2. ETA EVALUATIONS USING WEBER FUNCTIONS. Introduction

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1 . ETA EVALUATIONS USING WEBER FUNCTIONS Introduction So ar we have seen some o the methods or providing eta evaluations that appear in the literature and we have seen some o the interesting properties o eta quotients and how they relate to special values o L-series. However all the methods we have seen so ar have certain limitations. Whilst they may tell us that our eta quotients are units and which number ields they lie in or tell us that they relate to such and such an L-series or even give various Galois actions on them, the methods still do not tell us explicitly which units they are. One way o speciying units precisely is to give minimum polynomials or them. The methods that we demonstrate over the next ew pages do precisely that. In our earlier pages we also noted that many o the existing methods put severe limitations on the class numbers o the underlying imaginary quadratic number ields rom which the arguments o our eta evaluations are taken. In this chapter we will overcome this limitation and provide evaluations where the class number is an odd prime and not necessarily 3. We start by giving a short summary o the work o Weber who, using modular equations or his Weber unctions, was able to provide various eta evaluations at certain quadratic irrationals. His techniques do not have a limitation on the class number. Many o the evaluations that appear in the tables at the back o his Lehrbuch der Algebra [] have class numbers which are not a power o two. However Weber s methods do have other limitations. Firstly Weber can only evaluate eta quotients which are derived rom his Weber unctions. Secondly he limits himsel mostly to evaluations at values ω = m or a positive integer m. In numerous cases we manage to overcome the second limitation and provide evaluations at other quadratic irrationals. This can only be done because o a number o tricks which Weber could not have employed, mainly due to the act that computers did not exist in his day to provide the necessary numerical insight required to guess solutions to various equations. Despite this educated guesswork our methods are worth recording since they seem to work in more cases than one would initially expect and the inormation that is guessed is only slight and easily ollows once the inal result is proved. Thus, in this chapter, we are able to present quite a number o new explicit eta evaluations where the class number is ive and make progress towards evaluations where the class number is seven. We do however limit ourselves here to evaluations involving the Weber unctions. In later chapters, ater an examination o Weber s method or obtaining modular equations, we generalise these unctions and their modular equations and this allows us to provide urther eta evaluations which Weber could not provide. Strangely there we do not need any numerical inormation to guess solutions as we do in this chapter. 37

2 3. ETA EVALUATIONS USING WEBER FUNCTIONS 1. Weber s Own Evaluations Weber himsel provides numerous eta evaluations. He is usually interested however in evaluating one o his unction or 1 at m or some natural number m. In 1 o his Lehrbuch [] he uses previously calculated values o the unction γ (ω) = 3 j(ω) and the equation u 3 γ (ω)u 1 = 0 which has roots (ω), 1 (ω) and (ω) to calculate minimum polynomials or (ω) at certain values o ω. However this method works only or the situations where one can explicitly calculate the value γ (ω) and where it has a nice value, such as an integer. In act this happens or ω = p, where p is a prime or 1 and a discriminant with class number one. Thus Weber calculates minimum polynomials or ( p) when p = 1, 3, 7, 11, 19, 3, 7 and 13. He also calculates 1 ( ) in a similar way. In 19, by using transormations o second order (essentially derived rom the identity 1 (ω) (ω) = ) Weber is able to use the above results to compute values o 1 ( m) or the composite values m =,, 1, 1, and 3. By 130 Weber has realised that he can use both the Schläli modular equations and his modular equations o irrational orm to obtain Weber unction evaluations. We discuss both kinds o modular equation and the derivations Weber gives or them in a later chapter. For the reader not yet amiliar with these, a Schläli modular equation provides a polynomial relationship between u = (ω) and v = (nω) or between u = 1 (ω) and v = 1 (nω), usually where n is prime. These modular equations can be exploited by setting ω = m or some integer m so that we have a polynomial relationship between ( m) and ( n m), etc. This method is particularly useul or cases where the value o ( m) (or 1 ( m)) is known and is a nice value such as a root o an integer. This is certainly the case or m = 1,, 3, or 7. With this technique Weber is able to compute the values ( n m) where n m = 9, 5, 7, 9, 3, 75 and 175, and he computes 1 ( n m) where n m = 1, 3, 50 and 100. There are o course higher values o n m or which this technique works but one sees the limited applicability o it due to its reliance on a known simple value being given or ( m). Next Weber makes use o the act that m/m = 1/ m. It is easy to see that i one sets ω = m/m then (ω) = ( 1/ m) = ( m) = (mω). Using this inormation, a Schläli modular equation o degree m involving (ω) and (mω) becomes a minimum polynomial or ( m) once we set ω = m/m in it. Using this method Weber obtains the additional evaluations ( m) or the values m = 5, 13 and 17. Next Weber takes ω to be a root o x + rx + n where n is odd and r any integer. This has as root ω = ( r + m)/ where m = n r. O particular importance is the act that n/ω = r + m. The idea is to consider the modular equation o degree n relating (ω) and (ω/n). A simple calculation shows that these have the values exp( rπi/) /x and

3 . ETA EVALUATIONS USING WEBER FUNCTIONS 39 exp( rπi/)x respectively where x = 1 ( m) or x = ( m) depending on whether r is odd or even. Weber gives a bewildering array o values m and n or which this method works. He uses it to calculate ( 13) and 1 ( m) or m = 10, and. Lastly Weber sets ω to be the root ( r + m)/(n) o nx + rx + n where m = n r. This time he considers the values (ω), (ω/n) and (nω). Again depending on whether r is odd or even the last two o these can both be expressed in terms o x = 1 ( m) or x = ( m). Combining two Schläli modular equations o degree n in an obvious way one can then eliminate (ω) leaving only a relation between (ω/n) and (nω). This leaves o course an equation involving only the expression x. Weber uses this technique to evaluate ( 1). Now comes Weber s 131 where he makes use o the modular equations o irrational orm. These are modular equations that involve all three o his Weber unctions, 1 and. For example, consider Weber s modular equations o degree n 7 (mod ) o this orm. They are polynomial relations between unctions o the orm and A = (ω)(nω) + ( 1) n+1 (1 (ω) 1 (nω) + (ω) (nω)) B = 1 (ω) 1 (nω) + n+1 + ( 1) (ω) (nω) (ω)(nω). Now one sets ω = 1/ n so that nω = n. We set x = ( n). We also note that 1 (ω) 1 (nω) becomes ( n) 1 ( n) = /( n) = 1/x, etc. In act we have A = x + ( 1) m+1 /x and B = x + ( 1) m+1 /x, (where we correct a probable typographical error at this point in Weber). In other words the modular equation o irrational orm, being a polynomial relationship between the unctions A and B, induces a polynomial equation in the single value x which when actored provides a minimal polynomial or the value x = ( n)/. This method allows Weber to obtain new evaluations or ( n) where n = 3, 31, 7 and 71. We mention one inal method o Weber s involving his modular equations o irrational orm. His idea is somewhat limited in that it applies to a modular equation where the unction B does not appear. In the expression or A he picks the value o ω so that the term involving (ω) (ω/n) becomes a constant. The remaining terms involving and 1 he evaluates by making use o the ollowing identity o degree which he also derives in 131 (ω) (ω) + 1 (ω) (ω) = (ω) + / (ω). The reason that he can make use o this identity is that he picks ω very careully. He sets ω = ( r + m)/ a root o x +rx+n or some integer r and m = n r. The crucial beneit o this is that ω = n/ω r making the terms on the let hand side o the degree two identity equal to some power o the terms involving and 1 o his modular equation o irrational orm. So ater taking the appropriate power o his modular equation and substituting in this degree two identity Weber ends up with an expression containing only the value (ω) and various constants. But depending on whether or not r is even (ω)

4 0. ETA EVALUATIONS USING WEBER FUNCTIONS is, up to some constant actors, either ( m) or 1 ( m). Thus one obtains a polynomial eqution in one o these two values. Weber is able to use this process to obtain an evaluation o 1 ( ), since he can set m =, n = 3 and r = 0. There is another method which Weber gives in 131, however it makes use o his theory o modular equations o composite degree which we do not examine, so we omit a description o this method. It allows Weber to evaluate ( 39).. New Class Number Five Evaluations Using Weber Functions.1. Discriminant -7. We will calculate numerous eta quotients explicitly with arguments in the imaginary quadratic ( ield o discriminant d = 7. O special ) importance to us is the eta quotient which is related to the unit m1 o the previous chapter. This value is very easy to compute. However we will also calculate other eta quotients where the discriminant is 7, which are not so easy to get at, such as the unit m o the last chapter. Throughout this section we make extensive use o the Weber unction identities mentioned in the previous section and in addition we make use o the ollowing act without comment ( ) ( ) b + d b + d 1 = 1, a a where the bar denotes the complex conjugate and the arguments are quadratic irrationals. A similar result holds or the unction. These identities ollow straightorwardly rom the deinition o the Weber unctions and the q-series expression or the eta unction. Firstly we begin with Weber s own evaluation o x = ( 7) /. In his Lehrbuch [] he states in the tables at the end that x satisies the polynomial equation x 5 x 3 x x 1. But we notice that ( ) = / 1 (1 + 7) = / ( 7). Thus we can calculate this value rom the known value which Weber calculates. Next we notice that ( ) ( ) ( ) = = But now rom 1 = we calculate that ( ) = / ( ) = ( 7). ( ) Thus we have that x = / also satisies x 5 x 3 x x 1 = 0. This immediately gives us an evaluation o the unit m 1 o the last chapter or this discriminant. Next we notice that ( ) = / ( )

5 . ETA EVALUATIONS USING WEBER FUNCTIONS 1 This value is one that we know. In act we see that ( ) (1) y = / satisies y 5 + y + y 3 + y 1 = 0. ( ) What we will do now is relate this known value to / using a modular equation o degree three or Weber unctions. In act we will show later that this value is precisely the unit 1/m or the discriminant d = 7 which we spoke about in the last chapter. This is done as ollows. Firstly we ind ( ) and ( ) =, 1 ( ) ( ) = 1. 1 The arguments which appear here dier in pairs by a actor o three. We also note that ( ) ( ) =. 1 The type o modular equation which we will use is that o irrational orm spoken o by Weber in 75 o []. The degree three equation o this kind is (ω) (3ω) = 1 (ω) 1 (3ω) + (ω) (3ω). I we substitute in the value ω = then this becomes (3ω) = (3ω) 1 (3ω) + 1 (3ω) (3ω), where we use the act that 1 = to see that (ω) and (3ω) are conjugates. We wish to do away with the asymmetry on the right hand side o this equation. This we can do by repeatedly squaring the equation and rearranging so that all the asymmetry remains on the right hand side. Ater doing this twice (always replacing 1 (3ω) (3ω) with 1 (3ω) wherever it occurs) we obtain (3ω) 1 + 3/ (3ω) 1 (3ω) = 1 (3ω) (3ω) + (3ω) 1 (3ω) = 1 (3ω) ((3ω) 1 (3ω) ) + (3ω) ((3ω) (3ω) ) In other words we have shown = 1 (3ω) 1 (3ω) 1 + (3ω) 1. 1 (τ) 3/ (τ) = 1 (τ) 1 + (τ) 1, where τ = Then letting a = (τ), b = 1 (τ) and c = (τ) so that abc = we see that /(bc) (bc) = b 1 + c 1, which ater rearranging and taking the square root becomes /(bc) = b + c. It seems logical to now set l = a, m = b / and n = c /, since these are the actual units we will be dealing with. Now lmn = 1 and the equation above has become the very elegant identity () m 5 n + n 5 m = 1.

6 . ETA EVALUATIONS USING WEBER FUNCTIONS But the value n is precisely the value given in the equation (1) above. Thereore we only need to solve the previous equation in order to obtain the eta evaluation or m. At this point we happen to notice numerically that l = n + 1. Thus rom lmn = 1 we have that 1/m = n(n + 1). Substituting this value into the equation () and making use o the act that (n) = 0 where (x) is the minimum polynomial or n we ind that this value o m is indeed a root. Since the polynomial () only has one real root we must have the correct value or m. Now rom the minimal polynomial (x) or the value n we have that + n = (n) + + n. Ater expanding this out, the right hand side is divisible by n. Dividing through by n we obtain /n + 1 = n + n 3 + n + n + 1. Now the right hands side o this equation is a square. Thus we have /n + 1 = n + n + 1 = /m + 1. In other words, m = 1+ /n+1. Expanding this out leads to n = m /(m + 1). I we plug this into () and actorise we obtain a minimum polynomial or m (it is the only actor o degree 5). We ind in this way that m has minimum polynomial x 5 + x + x 3 x x 1. Now as we mentioned earlier this value is in act the unit 1/m spoken o in the previous chapter. This can be proved as ollows 3 m = h(, 1, ) /h(3, 1, ) = η((1 + 7)/) η((1 + 7)/) 3 = η((1 + 7)/) i( 1 + 7)/ η(( 1 + = / 1 ((1 + 7)/), 7)/) which is the inverse o the value we have calculated above. We can continue rom our last evaluation and obtain urther interesting ones. Firstly we see that ( ) ( ) = = / ( ) ( ) Thus z = / satisies z 5 + z + z 3 z z 1 = 0. Now we do an interesting side calculation. We have ( ) ( ) = = / ( ) = / ( ) = / ( ) = / ( ) ( ) Thus we have 1+ 7 =. Combining this with 1 = we ind ( ) = / ( ) ,

7 . ETA EVALUATIONS USING WEBER FUNCTIONS 3 Eta Evaluations (d = 7) Minimum Polynomial ( 7) / = 1 / ( ) 1+ 7 x 5 x 3 x x 1 ( ) = 1+ 7 / ( ) 1 / = 1+ 7 ( ) 1+ 7 / ( ) 1 / = x 5 + x + x 3 x x 1 ( ) = 1+ 7 / ( ) = ( ) 1+ 7 x Table.1 and thereore satisies the polynomial equation x 5 + x + x 3 x x 1 = 0. ( ) However since 1+ 7 / ( ) = we have that ( ) z = / satisies z 5 + z + z 3 z z 1 = 0. 3 We summarise the eta evaluations that we have completed or the discriminant d = 7 in the above table... Discriminant -79. We move on to the next discriminant o class number 5 which is d = 79. Here we particularly wish to evaluate the quantities m 1 = / ( ) and m = / ( ) Recall that the second o these quantities was not even proved to be a unit in the previous chapter. However this will be obvious rom its evaluation which we will obtain shortly. Unortunately Weber does not compute x = ( 79) /, however Ramanujan has essentially calculated it. According to the table o class invariants o 3. o [1] it satisies the equation x 5 3x + x 3 x + x 1 = 0. Now we note that ( ) = / ( ) = / ( 79). Also we have ( ) ( ) ( ) = = Thereore rom 1 = we ind that ( ) x = / satisies x 5 3x + x 3 x + x 1 = 0 Also we have ( ) = / ( ) Thus in particular ( ) y = / satisies y 5 y + y 3 y + 3y 1 = 0.

8 . ETA EVALUATIONS USING WEBER FUNCTIONS We now relate this value to another via a modular equation o degree 5. Firstly we have ( ) ( ) = 1 0 and ( ) ( ) = 0 and ( ) ( ) =. 0 We employ Weber s modular equation o irrational orm o degree 5. This is given in 75 o [] as (3) = (ω) (5ω) 1 (ω) 1 (5ω) (ω) (5ω). Once we substitute the particular value ω = this can be written = (5ω) (5ω) 1 (5ω) 1 (5ω) (5ω). We apply the same technique as we did or d = 7, putting the asymmetric parts to one side and squaring. Doing this just once we obtain (5ω) 1 1 (5ω) + = (5ω) 1 (5ω) + 1 (5ω) (5ω) + 1 (5ω) (5ω) = (5ω) ((5ω) (5ω) ) + 1 (5ω) ((5ω) 1 (5ω) ) + 3/ (5ω) = (5ω) 1 1 (5ω) 1 (5ω) 1 + 3/ (5ω). So writing l = (5ω), m = 1 (5ω) / and n = (5ω) so that lmn = 1 we have 1/(mn) = 1m + 1n 3(mn). But adding (mn) to both sides we end up with a square on the let and the right ((mn) /(mn) ) = (m + n ). It is not hard to determine that the correct square root to take is 1/(mn) (mn) = m + n which ater rearranging again and taking the correct square root gives that is 1/(mn) = m + n, () m 3 n + mn 3 = 1. Now we happen to note numerically that 1/n = m + 1. Substituting this into the previous equation and taking the resulting expression modulo the known minimum polynomial or n we see that this indeed gives a root o the equation (). Since the equation only has one real root, this must be it. Now we know the minimum polynomial or n. We can easily calculate the minimum polynomial or 1/n rom it and or 1/n 1 or that. But this must be the minimum polynomial or m. ( ) Thus we have that m = / satisies m 5 +m 3m m 1 = 0.

9 . ETA EVALUATIONS USING WEBER FUNCTIONS 5 Eta Evaluations (d = 7) Minimum Polynomial ( 79) / = 1 / ( ) x 5 3x + x 3 x + x 1 ( ) = / ( ) 1 / = ( ) / ( ) 1 / = x 5 + x 3x x 1 ( ) = ) x ( Table. There are a ew more evaluations that we can complete with this inormation. Firstly ( ) = / ( ) ( ) In other words z = / satisies z 5 + z + 3z 3 z 1 = 0. However we now compute ( ) ( ) = = / ( ) = / ( ) = / ( ) = / ( ) Thus we have that ( ) =. Thus combining this with the above we have that ( ) satisies x 5 + x 3x x 1 = 0. We summarise some o the evaluations we have obtained or d = 79 in the table above..3. Discriminant Again similar techniques work or d = 103 also o class number 5. Firstly we wish to evaluate the expression m 1 = η(( )/) η(( )/) = / ( ) = / ( ) = / ( ) However since both o the last expressions in the two rows o this equation are the same, then rom 1 = we see that these expressions are equal to the value ( ) = / ( ) / = ( 103). Now this evaluation is strangely not made in either [] or [1], however there does not seem to be any reason why it could not be calculated using Weber s method providing one accepts the use o a computer to calculate the q-series involved. We cheat o course and simply ask Pari to provide us with the minimum polynomial

10 . ETA EVALUATIONS USING WEBER FUNCTIONS using its algdep() unction. It does not take long to determine that m 1 has minimum polynomial x 5 + x + 3x 3 + 3x + x 1. Now it will be convenient to write m 1 in a slightly dierent orm. From the irst equation or m 1 above we see that m 1 = / ( ) ( ) 1 = /. For consistency we will denote this unit by n or consistency with the earlier cases we dealt with. Now we also wish to calculate the value m or the discriminant d = 103. In act we have 1/m = η(( )/) η(( )/) / = η(( )/) η(( )/) / ( ) = / ( ) = /. Again or consistency we denote this value by l. O course we also denote ( ) m = , so that lmn = 1. We happen to notice numerically that l = n + 1. Now in order to make use o a modular equation we must the values l, m and n in terms o values with argument τ = In act by making simple transormations o the expressions above we ind that m = (7τ), n = (7τ) / and l = 1 (7τ) /. Now we note that ( ) ( ) (τ) = = = (7τ). Similarly we ind that 1 (τ) = (7τ), (τ) = 1 (7τ). Now the appropriate modular equations is Weber s modular equation o irrational orm o degree 7. This is given by Weber in 75 o [] as (τ)(7τ) (τ)(7τ) 1 (τ) 1 (7τ) = 0, which, or the speciic value o τ that we have chosen, becomes (7τ)(7τ) (7τ) 1 (7τ) 1 (7τ) (7τ) = 0. We remove the asymmetry rom this equation as we did in earlier cases by repeatedly putting all the asymmetric terms on one side o the equation and squaring until all the asymmetric terms contain -th powers. These we replace in exactly the same manner as beore and we inally end up with (l + n ) = /m + m 5 5m. O course we can use the expression lmn = 1 to change this equation so that it is only in terms o two o the values l, m and n.

11 . ETA EVALUATIONS USING WEBER FUNCTIONS 7 Eta Evaluations (d = 103) Minimum Polynomial / ( ) / ( ) 1 = 103 x 5 + x + 3x 3 + 3x + x 1 ( ) = / ( ) ( ) 1 / = / x 5 3x + 5x 3 x + x 1 Table.3 Firstly we subtract /m = l m rom each side and multiply through by m. Ater rearranging we obtain m (l + n ) = (m 3 1)(m 3 ). Dividing through by m 9 and using 1/m = ln we obtain (ln) 5 (l n ) = (1 (ln) 3 )(1 (ln) 3 ). By putting l = n + 1 into this and using the minimum polynomial or n which we know, we see that indeed l = n + 1 is a root o this equation. However, i instead we substitute n = l 1 then we end up with a polynomial which l satisies. It has only two quintic actors (and some irrelevant actors o lower degree). One o these quintic actors we recognize as the minimum polynomial o n. Since l cannot equal n (otherwise l and n would be rational) and since n is the only real root o this quintic actor then l must have the other quintic actor as minimum polynomial. That is l satisies the quintic equation l 5 3l + 5l 3 l + l 1 = 0. We summarise these eta evaluations in Table.3... Discriminant -17. Finally or class number ive we look at the discriminant d = 17. The process or this discriminant is very similar to that or d = 79 so we simply summarise the steps involved. Firstly we wish to evaluate the unit which we call m 1 in the last chapter. It has the value m 1 = η((1 + 17)/) η((1 + 17)/ = / ( ) = / ( ) = / ( 17). But this is also equal to ((1 + 17)/) /. Likewise we ind that the unit m is given by 1/m = η((1 + 17)/) η((1 + 17)/ / ( ) = /. We call the irst o these values above n, the second m and to be consistent with what has gone beore we denote ( ) l = , so that as usual lmn = 1. Again neither Weber nor Ramanujan/Berndt evaluate ( 17), however we believe that Weber s method can still be applied. We ourselves ask Pari to do the evaluation numerically which has that n satisies n 5 n n 3 + n + 3n 1 = 0.

12 . ETA EVALUATIONS USING WEBER FUNCTIONS Eta Evaluations (d = 17) Minimum Polynomial / ( ) / ( ) 1 = 17 x 5 x x 3 + x + 3x 1 ( ) = / ( ) / x 5 x 3 x + x 1 = 0 Table. Again we happen to notice numerically that m = 1/l + 1 (we must by now suspect some kind o general relation o this orm or class number ive at least when the Weber unctions are involved, though we still do not have a speciic conjecture and certainly no idea o a proo o such a result). Even better we notice that n + 1/m = 1. Thus i this result were proved, rom the minimal polynomial or n by substituting n = 1 1/m we can obtain a minimal polynomial or m. However as usual, to veriy that the result is indeed true we go via a modular equation. We write ( ) m = / ( ), n = /, etc. Now we can make use o the modular equation o degree 11, since ( ) ( ) ( ) = 1, 1 and so on. = ( In a manner similar to our previous arguments, the modular equation or degree 11 becomes (11τ) (11τ) (11τ) 1 (11τ) 1 (11τ) (11τ) =, or the value τ = (7 + 17)/. Ater twice putting the asymmetric part on one side o the equation and squaring we obtain l l + 17l + 1/l /l = m + n. Ater rearranging we can take the square root l 3l + = l(m + n ). Now we replace l with 1/(mn), multiply through by (mn) and we have (mn) 3(mn) + 1 = (mn) 3 (m + n ). Substituting in m = 1/(1 n) and using the minimum polynomial or n we see that this is indeed a root o the equation. As usual we prove that our relation between m and n holds. Finally substituting n = 1 1/m into this equation gives us a minimum polynomial or m ater rearranging and actorizing. In act the resulting expression has only one quintic actor so we ind that m satisies m 5 m 3 m + m 1 = 0. Again we summarise our evaluations in a table. This is Table. above. ),

13 . ETA EVALUATIONS USING WEBER FUNCTIONS 9 3. Other Discriminants and Class Numbers Unortunately our run o luck with class number ive now comes to an end. The next two undamental discriminants with this class number are d = 131 and d = 179. However the units that are associated with these discriminants are no longer Weber unction eta quotients. It seems that we need some new sets o unctions which are eta quotients similar to the Weber unction but where the arguments o the numerator and denominator dier by a actor o 3, 5 or even 7. In a later chapter we ind exactly such sets o unctions and ind that they satisy modular equations. Using these we are able to provide new evaluations which the Weber unctions do not provide. Another direction we would like to take things is to higher class numbers. The irst discriminant o class number seven is d = 71. There are now three units which we would like to evaluate m 1 = η((1 + 71)/) η((1 + 71)/) = / ( ) ( ) = / ( ) = /, m = η((1 + 71)/) η((3 + 71)/) = / ( ) = / ( ) = / ( ) and m 3 = η((3 + 71)/) 3 η((1 + 71)/) = η((5 + 71)/) η((5 + 71)/1) = 3 = / 1 ( η(( )/) η(( )/) ) = / ( ) The irst o these units can be expressed in terms o ( 71). Firstly we note that ( ) ( ) = , and so ( ) = / ( ) = m 1. But now m 1 = / 1 ( ) = / ( 71). But Weber has computed this last value in the tables at the end o his Lehrbuch []. We see in act that m 1 has minimum polynomial x 7 +x x 5 x x 3 +x +x 1. Now we wish to relate m 1 and m with a modular equation o degree 5. We note that ( ) ( ) =, etc. 0

14 50. ETA EVALUATIONS USING WEBER FUNCTIONS The modular equation o degree ive has already been given as equation (3) and it must lead to the same relation () as beore except that this time we note that it must relate m 1 with 1/m, i.e. (5) m 3 1/m + m 1 /m 3 = 1. Numerically we note that m 1 = 1 m 1/m + 1/m. Substituting this in gives a prospective minimal polynomial or m, which is x 7 x + x 5 x + 5x 3 x + x 1. Forget our original deinition o m or a moment and deine it to be the (only) real root o this polynomial. But i this is so then 1/m = m m 5 + m m 3 + 5m m +. We square this to obtain a similar expression or 1/m. Now we plug these into m 1 = 1 m 1/m +1/m and reduce modulo the minimal polynomial or m. We obtain m 1 = m m 5 + m + m 1. Now m 1 is real and we easily veriy that it satisies the minimal polynomial or m 1. Thus it in act equals m 1. Thus we have proven that m 1 = 1 m 1/m + 1/m where m is the real root o x 7 x + x 5 x + 5x 3 x + x 1. But since this really is only the purported m let us now denote it m. Thus we have a proven expression or m 1 in terms o m. Plugging this into (5) we obtain a proven expression or m in terms o m. Since m is determined uniquely by this equation it is suicient to show that m = m is a solution. This we do, thus showing that m and m are one and the same. It is easy to see that these evaluations are not as easy as the class number ive ones. The situation does not improve much or the unit m 3. From the deinitions above, we see that m 3 is equal to / ( ) = / ( ) whilst m becomes the value / ( ) ( ) 1 = / ( ) = /. But now we note that ( ) ( ) =, etc. Thereore we can use the modular equation o degree three. Ater a struggle similar to that above we ind that m 3 has minimum polynomial x 7 + x + x 5 + x + x 3 x 3x 1.. Conclusion We have been able to evaluate eta quotient units when the class number is ive, by making use o interesting numerical coincidences. This works or the irst our discriminants o class number ive and it is interesting to note that it is the modular equations o degree 3, 5, 7 and 11 which are employed, in that order, or these cases. Ater this we run into trouble expressing our eta units in terms o Weber unctions. This gives the impetus or our development o generalizations o the Weber unctions and their modular equations, which are described in the later chapters.

15 . ETA EVALUATIONS USING WEBER FUNCTIONS 51 For class number seven, things are not so simple. It may be possible to do better by using our higher level Weber unctions o later chapters. We deer a urther discussion o eta quotient evaluations until then. Over the next ew chapters we look at Weber s development o modular equations or his Weber unctions and we inally generalize these and their modular equations in preparation or urther evaluation work. This generalization o the Weber unctions will in act be the workhorse o this document and represents the most signiicant part o the research that we undertake. For the next chapter however we include a short (light) article written or the Australian Mathematical Society Gazette which recapitulates in succinct orm the motivations which lead to the later work, as just described. Reerences [1] Bruce Berndt, Ramanujan s Notebooks Part V, Springer-Verlag, 199 [] Heinrich Weber, Lehrbuch der Algebra, Dritter Band, Third Edition. Chelsea N.Y.

16 5. ETA EVALUATIONS USING WEBER FUNCTIONS

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