Theta Functions V 0 1. Figure 1. The lattice Λ = Z τz and its fundamental parallelogram V = {z = t 1 + t 2 τ C : 0 t 1, t 2 < 1}.

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1 Irish Math. Soc. Bulletin 6 (27), Theta Functions MARINA FRANZ Abstract. On our analytic way to the group structure o an elliptic unction we meet so called theta unctions. These complex unctions are entire and quasi-periodic with respect to a lattice Λ. In the proo o Abel s theorem we use their properties to characterise all meromorphic unctions rom C/Λ to C. Finally we have a closer look at a very special and interesting Λ-periodic meromorphic unction, the Weierstraß -unction. This unction delivers an analytic way to give a group structure to an algebraic variety.. Introduction First o all, we want to analyse periodic complex unctions : C C with respect to a lattice Λ. So let us ix once and or all a complex number τ C, Im τ > and consider the lattice Λ := Z τz C. τ V + τ Figure. The lattice Λ = Z τz and its undamental parallelogram V = {z = t + t 2 τ C : t, t 2 < }. Lemma. An entire doubly-periodic complex unction is constant. To prove this lemma we need Liouville s Theorem, which we know rom complex analysis. It states that each entire and bounded complex unction : C C is constant.

2 92 Marina Franz Proo. The values o a doubly-periodic unction are completely determined by the values on the closure o the undamental parallelogram V = {z C : z = t + t 2 τ or some t, t 2 } which is a compact set. But a continuous unction on a compact set is bounded. Hence our unction is entire and bounded. Thereore it is constant by Liouville s Theorem. As we have seen, entire doubly-periodic unctions are not very interesting, so in the ollowing we will consider entire quasi-periodic unctions and use them to prove Abel s Theorem which says what meromorphic doubly-periodic unctions look like. 2. Theta Functions and Abel s Theorem Deinition. The basic theta unction is deined to be the unction θ : C C given by θ(z) := θ(τ)(z) : exp(πin 2 τ) exp(nz) Note. The unction θ depends on τ. So or each τ C with Im τ > we get a (not necessarily dierent) basic theta unction. Hence there is a whole amily o basic theta unctions {θ(τ)} τ C,Im τ>. But here we assume τ to be ixed, so we have only one basic theta unction. Remark. As the series in the deinition above is locally uniormly unordered convergent (without proo) our basic theta unction is an entire unction. Lemma 2. The basic theta unction is quasi-periodic. Proo. Consider θ(z + λ) or λ Λ, i.e. λ = pτ + q or p, q Z. For λ =, i.e. or p = and q = we have θ(z + ) de exp(πin 2 τ) exp(n(z + )) exp(πin 2 τ + nz + n) exp(πin 2 τ) exp(nz) exp(n) }{{} = or all n Z

3 Theta Functions 93 exp(πin 2 τ) exp(nz) de = θ(z) Hence the basic theta unction is periodic with respect to the x- direction. For λ = τ, i.e., or p = and q = we have θ(z + τ) de exp(πin 2 τ) exp(n(z + τ)) exp(πin 2 τ + nz + nτ) i we complete the square and rearrange the summands then exp ( πin 2 τ + nτ + πiτ πiτ +nz + z z) = exp( πiτ z) n Z exp(πi(n + ) 2 τ) exp((n + )z) i we make a simple index shit m = n + then = exp( πiτ z) m Z exp(πim 2 τ) exp(mz) de = exp( πiτ z)θ(z) Hence the basic theta unction is not periodic with respect to the τ-direction as in general exp( πiτ z). In the general case we obtain θ(z + λ) = θ(z + pτ + q) de exp(πin 2 τ) exp(n(z + pτ + q)) exp(πin 2 τ + nz + npτ + nq)

4 94 Marina Franz i we complete the square and rearrange the summands then exp ( πin 2 τ + npτ + πip 2 τ πip 2 τ +nz + pz pz + nq) = exp( πip 2 τ pz) [ exp(πi(n + p) 2 τ) exp((n + p)z) n Z exp(nq) }{{} = or all n Z = exp( πip 2 τ pz) exp(πi(n + p) 2 τ) exp((n + p)z) n Z i we make a simple index shit m = n + p then ] = exp( πip 2 τ pz) m Z exp(πim 2 τ) exp(mz) de = exp( πip 2 τ pz)θ(z) Hence the basic theta unction θ is quasi-periodic with θ(z + λ) = θ(z + pτ + q) = exp( πip 2 τ pz)θ(z) or all λ = pτ + q Λ and z C. Deinition. We deine e(λ, z) := exp( πip 2 τ pz) and call this the automorphy actor. Remark. We have e(λ + λ 2, z) = e(λ, z + λ 2 )e(λ 2, z) or all λ, λ 2 Λ.

5 Theta Functions 95 Let λ, λ 2 Λ, i.e. λ = p τ + q and λ 2 = p 2 τ + q 2 or some p, p 2, q, q 2 Z, and thus λ + λ 2 = (p + p 2 )τ + (q + q 2 ) Λ. Then e(λ + λ 2, z) = e((p + p 2 )τ + (q + q 2 ), z) de = exp( πi(p + p 2 ) 2 τ (p + p 2 )z)) = exp( πip 2 τ p p 2 τ πip 2 2τ p z p 2 z) de = exp( πip 2 τ p p 2 τ p z)e(λ 2, z) = exp( πip 2 τ p z p p 2 τ p q }{{} 2 ) exp(p q 2)= e(λ 2, z) = exp( πip 2 τ p (z + λ 2 ))e(λ 2, z) de = e(λ, z + λ 2 )e(λ 2, z) Summary. The basic theta unction θ : C C is entire and quasiperiodic with automorphy actor e, i.e., we have θ(z + λ) = e(λ, z)θ(z) = exp( πip 2 τ pz)θ(z) () or all λ = pτ + q Λ and all z C. Now we want to enlarge our category o theta unctions. So ar we have only one (basic) theta unction corresponding to the point C (and each point q Z C). Now, or our ixed τ, we will deine a new theta unction or each point in C. Thereore let s start with our old theta unction and translate z by a ixed ξ, i.e. consider θ(z + ξ) or ξ = aτ + b or some ixed a, b R: θ(z + ξ) = θ(z + aτ + b) de exp(πin 2 τ) exp(n(z + aτ + b)) exp(πin 2 τ + nz + naτ + nb) I we complete the square and rearrange the summands then we obtain

6 96 Marina Franz θ(z + ξ) exp ( πin 2 τ + naτ + πia 2 τ πia 2 τ +n(z + b) + a(z + b) a(z + b)) = exp( πia 2 τ a(z + b)) exp(πi(n + a) 2 τ) exp((n + a)(z + b)) n Z Note that the sum n Z exp(πi(n + a)2 τ) exp((n + a)(z + b)) looks very similar to the sum in the deinition o our basic theta unction above. Deinition. For ξ = aτ + b and a, b R the modiied theta unction is deined to be the unction θ ξ : C C given by θ ξ (z) := θ ξ (τ)(z) : exp(πi(n + a) 2 τ) exp((n + a)(z + b)) and ξ is called theta characteristic. Note. From the calculation above we obtain a relation between the basic theta unction and the modiied theta unction with characteristic ξ = aτ + b or some ixed a, b R: θ ξ (z) exp(πi(n + a) 2 τ) exp((n + a)(z + b)) (2) = exp(πia 2 τ + a(z + b))θ(z + ξ) (3) or all z C. Remark. As the series in the deinition is locally uniormly unordered convergent (without proo) the modiied theta unctions are entire unctions. Lemma 3. Modiied theta unctions are quasi-periodic unctions.

7 Theta Functions 97 Proo. Let a, b R such that ξ = aτ + b is the characteristic o the modiied theta unction θ ξ. Consider θ ξ (z + λ) or λ = pτ + q Λ. θ ξ (z + λ) (3) = exp(πia 2 τ + a(z + λ + b))θ(z + λ + ξ) () = exp(πia 2 τ + a(z + λ + b))e(λ, z + ξ)θ(z + ξ) (3) = exp(πia 2 τ + a(z + λ + b))e(λ, z + ξ) exp( πia 2 τ a(z + b))θ ξ (z) = exp(aλ) exp( πip 2 τ p(z + ξ))θ ξ (z) = exp(aλ πip 2 τ p(z + ξ))θ ξ (z) Hence the modiied theta unction θ ξ is quasi-periodic with θ ξ (z + λ) = θ aτ+b (z + pτ + q) = exp(aλ πip 2 τ p(z + ξ))θ ξ (z) or all λ = pτ + q Λ and z C. Deinition. Let a, b R be ixed and let ξ = aτ + b. We deine e ξ (λ, z) := exp(aλ πip 2 τ p(z + ξ)) and call this the automorphy actor. Remark. Let a, b R be ixed and let ξ = aτ + b. We have e ξ (λ + λ 2, z) = e ξ (λ, z + λ 2 )e ξ (λ 2, z) or all λ, λ 2 Λ. Let λ, λ 2 Λ, i.e. λ = p τ + q and λ 2 = p 2 τ + q 2 or some p, p 2, q, q 2 Z, and λ + λ 2 = (p + p 2 )τ + (q + q 2 ) Λ. Then e ξ (λ + λ 2, z) = e ξ ((p + p 2 )τ + (q + q 2 ), z) de = exp ( a(λ + λ 2 ) πi(p + p 2 ) 2 τ (p + p 2 )(z + ξ)) = exp ( aλ + aλ 2 πip 2 τ p p 2 τ πip 2 2τ p (z + ξ) p 2 (z + ξ))

8 98 Marina Franz de = exp(aλ πip 2 τ p p 2 τ p (z + ξ)) e ξ (λ 2, z) = exp ( aλ πip 2 τ p p 2 τ p q 2 }{{} exp(p q 2)= p (z + ξ)) e ξ (λ 2, z) = exp(aλ πip 2 τ p (z + λ 2 + ξ))e ξ (λ 2, z) de = e ξ (λ, z + λ 2 )e ξ (λ 2, z) Summary. Let ξ = aτ + b with a, b R ixed. The modiied theta unction with characteristic ξ is entire and quasi-periodic with automorphy actor e ξ, i.e. we have θ ξ (z + λ) = e ξ (λ, z)θ ξ (z) (4) = exp(aλ πip 2 τ p(z + ξ))θ ξ (z) (5) or all λ = pτ + q Λ and all z C. Now we want to determine all zeros o all theta unctions. Thereore we consider a special modiied theta unction, the theta unction with characteristic σ := 2 τ + 2. In this case the determination o zeros is very simple because the zeros are easy to describe. Lemma 4. is an odd unction, i.e. ( z) = (z) or all z C. In particular we have () =. Proo. We have ( z) = θ 2 τ+ ( z) 2 [ ( de exp πi ( ) n + 2 τ 2) ( ( exp n + ) ( z + ))] 2 2

9 Theta Functions 99 i we make a simple index shit m = n then = [ ( ( ) exp πi m 2) 2 τ m Z ( ( exp m ) ( z + ))] 2 2 = [ ( ( ) exp πi m + 2 τ 2) m Z ( ( exp m + ) ( z + ) ( m + ))] = [ ( ( ) exp πi m + 2 τ 2) m Z ( ( exp m + ) ( z + )) de = (z). 2 2 From complex analysis we know a simple way to count zeros and poles o a meromorphic unction : C C: = total number o zeros - total number o poles γ where γ is a piecewise smooth path that runs around each zero and each pole exactly one time. We will use this integral to determine all zeros o the theta unctions with σ = 2 τ + 2. Lemma 5. We have (z) = precisely or all z Λ and all zeros are simple zeros. Proo. Consider the undamental parallelogram V := {z C : z = t τ +t 2 or some t, t 2 < }. Choose w C such that the border o V w := w + V contains no zeros o and V w. Further consider the ollowing paths along the border o V w : : [, ] C; t w + t : [, ] C; t w + + tτ γ : [, ] C; t w + ( t) + τ δ : [, ] C; t w + ( t)τ

10 Marina Franz w + τ γ w + + τ δ w w + Figure 2 In the above igure w C is chosen such that the border o the parallelogram V w = w + V contains no zeros o and such that V w. The paths,, γ and δ run along the border o V w. Note that γ(t) = w + ( t) + τ = ( t) + τ and δ(t) = w + ( t)τ = ( t). We want to show that =. V w Thereore we will show that = and γ =. δ = (γ(t))γ (t) dt γ = = = = (( t) + τ)( ) dt (z + τ) dz e σ(τ, z) (z) + e σ (τ, z)(z) dz e σ (τ, z) (z) e σ(τ, z) e σ (τ, z) dz

11 Theta Functions when we use e σ (τ, z) = exp( 2τ πiτ (z +σ)) then the above expression becomes exp ( (z + σ)) exp( (z + σ)) dz = = θ σ dz = (δ(t))δ (t) dt δ = (( t) )( τ) dt V w = = = when we use then θ σ (z ) dz θ σ e σ(, z) (z) + e σ (, z)(z) dz e σ (, z) (z) e σ(, z) e σ (, z) dz (z) e σ (, z) = exp( 2 ) = = Then we have = =. exp ( πi) exp( πi) dz (z) θ σ(z) + + γ θ σ + δ θ σ

12 2 Marina Franz As is holomorphic in V w, i.e. it doesn t have any poles, we know that has a single zero. And by Lemma 4 this zero is in. Now consider V w + λ = V w+λ or some λ Λ. As (z + λ) = e σ (λ, z) (z) we obtain that has the only zero + λ = λ in V w+λ and this is a simple zero. But C = λ Λ V w+λ. Hence has zeros exactly in Λ and all zeros are simple. Corollary 6. Let ξ = aτ + b with a, b R. We have θ ξ (z) = precisely or all z σ ξ + Λ and all its zeros are simple. Proo. We know (z) = i and only i z Λ and all the zeros are simple. Hence θ ξ (z) = (3) exp(πia 2 τ + a(z + b))θ(z + ξ) = (3) exp ( πia 2 τ + a(z + b) ) ( ( ) 2τ exp πi 2 2 (z + ξ σ) = z + ξ σ Λ z σ ξ + Λ ( z + ξ 2 τ 2 + )) 2 In particular we have θ(z) = i and only i z σ + Λ. So ar we have considered entire quasi-periodic unctions. Now we want to use our knowledge about them to see what meromorphic doubly-periodic unctions with given zeros a i and poles b j o given order n i resp. m j and number n resp. m look like. Furthermore we will decide whether such a unction exists or not and whether it is unique or not. Abel s Theorem 7. There is a meromorphic unction on C/Λ with zeros [a i ] o order n i or i n and poles [b j ] o order m j or j m i and only i n i= n i = m j= m j and n i= n i[a i ] = m j= m j[b j ]. Moreover, such a unction is unique up to a constant actor. Proo. Let : C/Λ C be a meromorphic unction with zeros [a i ] o order n i and poles [b j ] o order m j. Choose w C such that V w = {w + z C : z = t τ + t 2 or some t, t 2 < } contains a representative a i resp. b j or every zero resp. pole o. Further

13 consider the paths Theta Functions 3 : [, ] C; t w + t : [, ] C; t w + + tτ γ : [, ] C; t w + ( t) + τ δ : [, ] C; t w + ( t)τ along the border o V w and the paths i : [, ] C; t a i + r i e t j : [, ] C; t b j + s j e t around the zeros resp. poles o where r i resp. s j is chosen small enough that D i = {z C : z a i < r i } resp. D j = {z C : z b j < s j } contains no other zeros or poles o. w + τ γ w + + τ D i a i i ai δ b j D j b j j a i2 w w + Figure 3 Here, w C is chosen such that the parallelogram V w = w + V contains a representative a i resp. b j or every zero resp. pole o. The paths,, γ and δ run along the border o V w, the paths i around the zero a i o and the path j around the pole b j o. First we show that n i= n ia i m j= m jb j Λ as ollows: n m n i a i m j b j = i= j= n i= = z i m + z j z Vw Λ j=

14 4 Marina Franz To establish the irst equality note that we can write (z) = c i (z a i ) ni h i (z) or a constant c i and with h i (a i ) = around a i and hence (z) = c i n i (z a i ) ni h i (z) with h i (a i ) =. We obtain z (z) = z n i h i (z) z a i h i with hi h i (a i ) =. Hence we have z i = n ia i by Cauchy s integral ormula or discs. The same holds or the poles o. The second equality is clear since V w contains a representative or every zero and pole o in C/Λ. To see, that z Vw is an element o Λ, note that γ z = = = = γ(t) (γ(t))γ (t) dt (( t) + τ) ((( t) + τ))( ) dt ( t) (( t)) dt τ (( t)) dt z τ

15 Theta Functions 5 and δ z = = = = δ(t) (δ(t))δ (t) dt (( t) ) ((( t) ))( τ) dt + ( t) (( t))τ dt (( t))τ dt z + hence z Vw = z + z + z + z γ δ = τ + Λ since, Secondly we show that Z. n m n i m j = Vw i= j= = Again the irst equality is clear, since V w contains a representative or every zero and pole o in C/Λ.

16 6 Marina Franz The second equality ollows rom: = (γ(t))γ (t) dt and hence γ δ = = = (( t) + τ)( ) dt = = Vw = =. = = (( t)) dt (δ(t))δ (t) dt (( t) )( τ) dt (( t))τ dt + + γ + δ Now let [a i ], [b j ] C/Λ and n i, m j N or i n and j m be such that n i=n n i = m j=m m j and n i= n i[a i ] = m j= m j[b j ]. We will construct a meromorphic unction : C/Λ C with zeros [a i ] o order n i and poles [b j ] o order m j. We choose representatives a i, b j C or [a i ] resp. [b j ] such that n i= n ia i = m j= m jb j and deine the unction g : C C; z n i= (z a i ) ni m j= (z b j ) mj

17 Theta Functions 7 where is the theta unction with characteristic 2 τ + 2. Obviously g is a meromorphic unction with zeros in a i +Λ o order n i and poles in b j +Λ o order m j. We have to show that g is doubly-periodic with respect to Λ. Thereore we have to show that g(z + λ) = g(z) or all λ Λ. It suices to show that g(z + ) = g(z) and g(z + τ) = g(z). g(z + ) = n i= (z + a i ) ni m j= (z + b j ) mj n = i= (z a i ) ni m j= (z b j ) mj = g(z) and g(z + τ) = = = = n i= (z + τ a i ) ni m j= (z + τ b j ) mj n i= (e σ(τ, z a i ) (z a i )) ni m j= (e σ(τ, z b j ) (z b j )) mj n i= e σ(τ, z a i ) ni n i= (z a i ) m j= e ni m σ(τ, z b j ) mj j= (z b j ) mj n i= e σ(τ, z a i ) ni m j= e σ(τ, z b j ) mj g(z) but n i= e σ(τ, z a i ) ni m j= e σ(τ, z b j ) mj n = i= exp( (z a i + σ)) ni m j= exp( (z b j + σ)) mj = n i= m j= exp( (z + σ))ni exp( (z + σ))mj n i= exp(a i) ni m j= exp(b j) mj = exp( (z + σ)) n i= ni exp( (z + σ)) m j= mj =. exp( n i= n ia i ) exp( m j= m jb j ) So g(z + τ) = g(z) as well. Hence g is doubly periodic w.r.t. Λ and the unction : C/Λ C with ([z]) = g(z) is well-deined and a solution.

18 8 Marina Franz Now suppose we are given two meromorphic unctions, g : C/Λ C with zeros [a i ] o order n i and poles [b j ] o order m j. Then g has no zeros or poles. Hence it is constant. 3. Weierstraß -unction Now we want to capitalize on our work above. Thereore we consider a very special periodic unction, the Weierstraß -unction. Deinition. The W eierstraß unction is deined to be the unction : C C given by (z) = z 2 + ( (z λ) 2 ) λ 2 λ Λ Proposition 8. (Without proo) is a Λ-periodic meromorphic unction with poles o order 2 exactly in Λ. The ollowing lemma gives a connection between the Weierstraß -unction and our well known theta unction with characteristic σ = τ. Lemma 9. There is a constant c C such that ( ) θ (z) = σ (z) + c Note. The quotient θ σ ( θ σ θσ ) is doubly-periodic. isn t doubly-periodic, but the derivative To see this consider θ σ θσ (z + λ) or some λ = pτ + q Λ. (z + λ) (5) = (e σ(λ, z) (z)) = e σ(λ, z) (z) + e σ (λ, z)(z) e σ (λ, z) (z) e σ (λ, z) (z) de = exp (πiλ πip 2 τ p(z + σ)) (z) + e σ (λ, z)θ σ(z) e σ (λ, z) (z) = pe σ(λ, z) (z) + e σ (λ, z)θ σ(z) e σ (λ, z) (z) = p + θ σ (z) θ σ (z) as in general p. From the equation θ σ θσ (z + λ) = p + θ σ θσ (z) ( ) θ σ above it ollows directly that is doubly-periodic.

19 Theta Functions 9 Proo. We know that is holomorphic and has its zeros precisely in the lattice points λ Λ. That means that the expansion o θ σ θσ in a Laurent series around looks like θ σ (z) = a z + a + a z + a 2 z 2 + a 3 z 3 + terms o higher order or some constants a i C. We can choose a neighborhood U o such that is the only zero o in U. As is a single zero we know that ( ) θ a = Res σ = = where : [, ] C; t re t or some suitable r. We conclude θ σ (z) = z + a + a z + a 2 z 2 + a 3 z 3 + terms o higher order and calculate ( ) θ σ (z) = z 2 + a + 2a 2 z + 3a 3 z 2 + terms o higher order ( θ σ ) I we add and then we obtain ( ) θ (z)+ σ (z) = ( λ Λ (z λ) 2 λ 2 ) +a +2a 2 z +3a 3 z ( ) θ σ From this sum we see directly that + doesn t have any poles ( ) θ σ in U. Hence + is holomorphic in a neighborhood o and thus holomorphic everywhere. As it is in addition doubly-periodic ( θ σ ) (since is as well as doubly-periodic) we know rom our very irst lemma that it must be constant. The Weierstraß -unction satisies a number o equations and dierential equation. This eature makes the Weierstraß -unction to be o interest. The most important dierential equation that is satisied by the Weierstraß -unction is the ollowing: Theorem. The Weierstraß -unction satisies the dierential equation (z) 2 = c 3 (z) 3 + c 2 (z) 2 + c (z) + c

20 Marina Franz where the constants c 3 = 4, c 2 =, c = 6 λ Λ λ 4 and c = 4 λ Λ depend on the lattice Λ. Proo. Consider (z) z = ( 2 λ Λ (z λ) 2 λ ). This unction 2 is holomorphic in a neighborhood o. We can expand the summands (z λ) 2 λ : 2 (z λ) 2 λ 2 = λ 2 ( ) ( ) z 2 λ ( = ) ( z ) 2 n λ 2 λ n= = ( λ 2 2 z ) λ + 3 z2 λ z3 λ z4 λ = 2 z λ z2 λ z3 λ z4 λ This sum is absolutely convergent or all z C with z < λ ; in particular in a neighbourhood o. To simpliy the big sum rom above we deine s n := λ Λ λ n+2 or n N. Note that s n = or all odd n N. We obtain (z) = z 2 + 2s z + 3s 2 z 2 + 4s 3 z 3 + 5s 4 z = z 2 + 3s 2z 2 + 5s 4 z 4 + 7s 6 z 6... which is true in a neigborhood o. With the constants c 3 = 4, c 2 =, c = 6 λ 4 and c = 4 we obtain hence λ Λ λ Λ (z) = z 2 c 2 z2 c 28 z4 + terms o higher order (z) = 2 z 3 c z c 7 z3 + terms o higher order λ 6 λ 6

21 and (z) 2 = 4 z 6 + 2c 5 Theta Functions z 2 + 4c + terms o higher order 7 (z) 3 = z 6 3c 2 z 2 3c + terms o higher order 28 Now consider (z) := (z) 2 c 3 (z) 3 c (z) c The series o has only positive powers o z. Hence is holomorphic around. Hence it is holomorphic everywhere. And as it is doublyperiodic, it is constant. But the constant part o the series is 4 7 c c c =. Hence =. We will mention one more equation that is satisied by the Weierstraß -unction: Remark. Remember that our lattice Λ is generated by and τ. Hence the set o zeros o is given by ( 2 + Λ) ( τ 2 + Λ) ( +τ 2 + Λ ). Set e := ( ) 2, e2 := ( ) τ 2, e3 := ( ) +τ 2 C. Then we have ( ) 2 = 4( e )( e 2 )( e 3 ) and e + e 2 + e 3 = e e 2 + e e 3 + e 2 e 3 = 4 c e e 2 e 3 = 4 c where c and c are the constants rom above. Finally we will see how to use the Weierstraß -unction to give a group structure to an elliptic curve. Remark. I we consider the elliptic curve C := {(x, y) C 2 such that y 2 = c 3 x 3 + c 2 x 2 + c x + c } or the constants c 3 = 4, c 2 =, c = 6 λ Λ λ 4 and c = 4 rom the theorem above then we have a bijection C/Λ \ {} C given by z ( (z), (z)) λ Λ In particular we can give the variety C the group structure o C/Λ. λ 6

22 2 Marina Franz This can be extended to an embedding o C/Λ into the projective plane. For more details see the article o M. Khalid [2]. Reerences [] A. Gathmann, Algebraic Geometry, Notes or a class taught at the University o Kaiserslautern 22/23, available at gathmann/de/pub.html [2] M. Khalid, Group Law on the Cubic Curve, this issue. [3] D. Mumord, Tata Lectures on Theta, Progress in Mathematics Vol 28, Birkhauser Verlag, 983. [4] G. Trautmann, Complex Analysis II, Notes or a class taught at the University o Kaiserslautern 996/997. Marina Franz, Fachbereich Mathematik, Technische Universität Kaiserslautern, Postach 349, Kaiserslautern, Germany ranz@mathematik.uni-kl.de Received in inal orm on 23 August 27.