aa r X i v : . [ m a t h . N T ] J un ON LINEAR FORMS IN JACOBI THETA-CONSTANTS
CARSTEN ELSNER AND VEEKESH KUMAR
Abstract.
Elsner, Luca and Tachiya [5] proved that the values of the Jacobi-theta constants θ ( mτ ) and θ ( nτ ) are algebraically independent over Q for distinct integers m, n under someconditions on τ . On the other hand, in 2018 Elsner and Tachiya [4] also proved that three values θ ( mτ ) , θ ( nτ ) and θ ( ℓτ ) are algebraically dependent over Q . In this article we prove the non-vanishing of linear forms in θ ( mτ ), θ ( nτ ) and θ ( ℓτ ) under various conditions on m, n, ℓ , and τ . Among other things we prove that for odd and distinct positive integers m, n > θ ( τ ), θ ( mτ ) and θ ( nτ ) are linearly independent over Q when τ is an algebraic numberof some degree greater or equal to 3. In some sense this fills the gap between the above-mentionedformer results on theta constants. A theorem on the linear independence over C ( τ ) of the functions θ ( a τ ) , . . . , θ ( a m τ ) for distinct positive integers a , . . . a m is also established. introduction For a complex number τ from the upper complex half plane H , the theta functions are definedas follows; θ ( τ ) = 2 ∞ X n =0 q ( n +1 / , θ ( τ ) = 1 + 2 ∞ X n =1 q n , and θ ( τ ) = 1 + 2 ∞ X n =1 ( − n q n , where q = e iπτ . For the sake of brevity we sometimes write θ i instead of θ i ( τ ), i = 2 , , j - function as follows; j ( τ ) = 256 ( λ − λ + 1) λ ( λ − , where λ = λ ( τ ) = θ θ , which is a modular function with respect to the group SL (2 , Z ).The motivation of this article comes from the following sources: In 2018, C. Elsner and Y. Tachiya[4] proved that for distinct integers ℓ, m and n , the functions θ ( ℓτ ) , θ ( mτ ) and θ ( nτ ) are alge-braically dependent over Q . Recently, in 2019, C. Elsner, F. Luca and Y. Tachiya [5] proved thefollowing: let τ be any complex number with Im( τ ) > e iπτ is algebraic. Let m, n ≥ θ ( mτ ) and θ ( nτ ) are algebraically independentover Q .Naturally the following two questions arise. Question 1.
Let m ≥ a , a , . . . , a m be distinct positive integers. Are the functions θ ( a τ ) , θ ( a τ ) , . . . , θ ( a m τ )linearly independent over C ( τ )?By Theorem 1.1 in [6] we know that for different positive integers m, n , and an algebraic number e iπτ with τ ∈ H , the two numbers θ ( mτ ) and θ ( nτ ) are algebraically independent over Q ,such that for algebraic numbers α and α , which do not vanish simultaneously, the linear form Mathematics Subject Classification.
Primary 11J72, Secondary 11J91.
Key words and phrases.
Linear independence, Jacobi theta-constants.
1N LINEAR FORMS IN JACOBI THETA-CONSTANTS 2 α θ ( mτ ) + α θ ( nτ ) does not vanish. So it seems natural to consider linear forms in three valuesof the theta constant θ . Question 2.
Let m ≥ a , a , . . . , a m be distinct positive real numbers. What are thevalues of τ and α , α , α such that the linear form L := α θ ( τ ) + α θ ( mτ ) + α θ ( nτ )does not vanish?In this article, we give the complete answer to the question 1 and answer the question 2 in thefollowing way: We consider linear forms with integers n > m >
1, certain numbers τ ∈ H , andalgebraic numbers α , α , α , and give conditions on α , α , α such that L = 0.We divide the remaining part of our article into three sections: In section 2 we will state ourtheorems, in section 3 we collect all the tools to prove our results, and in the last section we givethe proof of all the theorems from section 2.2. The results
The linear independence over C ( τ ) of the functions θ ( a τ ) , . . . , θ ( a m τ ) in τ . We begin with the following result on the linear independence over C ( τ ) of Jacobi-theta constants. Theorem 2.1.
Let a , a , . . . , a m be distinct positive integers. Then the m functions θ ( a τ ) , θ ( a τ ) , . . . , θ ( a m τ ) in τ ∈ H are linearly independent over C ( τ ) . On the linear independence of values of Jacobi-theta constants for θ ( τ ) , θ ( mτ ) , θ ( nτ ) with odd integers m, n .Theorem 2.2. Let ≤ n < m be two odd integers. If one of the following conditions holds, namely (1) τ is an algebraic number of degree ≥ , (2) τ ∈ H such that q = e iπτ is algebraic over Q ,then the three numbers θ ( τ ) , θ ( mτ ) θ ( nτ ) are Q - linearly independent. Results on linear forms α θ ( τ ) + α θ ( mτ ) + α θ ( nτ ) for mn ≡ undercertain restrictions on the coefficients. Let m, n be two different positive integers, and let τ ∈ H satisfying the conditions in Theorem 2.2.Since the numbers θ ( mτ ) and θ ( nτ ) are algebraically independent when e iπτ is algebraic byTheorem 1.1 in [6], we consider the linear relations α θ ( τ ) + α θ ( mτ ) + α θ ( nτ )with α = 0. In order to state our next result we introduce the following set. Let s ≥ M s := (cid:8) ± √ u, ± i √ u : u ∈ N ∧ s ≡ u ) (cid:9) . N LINEAR FORMS IN JACOBI THETA-CONSTANTS 3
Theorem 2.3.
Let m = 2 a s and n = 2 b s be two different integers with a, b ≥ and odd integers s , s ≥ . Let τ ∈ H such that e iπτ is an algebraic number. Then, the inequality α θ ( τ ) + α θ ( mτ ) + α θ ( nτ ) = 0 holds, if β ∈ { α α − , α α − } satisfies one of the following conditions: β M s ∩ M s , β Q , R n, ( β − ) = 0 where R n, ( X ) is the polynomial defined in (3.5) in Lemma 3.1 below. In the case when additionally s and s are coprime odd integers, we have M s ∩ M s = {± , ± i } .Then we obtain from Theorem 2.3 the following corollary. Corollary 2.1.
Let m = 2 a s and n = 2 b s be two different integers with a, b ≥ and odd coprimeintegers s , s ≥ . Let τ ∈ H such that e iπτ is an algebraic number. Then, the inequality α θ ( τ ) + α θ ( mτ ) + α θ ( nτ ) = 0 holds for all algebraic numbers α , α , α , where either α /α or α /α is no unit of the Gaussianintegers Z [ i ] . For any positive integer n let ψ ( n ) := n Y p | n (cid:18) p (cid:19) , where p runs through all primes dividing n . Theorem 2.4.
Let m = 2 a s and n be two integers with a ≥ and odd integers n, s ≥ . Let τ ∈ H be as in Theorem 2.2. Then, the inequality α θ ( τ ) + α θ ( mτ ) + α θ ( nτ ) = 0 holds, if β := α α − satisfies one of the following conditions: deg Q (cid:0) β (cid:1) > ψ ( n ) , S n, ( n β − ) S n,d n ( n β − ) = 0 , where S n, ( X ) and S n,d n ( X ) are polynomials defined in (3.2) in Theorem 3.3 below. The preceding Theorems do not treat θ ( τ ) , θ (2 τ ), and θ (3 τ ) simultaneously. For this situationwe cite a result from [4, Example 1.5]: Let τ ∈ H , and define P ( X, Y, Z ) := 27 X − X Y − X Y Z + 64 X Y Z − X Z − Z . Then P ( X , Y , Z ) = 0 holds for X := θ (3 τ ) , Y := θ (2 τ ) , Z := θ ( τ ) . This shows that θ ( τ ) , θ (2 τ ) , θ (3 τ ) are homogeneously algebraically dependent of degree 8. Proposition 1.
Let m ≥ be an integer. Let τ ∈ H be as in Theorem 2.2. Then the three numbers θ (2 m τ ) , θ (2 m +1 τ ) , θ (2 m +2 τ ) are Q - linearly independent. N LINEAR FORMS IN JACOBI THETA-CONSTANTS 4
In Proposition 1 it is not possible to avoid the condition deg Q ( τ ) ≥
3. This follows for τ = i and m = 0 from the nontrivial relation q √ θ ( i ) − θ (2 i ) = 0due to Ramanujan, cf. [1, p. 325]. We can also find similar linear relations for the theta-constants θ and θ : q − √ θ ( i ) − √ θ (2 i ) = 0 , √ θ ( i ) − θ (2 i ) = 0 . Moreover, we have for τ = 1 + i √ q = − e − π √ ) the following linear identity involving valuesof θ and θ , 2 θ (cid:0) i √ (cid:1) − (1 + i ) q − √ θ (1 + i √
3) = 0 . Main Tools towards the proof of our results
Theorem 3.1. [[10] , page 5 ] Let τ ∈ H be an algebraic number of some degree not equal to 2. Then j ( τ ) is transcendental. Theorem 3.2. [[2] , Theorem 4 ] For any τ ∈ H , if q = e iπτ is an algebraic number, and for integers j, k, ℓ ∈ { , , } with j = k ,the three values θ j ( τ ) , θ k ( τ ) and Dθ ℓ ( τ ) are algebraically independent over Q . Here, D := 1 πi ddτ is a differential operator. Theorem 3.3. [[9] , Theorem 1, Corollary 4 ] For any odd integer n ≥ there exists an integer polynomial P n ( X, Y ) with deg X P n ( X, Y ) = ψ ( n ) such that P n (cid:18) n θ ( nτ ) θ ( τ ) , θ ( τ ) θ ( τ ) (cid:19) = 0 holds for all τ ∈ H . Moreover, the polynomial P n ( X, Y ) is of the form P n ( X, Y ) := X ψ ( n ) + R ( Y ) X ψ ( n ) − + · · · + R ψ ( n ) − ( Y ) X + R ψ ( n ) ( Y ) , (3.1) where R j ( Y ) ∈ Z [ Y ] for j = 1 , . . . , ψ ( n ) , and deg R k ( Y ) ≤ k · n − n (cid:0) ≤ k ≤ ψ ( n ) (cid:1) . Moreover, P n ( X, Y ) can be written as P n ( X, Y ) = d n X j =0 S n,j ( X ) Y j , (3.2) where S n,j ( X ) ∈ Z [ X ] with ≤ j ≤ d n and d n > , such that S n,j (0) = 0 (1 ≤ j ≤ d n ) , S n,d n ( X ) , (3.3) S n, (0) = P n (0 , ∈ Z \ { } . (3.4) N LINEAR FORMS IN JACOBI THETA-CONSTANTS 5
The properties (3.2) to (3.4) of P n ( X, Y ) follow from the proof of Lemma 7 in [5]; cf. formula(9).
Remark 1.
From (3.1), we can observe that for any complex number α , the polynomial P n ( X, α )is non-zero.The following Lemmas are very crucial for the proof of our results.
Lemma 3.1. [[4] , Lemma 2.5 ] Let n = 2 a s be an integer with a ≥ and an odd integer s ≥ . Then there exists an integerpolynomial Q n ( X, Y ) such that Q n (cid:18) θ ( nτ ) θ ( τ ) , θ ( τ ) θ ( τ ) (cid:19) = 0 holds for all τ ∈ H . Moreover, the polynomial Q n ( X, Y ) is of the form Q n ( X, Y ) = c a n Y a ψ ( s ) + a ψ ( s ) − X j =0 R n,j ( X ) Y j , (3.5) where c n is a non-zero integer. Moreover, deg R n,j ( X ) ≤ α ψ ( s ) − j (cid:0) ≤ j < a ψ ( s ) (cid:1) , (3.6) and Q n (0 , Y ) = c a n Y a ψ ( s ) , (3.7) R n, ( X ) = Q n ( X,
0) = 2 a − ψ ( s ) X (2 a − ψ ( s ) P s ( s X, . (3.8) Proof.
Apart from formula (3.8) the statements are given in [4, Lemma 2.5]. It remains to prove(3.8). We proceed by induction with respect to a and follow the lines of the proof of Lemma 2.5in [4]. Set Q s ( X, Y ) := P s ( s X, Y ). As in the first part of the proof of Lemma 2.5 (correspondingto a = α = 1) we construct the polynomials B s ( X, Y ), ˜ Q s ( X, Y ), and Q s ( X, Y ), where˜ Q s ( X , Y ) = B s ( X, Y ) B s ( X, iY )and Q s ( X, Y ) = ˜ Q s ( X, − Y ) . From the proof of Lemma 2.5 we obtain B s ( X,
1) = 2 ψ ( s ) P s ( s X , , B s ( X, i ) = 2 ψ ( s ) X ψ ( s ) . This gives ˜ Q s ( X ,
1) = B s ( X, B s ( X, i ) = 2 ψ ( s ) X ψ ( s ) P s ( s X , , and, consequently, Q s ( X,
0) = ˜ Q s ( X,
1) = 2 ψ ( s ) X ψ ( s ) P s ( s X, . This shows that (3.8) holds for a = 1. Next, let (3.8) be already proven for some fixed a ≥
1. Forthe induction step we construct the polynomials B a +1 s ( X, Y ), ˜ Q a +1 s ( X, Y ), and Q a +1 s ( X, Y ),where ˜ Q a +1 s ( X , Y ) = B a +1 s ( X, Y ) B a +1 s ( X, iY ) , Q a +1 s ( X, Y ) = ˜ Q a +1 s ( X, − Y ) . Since deg X Q a s ( X, Y ) = 2 a ψ ( s ), we obtain by applying the induction hypothesis, B a +1 s ( X,
1) = 2 · a ψ ( s ) Q a s ( X ) = 2 a +1 ψ ( s ) (cid:0) a − ψ ( s ) X a − ψ ( s ) P s ( s X , (cid:1) ,B a +1 s ( X, i ) = 2 · a ψ ( s ) X · a ψ ( s ) = 2 a +1 ψ ( s ) X · a ψ ( s ) . N LINEAR FORMS IN JACOBI THETA-CONSTANTS 6
Therefore, it turns out that˜ Q a +1 s ( X ,
1) = B a +1 s ( X, B a +1 s ( X, i )= 2 · a +1 ψ ( s )+4(2 a − ψ ( s ) X · a ψ ( s )+4(2 a − ψ ( s ) P s ( s X , a +1 − ψ ( s ) X a +1 − ψ ( s ) P s ( s X , . We complete the proof of the lemma by observing that Q a +1 s ( X,
0) = ˜ Q a +1 s ( X,
1) = 2 a +1 − ψ ( s ) X (2 a +1 − ψ ( s ) P s ( s X, , which is the identity in (3.8) with a replaced by a + 1. (cid:3) Lemma 3.2.
Let s ≥ be an odd integer. Then we have P s ( s X,
0) = s ψ ( s ) Y u | su ≥ (cid:0) X − u − (cid:1) w ( s/u,u ) , where w ( a, b ) is defined by the number of integers k with ≤ k < b and gcd ( a, b, k ) = 1 . Proof.
This follows from the identity given in Lemma 4 in [5], namely, P s ( X,
0) = Y u | su ≥ (cid:0) X − u (cid:1) w ( u,s/u ) . (cid:3) Lemma 3.3.
Let τ ∈ H be as in Theorem 2.2. Then the numbers θ /θ and θ /θ are transcen-dental.Proof. Case 1. τ is an algebraic number of degree ≥ j ( τ ) = 256 ( λ − λ + 1) λ ( λ − is transcendental. This implies that λ = θ /θ is transcendental, and so is θ /θ . Now by usingthe identity θ + θ = θ , we conclude that the number θ /θ is transcendental. Case 2. τ ∈ H such that q = e iπτ is algebraic over Q Since q = e iπτ is an algebraic number, θ and θ are algebraically independent as well as θ and θ (cf. Theorem 3.2). Therefore, the numbers θ /θ and θ /θ are transcendental. By Case 1 and 2,we complete the proof of the lemma. (cid:3)(cid:3) Lemma 3.4.
Let m ≥ be an integer which is either odd or it is an even number of the form a s , where a ≥ , and s ≥ is an odd integer. Then, for any τ ∈ H satisfying the conditions inTheorem 2.2, the number θ ( mτ ) θ ( τ ) is transcendental.Proof. We assume that θ ( mτ ) /θ ( τ ) is algebraic. By Theorem 3.3 and Lemma 3.1, there exists aninteger polynomial T m ( X, Y ) defined by T m ( X, Y ) := (cid:26) P m ( m X, Y ) , if m ≡ ,Q m ( X, Y ) , if m ≡ , N LINEAR FORMS IN JACOBI THETA-CONSTANTS 7 such that T m (cid:18) θ ( mτ ) θ ( τ ) , θ ( τ ) θ ( τ ) (cid:19) = 0 . (2.5)Now we consider the polynomial R m ( Y ) = T m (cid:16) θ ( mτ ) θ ( τ ) , Y (cid:17) ∈ Q [ Y ] having algebraic coefficientsby our assumption. The polynomial R m ( Y ) does not vanish identically: for odd integers m thisfollows from Lemma 2.1 in [4], for even m this is a consequence of Lemma 3.1, cf.(3.5). Hence, by(2.5), we obtain R m (cid:18) θ ( τ ) θ ( τ ) (cid:19) = T m (cid:18) θ ( nτ ) θ ( τ ) , θ ( τ ) θ ( τ ) (cid:19) = 0 . This implies that θ /θ is algebraic, which is a contradiction to Lemma 3.3. Therefore, we concludethat the number θ ( mτ ) /θ ( τ ) is transcendental. (cid:3) Proof of our results
Proof of Theorem 2.1.
Suppose that these functions are linearly dependent over C ( τ ). Thenthere exist c ( τ ) , . . . , c m ( τ ) ∈ C [ τ ], not all zero and with minimal degree, such that c ( τ ) θ ( a τ ) + · · · + c m ( τ ) θ ( a m τ ) = 0 for all τ ∈ H . (4.1)Notice that for all i = 1 , , . . . , m , θ ( a i ( τ + 2)) = 1 + 2 ∞ X n =1 e iπa i τn e a i πi = 1 + 2 ∞ X n =1 e iπa i τn = θ ( a i τ ) . Hence, the functions θ ( a τ ) , θ ( a τ ) , . . . , θ ( a m τ ) are periodic.Replacing τ by τ + 2 and using the periodicity, we have c ( τ + 2) θ ( a τ ) + · · · + c m ( τ + 2) θ ( a m τ ) = 0 for all τ ∈ H . (4.2)Thus, from (4.1) and (4.2), we obtain( c ( τ ) − c ( τ + 2)) θ ( a τ ) + · · · + ( c ( τ ) − c m ( τ + 2)) θ ( a m τ ) = 0 for all τ ∈ H . Note that the degree of the polynomial c i ( τ +2) − c i ( τ ) is strictly less than the degree of the polyno-mial c i ( τ ). Therefore, by the minimality of the polynomials c ( τ ) , . . . , c m ( τ ), we get c i ( τ +2) = c i ( τ )for all i = 1 , , . . . , m , which in turns implies that c ( τ ) , . . . , c m ( τ ) are constant polynomials. Hence,in order to prove that these functions are linearly independent over C ( τ ), it suffices to prove thelinear independence over C .Therefore we can consider the identity c θ ( a τ ) + · · · + c m θ ( a m τ ) = 0 , for all τ ∈ H and fixed c i ∈ C . This can be rewritten as c ∞ X n =1 e iπτa n ! + · · · + c m ∞ X n =1 e iπτa m n ! = 0 for all τ ∈ H . (4.3)Putting τ = iX and letting X → ∞ in the above equality, we have( c + · · · + c m ) + 2 lim X →∞ c ∞ X n =1 e − πXa n + · · · + c m ∞ X n =1 e − πXa m n ! = 0 . N LINEAR FORMS IN JACOBI THETA-CONSTANTS 8
Since lim X →∞ (cid:16)P ∞ n =1 e − πXa i n (cid:17) = 0 for all i = 1 , , . . . , m , we have c + c + · · · + c m = 0 . Therefore (4.3) becomes c ∞ X n =1 e − πXa n + · · · + c m ∞ X n =1 e − πXa m n = 0 for all X > . (4.4)Without loss of generality we can assume that a < a < · · · < a m . Multiplying the above equalityby e a πX , we get − c = c ∞ X n =2 e − πXa n + πXa + (cid:16) c ∞ X n =1 e − πXa n + πXa + · · · + c m ∞ X n =1 e − πXa m n + πXa (cid:17) (4.5)Since − πa n + πa < n ≥ − πa i n + πa < i = 2 , , . . . , m , we see that theright-hand side of (4.5) tends to zero as X → ∞ . Therefore, we conclude that c = 0, and (4.4)becomes c ∞ X n =1 e − πXa n + · · · + c m ∞ X n =1 e − πXa m n = 0 for all X ∈ N . Now we multiply the above equality by e a πX and proceed by the same process in order to get c = 0. Hence, by continuing this process, we get c = c = · · · = c m = 0, which gives acontradiction. This proves the theorem. (cid:3) Proof of Theorem 2.2.
It is suffices to prove that the three numbers 1 , θ ( mτ ) θ ( τ ) , θ ( nτ ) θ ( τ ) are Q -linearly independent. Suppose that these numbers are Q - linearly dependent. Then, there existalgebraic integers α , α , α not all zero such that α + α θ ( mτ ) θ ( τ ) + α θ ( nτ ) θ ( τ ) = 0 . (4.6)It is clear that neither α nor α vanishes, sine otherwise (when α = 0, α = 0 or α = 0, α = 0) there is a contradiction to Lemma 3.4, since both the numbers θ ( mτ ) θ ( τ ) and θ ( nτ ) θ ( τ ) aretranscendental. This implies that both, α and α , are non-zero. Then, when α = 0, we get acontradiction to Theorem 1.1 in [6]. Thus it is α α α = 0. Then from (4.6) and Theorem 3.3, wehave P m m (cid:18) − α α − α α θ ( nτ ) θ ( τ ) (cid:19) , θ ( τ ) θ ( τ ) ! = 0 . (4.7)By the explicit form of the polynomials P m ( X, Y ) and P n ( X, Y ), we see that the polynomials H m ( X ) = P m m (cid:18) − α α − α α X (cid:19) , θ ( τ ) θ ( τ ) ! and S n ( X ) = P n (cid:18) n X , θ ( τ ) θ ( τ ) (cid:19) are non-zero. The polynomials P m m (cid:18) − α α − α α θ ( nτ ) θ ( τ ) (cid:19) , Y ! and P n (cid:18) n θ ( nτ ) θ ( τ ) , Y (cid:19) have the same common root Y := θ ( τ ) /θ ( τ ). Hence, the resultant R ( X ) := Res Y P m m (cid:18) − α α − α α X (cid:19) , Y ! , P n ( n X , Y ) ! (4.8) N LINEAR FORMS IN JACOBI THETA-CONSTANTS 9 is given by the determinant of a square matrix where the dimensions and elements of the corre-sponding Sylvester matrix depend on the coefficients of the polynomials P m (cid:18) m (cid:18) − α α − α α (cid:19) X, Y (cid:19) and P n ( n X, Y ). By Lemma 3.4 we know that θ ( nτ ) /θ ( τ ) is transcendental. Then, from S n,d n S m,d m (cid:16) m (cid:16) − α α − α α · θ ( nτ ) θ ( τ ) (cid:17) (cid:17) = 0 and S n,d n (cid:16) n θ ( nτ ) θ ( τ ) (cid:17) = 0 . Hence, there is some real number δ > n, m, α , α , α , and τ such that S m,d m (cid:16) m (cid:16) − α α − α α X (cid:17) (cid:17) = 0 and S n,d n (cid:16) n X (cid:17) = 0hold for (cid:12)(cid:12)(cid:12) X − θ ( nτ ) θ ( τ ) (cid:12)(cid:12)(cid:12) < δ . Then, for all X from this circle, we havedeg Y P m m (cid:18) − α α − α α X (cid:19) , Y ! = deg Y P m m (cid:18) − α α − α α · θ ( nτ ) θ ( τ ) (cid:19) , Y ! = d m , and, similarly, deg Y P n (cid:0) n X , Y (cid:1) = deg Y P n (cid:16) n θ ( nτ ) θ ( τ ) , Y (cid:17) = d n . For X restricted to the above circle, R ( X ) can be considered as a polynomial in X depending onthe elements of a Sylvester matrix with fixed dimensions d n + d m . On the scale of things R ( X ) issome polynomial with algebraic coefficients such that R (cid:16) θ ( nτ ) θ ( τ ) (cid:17) = 0 , (4.9)since Y is a common root of the polynomials under consideration. First we note that the polyno-mial R ( X ) is not identically zero. We assume the contrary, namely R ( X ) ≡ . (4.10)Then, by (4.8) and (4.10),Res Y P m m (cid:18) − α α − α α X (cid:19) , Y ! , P n ( n X , Y ) ! = R ( X ) ≡ , and so there exists a common factor H ( X, Y ) ∈ C [ X, Y ] with positive degree in Y of the polyno-mials P m m (cid:18) − α α − α α X (cid:19) , Y ! and P n ( n X , Y ) . Let P m m (cid:18) − α α − α α X (cid:19) , Y ! = H ( X, Y ) G ( X, Y ) . N LINEAR FORMS IN JACOBI THETA-CONSTANTS 10
By substituting Y = λ ( τ ) into the above equation, we have P m m (cid:18) − α α − α α X (cid:19) , λ ( τ ) ! = H ( X, λ ( τ )) G ( X, λ ( τ )) . (4.11)By deg X T ( X, Y ) and deg Y T ( X, Y ) we denote the degree of the polynomial T ( X, Y ) with respectto X and Y , respectively; and deg T ( X, Y ) denotes the total degree of the polynomial T ( X, Y ).Since, by Theorem 3.3,deg Y R k ( Y ) ≤ k · m − m < k, (cid:0) ≤ k ≤ ψ ( m ) (cid:1) , such that we havedeg X P m m (cid:18) − α α − α α X (cid:19) , λ ( τ ) ! = ψ ( m ) = deg P m m (cid:18) − α α − α α X (cid:19) , Y ! . Hence, by the above identities, we obtaindeg X H (cid:0) X, λ ( τ ) (cid:1) + deg X G (cid:0) X, λ ( τ ) (cid:1) = deg H ( X, Y ) + deg G ( X, Y ) . Additionally, we have the obvious inequalitiesdeg X H (cid:0) X, λ ( τ ) (cid:1) ≤ deg H ( X, Y ) and deg X G (cid:0) X, λ ( τ ) (cid:1) ≤ deg G ( X, Y ) . Thus, we obtain deg X H ( X, λ ( τ )) = deg H ( X, Y ), and consequentlydeg X H ( X, λ ( τ )) ≥ deg Y H ( X, Y ) ≥ . (4.12)By [5, Lemma 1], the polynomial P m ( X, λ ( τ )) is irreducible, which implies that the polynomial P m m (cid:18) − α α − α α X (cid:19) , λ ( τ ) ! is also irreducible. Thus, from (4.11) and (4.12), we obtain P m m (cid:18) − α α − α α X (cid:19) , θ ( τ ) θ ( τ ) ! = β H ( X, λ ( τ ))for some non-zero complex number β . Similarly, there exists a non-zero complex number β suchthat P n ( n X , λ ( τ )) = β H ( X, λ ( τ )) , and hence P m m (cid:18) − α α − α α X (cid:19) , λ ( τ ) ! = cP n ( n X , λ ( τ )) , c := β /β . This polynomial identity holds for all complex numbers τ ∈ H . We know that for τ → i ∞ the function λ ( τ ) tends to zero. Hence, taking τ → i ∞ into the above equality, we have by [5,Lemma 2], Y d | m m (cid:18) − α α − α α X (cid:19) − d ! ω ( d,m/d ) = c Y d | n ( n X − d ) ω ( d,n/d ) . Then, comparing the multiplicity of the zero of these polynomials at X = − ( α + α / √ m ) /α (and d = 1 on the left-hand side), we obtain m = ω (1 , m ) ≤ max d ω ( d, n/d ) ≤ n, which is a contradiction to the condition n < m from the Theorem. Hence, the polynomial R ( X )is non-zero. Therefore, it follows from (4.9) that the number θ ( nτ ) /θ ( τ ) is algebraic, which is N LINEAR FORMS IN JACOBI THETA-CONSTANTS 11 a contradiction to the fact from Lemma 3.4 that the number θ ( nτ ) /θ ( τ ) is transcendental. Thisproves the assertion. (cid:3) Proof of Theorem 2.3.
Let m = 2 a s and n = 2 b s be two different integers with a, b ≥ s , s ≥
3. By Lemma 3.1 there exist integer polynomials Q m ( X, Y ) and Q n ( X, Y )such that Q m (cid:18) θ ( mτ ) θ ( τ ) , θ ( τ ) θ ( τ ) (cid:19) = 0 and Q n (cid:18) θ ( nτ ) θ ( τ ) , θ ( τ ) θ ( τ ) (cid:19) = 0 . We assume that the linear equation (4.6) holds, where α , α , α are algebraic numbers satisfyingthe hypothesis in Theorem 2.3. As in the proof of Theorem 2.2 we have α α α = 0. By thehypotheses of the Theorem we may assume without loss of generality that β := α /α M s , ordeg Q ( β ) ≥
3, or R n, ( β ) = 0. Then we obtain Q m (cid:18) − α α − α α θ ( nτ ) θ ( τ ) (cid:19) , θ ( τ ) θ ( τ ) ! = 0 . By the explicit form of the polynomials Q m ( X, Y ) and Q n ( X, Y ), we see that the polynomials Q m (cid:18) − α α − α α X (cid:19) , θ ( τ ) θ ( τ ) ! and Q n (cid:18) X , θ ( τ ) θ ( τ ) (cid:19) are non-zero. Hence the polynomials Q m (cid:18) − α α − α α θ ( nτ ) θ ( τ ) (cid:19) , Y ! and Q n (cid:18) θ ( nτ ) θ ( τ ) , Y (cid:19) have the same common root Y = θ ( τ ) /θ ( τ ).Let H m ( X, Y ) := Q m (cid:18) − α α − α α X (cid:19) , Y ! and W ( X ) := Res Y (cid:0) H m ( X, Y ) , Q n ( X , Y ) (cid:1) ∈ Q [ X ] . From Lemma 3.1 we know that both, deg Y Q m ( X, Y ) and deg Y Q n ( X, Y ), do not depend on X ,since the coefficients of the leading terms with respect to Y are non-zero integers. Thus, W ( X )can be considered as a polynomial for all X .In order to show that the polynomial W ( X ) does not vanish identically, we shall prove the existenceof a number η satisfying W ( η ) = 0, or, equivalently, that the polynomials H m ( η, Y ) and Q n ( η , Y )are coprime. Let η := − β = − α α . On the one side, by using (3.7), we obtain H m ( η, Y ) = Q m (0 , Y ) = c a m Y a ψ ( s ) . Therefore, H m ( η, Y ) is a nonvanishing polynomial in Y having exclusively a multiple root at Y = 0.On the other side, by applying formulas (3.5) and (3.6) in Lemma 3.1, we have Q n ( η , Y ) = c b n Y b ψ ( s ) + b ψ ( s ) − X j =0 R n,j ( η ) Y j N LINEAR FORMS IN JACOBI THETA-CONSTANTS 12 with c n ∈ Z \ { } and R n, ( X ) β and by (3.8)and Lemma 3.2 we conclude that R n, ( η ) = 0, since0 = η = 1 β = α α = 1 u with u | s and u ≥ . Consequently, we have Q n ( η ,
0) = R n, ( η ) = 0.Altogether, the polynomials H m ( η, Y ) and Q n ( η , Y ) have no common root. More precisely, weobtain for W ( X ), W ( η ) = Res Y (cid:0) H m ( η, Y ) , Q n ( η , Y ) (cid:1) = 0 . This shows that W ( X ) does not vanish identically. By construction, we know that W ( X ) vanishesfor X := θ ( nτ ) θ ( τ ) , which implies the algebraicity of θ ( nτ ) /θ ( τ ), a contradiction to Lemma 3.4.This finally shows that the linear relation (4.6) cannot hold. (cid:3) Proof of Theorem 2.4.
Let m = 2 a s and n be two integers with a ≥ n, s ≥ P n ( X, Y ) and Q m ( X, Y ) such that Q m (cid:18) θ ( mτ ) θ ( τ ) , θ ( τ ) θ ( τ ) (cid:19) = 0 and P n (cid:18) n θ ( nτ ) θ ( τ ) , θ ( τ ) θ ( τ ) (cid:19) = 0 . We assume that the linear equation (4.6) holds. As in the proof of Theorem 2.2 we have α α α = 0.By the hypotheses of the Theorem we may assume that β := α /α satisfies either deg Q ( β ) > ψ ( n ),or S n, ( n β − ) S n,d n ( n β − ) = 0. Then we obtain Q m (cid:18) − α α − α α θ ( nτ ) θ ( τ ) (cid:19) , θ ( τ ) θ ( τ ) ! = 0 . By the explicit form of the polynomials Q m ( X, Y ) and P n ( X, Y ) given by Theorem 3.3 and Lem-ma 3.1, we see that the polynomials Q m (cid:18) − α α − α α X (cid:19) , θ ( τ ) θ ( τ ) ! and P n (cid:18) n X , θ ( τ ) θ ( τ ) (cid:19) are non-zero. Hence the polynomials Q m (cid:18) − α α − α α θ ( nτ ) θ ( τ ) (cid:19) , Y ! and P n (cid:18) n θ ( nτ ) θ ( τ ) , Y (cid:19) have the same common root Y = θ ( τ ) /θ ( τ ). Let H m ( X, Y ) := Q m (cid:18) − α α − α α X (cid:19) , Y ! and W ( X ) := Res Y (cid:0) H m ( X, Y ) , P n ( n X , Y ) (cid:1) . From Lemma 3.1, formula (3.5), we know that deg Y Q m ( X, Y ) does not depend on X , since thecoefficient of the leading term with respect to Y is the non-zero integer c a m . For all real numbers X which are not a root of the polynomial S n,d n ( X ) in (3.2), the leading term of P n ( X, Y ) withrespect to Y does not vanish. Consequently, W ( X ) is given by the same polynomial for these X ,since deg Y H m ( X, Y ) and deg Y P n ( n X , Y ) do not change. Note that S n,d n ( X ) W ( X ) does not vanish identically for X with S n,d n ( X ) = 0, we shall prove N LINEAR FORMS IN JACOBI THETA-CONSTANTS 13 the existence of a number η satisfying W ( η ) = 0, or, equivalently, that the polynomials H m ( η, Y )and P n ( n η , Y ) are coprime. Let η := − β = − α α . On the one side, by using (3.7), we obtain H m ( η, Y ) = Q m (0 , Y ) = c a m Y a ψ ( s ) . Therefore, H m ( η, Y ) is a nonvanishing polynomial in Y having exclusively a multiple root at Y = 0.On the other side, by the hypothesis on β in Theorem 2.4 and bydeg X S n,d n ( X ) ≤ deg X P n ( X, Y ) = ψ ( n )(cf. (3.2) in Theorem 3.3), we know that S n,d n (cid:0) n η (cid:1) = S n,d n (cid:16) n β (cid:17) = S n,d n (cid:16) n α α (cid:17) = 0 . This shows that the degree with respect to Y of the polynomial on the right-hand side of (3.2) doesnot change for the particular choice of X = n η . Moreover, it follows from (3.4) that S n, X S n, ( X ) ≤ deg X P n ( X, Y ) = ψ ( n )and the conditions on β imply that S n, (cid:0) n η (cid:1) = S n,d n (cid:16) n β (cid:17) = S n, (cid:16) n α α (cid:17) = 0 . Thus, again the application of (3.2) gives P n (cid:0) n η , (cid:1) = 0 . Altogether, the polynomials H m ( η, Y )and P n ( n η , Y ) have no common root. More precisely, we obtain for W ( X ), W ( η ) = Res Y (cid:0) H m ( η, Y ) , P n ( n η , Y ) (cid:1) = 0 . This shows that W ( X ) does not vanish identically for all X satisfying S n,d n ( n X ) = 0. By con-struction, we know that W ( X ) vanishes for X := θ ( nτ ) θ ( τ ) , and since θ ( nτ ) /θ ( τ ) is transcendentalby Lemma 3.4, we have by (3.3) that S n,d n (cid:16) n θ ( nτ ) θ ( τ ) (cid:17) = 0 . Thus, X = X is a zero of the function W ( X ), which restricted to all values X satisfying S n,d n ( n X ) = 0 results in the same nonvanishing polynomial W ( X ). This implies the alge-braicity of θ ( nτ ) /θ ( τ ), a contradiction to Lemma 3.4. This finally shows that the linear relation(4.6) cannot hold. (cid:3) Proof of Proposition 1.
Replacing τ by 2 m τ , it suffices to prove the assertion for the three numbers θ ( τ ) , θ (2 τ ) , θ (4 τ ) . We have the following identies: 2 θ (2 τ ) = θ + θ , θ (4 τ ) = θ + θ . (4.13)Suppose there exist algebraic numbers α, β , γ not all zero such that2 αθ ( τ ) + 2 βθ (2 τ ) + 2 γθ (4 τ ) = 0 . (4.14) N LINEAR FORMS IN JACOBI THETA-CONSTANTS 14
Substituting (4.13) into (4.14), we get2 αθ + 2 β r θ + θ γ ( θ + θ ) = 0 . By rearranging this formula, we get (cid:0) (2 α + γ ) − β (cid:1) θ + ( γ − β ) θ + 2 γ (2 α + γ ) θ θ = 0 . (4.15)Dividing (4.15), by θ , we obtain (cid:0) (2 α + γ ) − β (cid:1) + ( γ − β ) (cid:18) θ θ (cid:19) + 2 γ (2 α + γ ) θ θ = 0 . Hence, by Lemma 3.3, we have(2 α + γ ) − β = 0 , γ − β = 0 , γ (2 α + γ ) = 0 . Thus, we conclude that α = β = γ = 0. This proves Proposition 1. (cid:3) Concluding remark.
As we have seen in the proof of Theorem 2.1, the assumption that thenumbers a , . . . , a m are integers, only using to reduce linear independence of m -theta functions θ ( a τ ) , . . . , θ ( a m τ ) over C . Therefore, by using the similar approach as in Theorem 2.1, one canprove the following: Let α , . . . , α m be distinct positive real numbers. Then the m -theta functions θ ( a τ ) , . . . , θ ( a m τ ) are linearly independent over C . Acknowledgements.
We would like to express our deep gratitude to Professor Y. Tachiya forhis useful comments. The second author would like to acknowledge the Department of AtomicEnergy, Govt. of India, for providing the research grant.
References [1] Berndt, B.C.,
Ramanujan’s Notebook , part V, Springer-Verlag, New York, 1998.[2] Bertrand, D.,
Theta functions and transcendence , Ramanujan J. , no. 4 (1997), 339-350.[3] Elsner, C., Algebraic independence results for values of theta-constants , Functiones et Approximatio (2015), 7-27.[4] Elsner, C., Tachiya, Y.,
Algebraic results for certain values of the Jacobi Theta - Constant θ ( τ ), Mathe-matica Scandinavica , no. 2 (2018), 249-272.[5] Elsner, C., Luca, F. and Tachiya, Y., Algebraic results for the values θ ( mτ ) and θ ( nτ ) of the Jacobi-thetaconstant , Mosc. J. Comb. Number Theory, vol 8, (2019), 71-79.[6] Elsner, C., Kaneko, M. and Tachiya, Y., Algebraic independence results for the values of the theta-constantsand some identities , to appear in
Journal of the Australian Mathematical Society .[7] Nesterenko, Yu. V.,
Algebraic independence , Tata Institute of Fundamental Research, Narosa PublishingHouse, New Delhi, 2009.[8] Nesterenko, Yu. V.,
On Arithmetic properties of values of theta-constants , Journal of Mathematical Sciences,vol 146, no. 2, (2007).[9] Nesterenko, Yu. V.,
On some identities for theta-constants , Diophantine analysis and related fields 2006,151–160,
Sem. Math. Sci. , Keio Univ., Yokohama, 2006.[10] Schneider, Th., Arithmetische Untersuchungen elliptischer Integrale , Math. Annalen (1937), 1-13.
Institute of Computer Sciences, FHDW University of Applied Sciences, Freundallee 15, 30173Hannover, Germany,
E-mail address , Carsten Elsner: [email protected]
Institute of Mathematical Sciences, Homi Bhabha National Institute, C.I.T Campus, Taramani,Chennai 600 113, India,
E-mail address , Veekesh Kumar:, Veekesh Kumar: