramanujan used interesting arithmetic properties (the analytic slightly different functions in place of the Tq(n). ∞ formula of the last section certainly does nothing As a result, their analogue of the series q=1 Tq(n) to change this opinion). There is nothing, for was divergent, although still useful.) Moreover, example, which would lead us to think that Rademacher computed precisely the error incurred p(n) should exhibit a preference to be even by truncating this series after Q terms. In partic- rather than odd. A natural suspicion, therefore, ular, there exist explicit constants A and B such might be that the values of p(n) are distributed that √ evenly modulo 2. A quick computation of the An B first 10,000 values confirms this suspicion: of p(n) − Tq(n) < . n1/4 q=1 these 10,000 values, exactly 4,996 are even and 5,004 are odd. This pattern continues with Since p(n) is an , this determines the exact 2 replaced by 3: of the first 10,000 values, 3,313; value of p(n) for large n. The rate at which 3,325; and 3,362 (in each case almost exactly

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one-third) are congruent respectively to 0, 1, and modulo 5 and 7 are quite ingenious but are not 2 modulo 3. When we replace 3 by 5, however, terribly difficult, while the proof of the something quite different happens: we discover modulo 11 is much harder). In these same that 3,611 (many more than the expected papers he sketched proofs of extensions of one-fifth) of the first 10,000 values of p(n) are these congruences. For example, we have divisible by 5. What is the explanation for this p(25n +24)≡ 0 (mod 25), aberration? ≡ The answer must have been clear to Ramanu- p(49n + 47) 0 (mod 49). jan when he saw MacMahon’s table of values of Ramanujan noticed the beginnings of other p(n). So Ramanujan would have seen something patterns in these first 200 values: like the following. p(116) ≡ 0 (mod 121),p(99) ≡ 0 (mod 125). 1123 5 From such scant evidence he made the following conjecture: 7 111522 30 If δ =5a7b11c and 24λ ≡ 1 (mod δ), 4256 77 101 135 then p(δn + λ) ≡ 0 (mod δ). 176 231 297 385 490 When δ =125, for example, we have λ =99. So 627 792 1002 1255 1575 Ramanujan’s conjecture is that

1958 2436 3010 3718 4565 p(125n + 99) ≡ 0 (mod 125).

What is striking, of course, is that every entry in We note that the general conjecture follows easily the last column is a multiple of 5. This phenome- from the cases when the moduli are powers of 5, non, which persists, explains the apparent aber- 7, or 11. ration above and was the first of Ramanujan’s It is remarkable that Ramanujan was able to ground-breaking discoveries on the arithmetic of formulate a general conjecture based on such p(n). Here is his own account. little evidence and therefore unsurprising that the conjecture was not quite correct (in the 1930s I have proved a number of arithmetic Chowla and Gupta discovered the counterexample properties of p(n)…in particular that p(243) ≡ 0 (mod 73) ). Much to Ramanujan’s p(5n +4)≡ 0 (mod 5), credit, however, a slightly modified version of his conjecture is indeed true; in particular, we p(7n +5)≡ 0 (mod 7). now know the following: …I have since found another method If δ =5a7b11c and 24λ ≡ 1 (mod δ), which enables me to prove all of these a b +1 c properties and a variety of others, of then p(δn + λ) ≡ 0 (mod 5 7 2 11 ). which the most striking is The task of assigning credit for the proofs of these conjectures when the modulus is a power of p(11n +6)≡ 0 (mod 11). 5 or 7 poses an interesting historical challenge. Typically, the proofs have been attributed to There are corresponding properties in G. N. Watson. Recently, however, the nature of which the moduli are powers of 5, 7, or Ramanujan’s own contributions [R] has been 11…. It appears that there are no greatly clarified. Indeed, a complete outline of the equally simple properties for any proof modulo powers of 5 and a much rougher moduli involving primes other than sketch for powers of 7 (so rough that it did not yet these three. reveal his error in the statement of the conjec- Ramanujan proved these congruences in a ture) are given by Ramanujan in a long manuscript series of papers (the proofs of the congruences which he wrote in the three years preceding his

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1 death. In typical fashion, Ramanujan provides in R(5n +4,m,5) = · p(5n + 4) for 0 ≤ m ≤ 4, neither case complete details for all of his asser- 5 1 tions. This manuscript was apparently in Watson’s R(7n +5,m,7) = · p(7n + 5) for 0 ≤ m ≤ 6. possession from 1928 until his death in 1965. 7 Indeed, a copy of the manuscript in Watson’s The truth of these conjectures would provide a handwriting (the whereabouts of the original is simple and elegant combinatorial explanation for unknown) resides in the library of Oxford’s Math- Ramanujan’s congruences. Dyson’s speculation ematical Institute. In any event, it seems clear was confirmed ten years later by Atkin and that Ramanujan deserves more credit than he H. P. F. Swinnerton-Dyer in a wonderful paper has historically been granted for these cases. By which combines classical combinatorial argu- contrast, the case of powers of 11 is much more ments with techniques from the theory of mod- difficult; the first published proof of Ramanujan’s ular functions. conjectures in this case was given by A. O. L. Atkin Unfortunately, Dyson’s rank does not seem in 1967. to enjoy such simple properties for primes other than 5 and 7. However, he conjectured the Dyson’s Rank and Crank existence of another natural statistic, the crank, The celebrated physicist Freeman Dyson, when he which explains the congruence was a college student in 1944, initiated an impor- ≡ tant subject in partition theory by discovering a p(11n +6) 0 (mod 11). delightfully simple phenomenon which appeared In the late 1980s George E. Andrews and Frank to explain why Garvan found such a crank [A-G], [G]. Further work of Garvan, Dongsu Kim, and Dennis Stan- ≡ p(5n +4) 0 (mod 5) ton [G-K-S] has produced, for the congruences with moduli 5, 7, 11, and 25, combinatorial in- and terpretations which are rooted in the modular p(7n +5)≡ 0 (mod 7). representation theory of the symmetric .

Dyson defined the rank of a partition to be the Atkin’s Examples largest summand minus the number of summands. We return to Ramanujan’s intuition that there Here, for example, are the partitions of 4 and their are no simple arithmetic properties for p(n) ranks: when the modulus involves primes greater than Partition Rank 11. Ramanujan seems to have been correct in this 44− 1 ≡ 3 (mod 5), claim; no new congruence as simple as the 3+1 3− 2 ≡ 1 (mod 5), originals has ever been found (although it has 2+2 2− 2 ≡ 0 (mod 5), not been proved that none exists). The 1960s, 2+1+1 2− 3 ≡ 4 (mod 5), however, witnessed tantalizing discoveries of 1+1+1+1 1− 4 ≡ 2(mod 5) . further examples (notably by Atkin, Newman, and O’Brien). Atkin, for example, found elegant infinite families of congruences modulo 5, 7, and Notice that the ranks of these partitions repre- 13 which are quite different from those previ- sent each residue class modulo 5 exactly once. ously known. A simple example of these is the After computing many more examples, Dyson congruence observed that, without exception, numbers of the 3 · ≡ form 5n +4(respectively 7n +5) have the property p(11 13n + 237) 0 (mod 13). that their ranks modulo 5 (respectively modulo 7) Atkin also gave more examples, though not so are equally distributed. More precisely, if systematic, with moduli 17, 19, 23, 29, and 31. 0 ≤ m

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seen so far, we will give a brief description here. large role in number theory; their importance, of Let SL2(Z) be the of 2 × 2 integer matrices with course, has been underscored by their central po- determinant equal to 1. Then, if N is an integer, sition in the proof of Fermat’s Last Theorem. The define the congruence Γ0(N) by crux of Wiles’ proof is to show that elliptic curves are “modular”; in other words, their arithmetic is ab dictated in part by certain modular forms to which Γ0(N):= ∈ SL2(Z):c ≡ 0 (mod N) . cd they are related. What has been learned recently is that the partition function does not escape the ab web of modularity; its arithmetic, too, is intimately An γ = acts on the upper half-plane cd connected to the behavior of a certain family of H of complex numbers via the linear fractional az+b modular forms. This connection has allowed the transformation γz = cz+d . By definition, a modular application of deep methods of Deligne, Serre, and function on Γ0(N) is a function f on H which Shimura to the study of p(n). These theories (some satisfies f (γz)=f (z) for all γ ∈ Γ0(N) and which of the most powerful of the last half-century) have in is meromorphic on H and “at the important ramifications for p(n); in particular, cusps”. When N is small, the of these functions properly applied, they imply that p(n) satisfies lin- is relatively simple; therefore, given several func- ear congruences for every prime $ ≥ 5. We shall tions in such a field, one expects to find nontrivial discuss in more detail how modular forms enter relations among them. If the right functions are in- the picture in the next section; let us first indicate volved, then such a relation may give information what they enable us to prove. about values of p(n). For Atkin’s examples when The second author (inspired by some formulae $ =5, 7, or 13, the relevant function fields have a of Ramanujan) was the first to notice these con- single generator; this is responsible for the infinite nections; as a result [O] he proved the following: families of congruences. As $ increases, however, ≥ things rapidly become more complicated. Atkin’s For any prime $ 5, there exist work is interesting for another reason: it marks an inf initely many congruences of the f orm early use of sophisticated in mathemat- p(An + B) ≡ 0 (mod $). ics. As he says, “It is often more difficult to discover results in this subject than to prove (We note that if the arithmetic progression An + B them, and an informed search on the machine may gives rise to such a congruence, then so do any of enable one to find out precisely what happens.” its infinitely many subprogressions; we do not count these as new when we speak of “infinitely A Problem of Erdo˝s many congruences”.) Shortly thereafter the first au- Even after all of the beautiful discoveries described thor [Ahl] extended this result by showing that the above, the general arithmetic properties of p(n) prime $ may in fact be replaced by an arbitrary must seem rather mysterious. Indeed, we have prime power $k; from this it can be shown that $ said nothing for any prime modulus $ greater than may in fact be replaced by any modulus M which 31, let alone for a general prime modulus. In this is coprime to 6. An immediate consequence of context we mention a conjecture of Erdo˝s from the these results is the following: 1980s. If $ ≥ 5 is prime, then a positive If $ is a prime, then there exists an n proportion of natural numbers n such that p(n) ≡ 0 (mod $). have p(n) ≡ 0 (mod $). If we reflect on this conjecture for a moment, we This provides a very convincing proof of the are struck by its weakness: it asserts only that conjecture of Erdo˝s mentioned above. every prime divides at least one value of the partition function. On the other hand, (until very More recently, the two authors [Ahl-O] have recently) the known results were even weaker; the shown that congruences for p(n) are even more best was a theorem of Schinzel and Wirsing, who widespread than these theorems indicate. To proved the existence of a constant c such that, for explain this, let us return to Ramanujan’s original large X, the number of primes $

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≡ H If $ =5, 7,or 11,thenp($n + λ$) 0 (mod $). In addition, f is required to be meromorphic on and at the cusps; if f is also holomorphic on H and ≥ Now, for any prime $ 5 and any exponent k, vanishes at the cusps, then we call f a cusp form. the results above guarantee the existence of We allow k to be an integer or half an integer (extra infinitely many progressions An + B such that care must be taken in the latter case); note that the ≡ k p(An + B) 0 (mod $ ) . An important feature modular functions introduced above are just mod- of the method used to prove the theorems above ular forms of weight zero. Every is that in every case, the progression An + B which f (z) has a Fourier expansion in powers of x = e2πiz; it produces is a subprogression of $n + λ$ (in other if f is a cusp form, then this expansion takes the words, $ | A and B ≡ λ (mod $)). As an example, $ form ∞ one of the simplest congruences guaranteed by n f (z)= af (n)x . this theorem is n=1 4 · ≡ p(59 13n + 111247) 0 (mod 13); When the weight k of a cusp form is integral, in this case we have 111247 ≡ 1/24 (mod 13). then the theory of Deligne and Serre is available What the authors have shown recently is that for the study of the Fourier af. In par- congruences are not confined to this single pro- ticular, there is a natural family of operators (the gression modulo $. In fact, we now know that if so-called Hecke operators) that act on spaces of $ ≥ 5 is prime and k is any exponent, then infinitely modular forms. If f is a normalized eigenform for this family, then Serre conjectured and Deligne many congruences p(An + B) ≡ 0 (mod $k) exist proved the existence of a representation within each of ($ +1)/2 progressions modulo $. In other words, for each prime, slightly more than ρf : Gal(Q/Q) → GL2(K) half of such progressions contain congruences. When $ =11, for example, the relevant progressions (for some field K) such that for all but finitely are many primes Q we have 11n +1, 11n +2, 11n +3, Trace(ρf (FrobQ)) = af (Q). 11n +5, 11n +6, 11n +8. Here FrobQ denotes a Frobenius element at the Of these, only Ramanujan’s own 11n +6had been prime Q. This result is extraordinarily powerful; distinguished by the previous theory. The latest it allows us to study the Fourier coefficients of mod- result provides a theoretical framework which ular forms using the structure of Galois groups. explains every known partition function congru- If the weight k of a cusp form is half-integral, ence. then we do not have the results of Deligne and Serre at our disposal. There is, however, a correspon- Modular Forms dence due to Shimura between cusp forms of half We will try to indicate briefly how the theory of integral weight and certain forms of integral weight; modular forms can be applied to the study of p(n) the Shimura correspondence is quite explicit and in order to yield the results of the preceding sec- commutes in the best possible way with the action tion. At the heart of the matter are the generating of the Hecke operators on the respective spaces. function We saw above that the expansion ∞ ∞ n 1 ∞ p(n)x = n +1 − 1 − xn 1/η(24z)= p xn = x 1 + x23 + ... n=0 n=1 24 n=−1 and Dedekind’s eta function contains every value of the partition function. Now ∞  1/η(24z) is a modular form on Γ (576). However, 1/24 − n 2πiz 0 η(z)=x (1 x ) (here x := e ). it has two major deficiencies: the weight is −1/2, n=1 and it has a pole at every cusp. So none of the the- Combining the last two formulae gives ories above seem to apply. It turns out, however, that starting with this expansion, one can con- ∞ n +1 − struct half-integral weight cusp forms which still 1/η(24z)= p xn = x 1 + x23 + ... . 24 preserve much information about the values of p(n) n=−1 modulo powers of primes. From these cusp forms Loosely speaking, a modular form of weight k the theory of Deligne and Serre, filtered through on the subgroup Γ0(N) is a function f on the upper Shimura’s correspondence, yields the results of half-plane H which satisfies a transformation prop- the preceding section. erty of the form L-Functions and Arithmetic az + b k ab Since modular forms play such an important role f =(cz + d) f (z) for all ∈ Γ0(N). cz + d cd in partition congruences, it is natural to suspect

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that there may be deeper connections between Conjecture (a vast generalization of the Birch and partitions and “modular” objects. As it turns out, Swinnerton-Dyer Conjecture) then implies that this is indeed the case. L(M , ($ − 3)/2)=Ω · #X(M ). To motivate the connection, consider the fol- D,$ D,$ D,$ lowing classical Diophantine question (already of Assuming the Bloch-Kato Conjecture, it can be interest to ancient Greek and Arab scholars): shown, for many n, that ≡ ⇒ X ≡ Which integers D are areas of right p(n) 0 (mod $)= # (MD,$) 0 (mod $), with sidelengths? where D depends on n. These two conditions are Such numbers D are known as congruent numbers. probably equivalent, and so it is likely that the Simple arguments show that a number D is con- divisibility of p(n) often dictates the presence of gruent precisely when there are infinitely many elements of order $ in these Tate-Shafarevich rational points (x, y) on the elliptic curve groups. So, perhaps surprisingly, it seems that congruences like Ramanujan’s are connected to 2 3 2 ED : y = x − D x. some highly abstract creations of modern number theory. How does one determine whether such a curve has infinitely many points? The Birch and The Future? Swinnerton-Dyer Conjecture, one of the main The beginnings of the partition function are outstanding conjectures in number theory (and a extraordinarily humble; after all, what could be million-dollar Clay Mathematics Institute prob- simpler than addition and counting? Despite its lem), provides the solution. humble start, the history of the partition function Let L(ED,s) denote the Hasse-Weil L-function includes connections to many central areas of attached to ED; this is an analytic function whose number theory, from the work of Euler to the definition depends on the behavior of ED modulo birth of the circle method to the modern theory of primes p. For the congruent number problem the modular forms and L-functions. It will be quite conjecture implies that interesting to see what further connections the L(ED, 1) = 0 ⇐⇒ D is congruent. future will reveal. In addition, the conjecture gives a precise formula References dictating the analytic behavior of L(ED,s) at s =1. [Ahl] S. AHLGREN, The partition function modulo com- For instance, if L(ED, 1) =0 , then the conjecture posite integers M, Math. Ann. 318 (2000), 795–803. asserts that [Ahl-O] S. AHLGREN and K. ONO, Congruence properties for · X the partition function, Proc. Nat. Acad. Sci. U.S.A., L(ED, 1) = ΩD # (ED). to appear. Here ΩD is an explicit transcendental number, and [A] G. E. ANDREWS, The Theory of Partitions, Cambridge Univ. Press, 1998. X(ED) is the Tate-Shafarevich group of ED. (The [A-G] G. E. ANDREWS and F. GARVAN, Dyson’s crank of a Tate-Shafarevich group is a certain Galois partition, Bull. Amer. Math. Soc. (N.S.) 18 (1988), cohomology group which measures the extent to 167–171. which the local-global principle fails for ED.) [R] B. C. BERNDT and K. ONO, Ramanujan’s unpublished In the early 1980s Jerrold Tunnell, using the manuscript on the partition and tau functions with works of Shimura and Waldspurger (see [K] for a commentary, The Andrews Festschrift (D. Foata and good account), constructed two modular forms G. N. Han, eds.), Springer-Verlag, 2001, pp. 39–110. of weight 3/2 whose coefficients “interpolate” the [G] F. GARVAN, New combinatorial interpretations of square roots of the L(E , 1). Together with the Ramanujan’s partition congruences mod 5, 7 and 11, D Trans. Amer. Math. Soc. 305 (1988), 47–77. Birch and Swinnerton-Dyer Conjecture, these [G-K-S] F. GARVAN, D. KIM, and D. STANTON, Cranks and modular forms provide a complete solution to t-cores, Invent. Math. 101 (1990), 1–17. the congruent number problem. [G-O] L. GUO and K. ONO, The partition function and the Recently, Li Guo and the second author [G-O] arithmetic of certain modular L-functions, Internat. have shown that if 13 ≤ $ ≤ 31 is prime, then Math. Res. Notices 21 (1999), 1179–1197. certain half-integral weight modular forms [K] N. KOBLITZ, Introduction to Elliptic Curves and whose coefficients interpolate values of p(n) Modular Forms, Springer-Verlag, 1993. [O] K. ONO, Distribution of the partition function modulo $ behave in a manner somewhat similar modulo m, Ann. of Math. 151 (2000), 293–307. to Tunnell’s modular forms. In particular, they showed that there are modular motives MD,$ (these may be viewed as analogs of elliptic curves) whose L-functions L(MD,$,s) have the property that the Note: Photograph of L. Euler courtesy of the − square roots of L(MD,$, ($ 3)/2) are related in a Institut Mittag-Leffler. The photograph of predictable way to the coefficients of these H. Rademacher was provided by Bruce Berndt. modular forms. The truth of the Bloch-Kato

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