FIRST YEAR PHD REPORT

FIRST YEAR PHD REPORT VANDITA PATEL Abstract. We look at some potential links between totally real number fiels an some theta expansions (these being moular forms). The literature relate to moular forms is rich, an any links mae to totally real number fiels coul help us to unerstan the number fiel better. Contents 1. Introuction. Construction of a generalise Theta expansion 4 3. A Worke Example - Quaratic Fiels 5 3.1. Case 1 (mo 4) 5 3.. Case 1 (mo 4) 7 4. Backgroun Material 10 4.1. Some Basic Definitions an Notation 10 4.. Relating Quaratic Forms an Moular Forms 11 References 13 5. Part [B] - escription of acaemic activities 14 Date: June 19, 014. 1

FIRST YEAR PHD REPORT 1. Introuction The four squares problem, first pose in the Arithmetica of Diophantus, states that any positive integer n can be represente as the sum of four integer squares. In mathematical notation, we write that there exists integers w, x, y an z such that n = w + x + y + z. In 1770, Lagrange gave a concrete proof for this conjecture. In 1834, Carl Gustav Jacobi looke further into the four squares problem an foun an exact formula for the total number of ways a positive integer n can be represente as the sum of four squares. His formula can be seen below: r 4 (n) = 8 m. m n,4 m Naturally we are then le to asking the question: given positive integers n an k, in how many ways can n be written as a sum of k integer squares? In other wors we are aske to fin, r k (n) = #{(x 1,..., x k ) Z k : x 1 + + x k = n}, where we call the r k (n) the representation number of n. Lagrange i not have access to the theory of moular forms uring his breakthrough, an his proof arose through the use of classical methos. However, with the use of moular forms, one can fin some very nice formulae for the representation numbers, r k (n). The Jacobi Theta Function (name after Carl Gustav Jacobi since he was primarily the one investigating them), is efine as the following, Θ(z) = q n n Z where q = e πiz. We can exten this efinition to associate a theta function to more generalise representation numbers, namely by the following construction, Θ k (z) = r k (n)q n n=0 an it turns out that these constructions are inee moular forms. The next natural question to ask is whether we can generalise this further to any number fiel F, where we look at the representation numbers, R F (n) as the set of integral solutions to some positive efinite quaratic form, which is constructe with respect to the number fiel F. In mathematical notation, we can write, R F (n) = #{(x 1,..., x k ) Z k : Q(x 1,..., x k ) = n}

FIRST YEAR PHD REPORT 3 where n an k are positive integers, an k is efine to be the egree of the number fiel. We also have Q as a positive efinite quaratic form in k variables, with integral coefficients. Analogous to the constructions of Jacobi, we can construct further theta series by the following, Θ F (z) = R F (n)q n n=0 where once again, we have q = e πiz. The main question pose now is: given a number fiel, F = Q(θ) for some algebraic integer θ, can we fin a unique moular form associate to it? Moreover given a moular form, can we then fin the unique number fiel associate to it? The motivation behin such a question is that in forming such links, shoul they have a one-to-one corresponence, we can make use of the extensive literature an theory relate to moular forms to then perhaps hope to unerstan generalise number fiels a bit better. However, shoul one not be able to establish such links, then the question woul then be whether any links between generalise number fiels an moular forms can tell us anything at all about initial number fiel i.e. can we perhaps group certain number fiels together, as if almost classifying them in some sense. In this report, we shall be looking primarily at three main topics in number theory. We cover some algebraic number theory, some theory about integral solutions to quaratic forms, an some theory about moular forms. We primarily use [Stewart an Tall(1987)] as our main source to recall efinitions an notation relate to algebraic number theory. We shall also be neeing some theory relate to moular forms. [Diamon an Shurman(005)] is an excellent text for a beginner to the course, an [Hanke(013)] provies us with some useful results to relate quaratic forms an moular forms. We shall summarise some of the main results from both of these sources in Section 4 - Backgroun Material.

FIRST YEAR PHD REPORT 4. Construction of a generalise Theta expansion We let K be a totally real number fiel of egree n. Then K = Q(θ) for some algebraic integer θ. We enote the ring of integers of the fiel K by O K. Now let σ 1,..., σ n be the n istinct monomorphisms of K such that σ i : K R for i = 1,..., n. We can construct a positive efinite quaratic form which we enote as Q as state below using an element α O K : Q = [σ 1 (α)] + [σ (α)] + + [σ n (α)]. Note here that we write the element α in terms of a Z-basis, i.e. we let {ζ 1, ζ,..., ζ n } be a set of generators of O K an so we have α = a 1 ζ 1 + a ζ + + a n ζ n where a i Z for all i = 1,..., n. Thus the quaratic form Q has variables a 1,..., a n an the coefficients of Q are in Z. We can count the number of integer solutions to the quaratic form Q = m where m N. We enote the number of integral solutions as R Q (m), i.e. R Q (m) = #{a Z m Q(a) = m}. We now look at a series expansion where we let R Q (m) be the coefficients. Θ(z) = R Q (m) q m m=0 where q = e πiz. This is in fact a moular form, an we shall call it a theta expansion from here on.

FIRST YEAR PHD REPORT 5 3. A Worke Example - Quaratic Fiels In this section, we shall explicitly look at K = Q( ) where is squarefree an a strict positive integer. Here, K is a totally real number fiel of egree. To o so, we follow the steps outline in the introuction to attempt to arrive at the theta series for the totally real number fiel K. We shall nee a very useful theorem, as state an prove in [Stewart an Tall(1987)], page 67. Theorem 3.1. Let be a squarefree rational integer (i.e. Z). Then the integers of K = Q( ), which we enote as O K, are: (a) Z[ ] if 1 (mo 4) (b) Z [ 1 + 1 ] if 1 (mo 4). We shoul remark at this stage that we shall call an element of Z a rational integer, an an element of O K an integer. 3.1. Case 1 (mo 4). First, we shall consier the totally real number fiel K = Q( ) in the case where 1 (mo 4). In this case, we have O K = 1, an so any element α O K can be written as α = a 1 + a with a 1, a Z. Recall that the istinct monomorphisms of K are: σ 1 : K R σ 1 (a + b ) = a + b σ : K R σ 1 (a + b ) = a b where a, b Q an the embeings map to R since we are looking at a totally real number fiel. Now we can construct a generalise quaratic form Q as follows: Q(a 1, a ) = [σ 1 (α)] + [σ (α)] = [a 1 + a ] + [a 1 a ] = a 1 + a = [Tr(α)] Nm(α). (3.1) where we efine the Trace (Tr) an Norm (Nm) as follows:

FIRST YEAR PHD REPORT 6 Definition 3.1. Let α K where K is a number fiel of egree n. Let σ i for i = 1,..., n be the n istinct monomorphisms of K. Then we efine the Norm an Trace as follows: Tr(α) = i σ i (α) Nm(α) = i σ i (α). In this specific case, we can calculate the norm an trace of an arbitrary element α O K to euce that the ientity given in Equation (3.1) is true. Tr(α) = σ i (α) i=1 = a 1 + a + a1 a = a 1. Nm(α) = σ i (α) i=1 = (a 1 + a )(a1 a ) = a 1 a. Next, we are to calculate the number of integer solutions to Q(a 1, a ) = m for all rational integers m, i.e. we nee to calculate the R Q (m). Let us look at a specific example now. Let us choose = 3, then we are looking for the number of integer solutions to the equation: which we can rewrite as: a 1 + 6a = m a 1 + 3a = µ. We show the first few terms for R Q (µ) in the table below.

FIRST YEAR PHD REPORT 7 Table 1. The first few terms for R Q (µ) Q(a 1, a ) = µ R Q (µ) a 1 + 3a = 0 1 a 1 + 3a = 1 a 1 + 3a = 0 a 1 + 3a = 3 a 1 + 3a = 4 6 a 1 + 3a = 5 0 a 1 + 3a = 6 0 a 1 + 3a = 7 4 a 1 + 3a = 8 0 a 1 + 3a = 9 a 1 + 3a = 10 0 Notice that we have the relation R Q (µ) = R Q (m), an of course R Q (m) = 0 when m is o. We can construct the function: Θ Q (z) := R Q (m)q m = m=0 R Q (µ)q µ µ=0 1 + q + 0q 4 + q 6 + 6q 8 + 0q 10 + 0q 1 + 4q 14 + 0q 16 + q 18 + 0q 0 + This is inee a moular form, of weight k = 1 an Level N. We shall see a rigorous argument as to why this is in Section 4 - Backgroun Material. 3.. Case 1 (mo 4). We now move on to consier the totally real number fiel K = Q( ) in the case where 1 (mo 4). In this case, we have O K = 1, 1 + 1 an so any element α OK can be written as ( 1 α = a 1 + a + 1 ) with a 1, a Z. The istinct monomorphisms of K remain the same as the case where 1 (mo 4) an we shall still call them σ 1 an σ.

FIRST YEAR PHD REPORT 8 Now we can construct a generalise quaratic form Q as follows: Q(a 1, a ) = [σ 1 (α)] + [σ (α)] = [ a 1 + a ( 1 + 1 ) ] + [ ( 1 a 1 + a 1 ) ] = a (1 + ) 1 + a 1 a + a = [T r] Nm. (3.) where we note that Q(a 1, a ) is a quaratic form with integer coefficients. Since 1 (mo 4), this implies that (1 + )/ Z. Again, we can calculate the norm an trace of an arbitrary element α O K to euce that the ientity given in Equation (3.) is true. Tr(α) = σ i (α) i=1 = a 1 + a ( = a 1 + a. 1 + ) ( 1 ) + a 1 + a Nm(α) = = σ i (α) i=1 (a 1 + a ( = a 1 + a 1 a + 1 + )) ( 1 (a )) 1 + a (1 ) a 4. To summarise, we have the following: Let K be a totally real number fiel, with K = Q( ) with > 0 being a squarefree rational integer. Then the quaratic form associate to K is Q(a 1, a ) = [T r] Nm. Next, we are to calculate the number of integer solutions to Q(a 1, a ) = m for all rational integers m, i.e. we nee to calculate the R Q (m).

FIRST YEAR PHD REPORT 9 Let us look at a specific example now. Let us choose = 5, then we are looking for the number of integer solutions to the equation: a 1 + a 1 a + 3a = m. We show the first few terms for R Q (m) in the table below. Table. The first few terms for R Q (m) Q(a 1, a ) = m R Q (m) a 1 + a 1 a + 3a = 0 1 a 1 + a 1 a + 3a = 1 0 a 1 + a 1 a + 3a = a 1 + a 1 a + 3a = 3 4 a 1 + a 1 a + 3a = 4 0 a 1 + a 1 a + 3a = 5 0 a 1 + a 1 a + 3a = 6 0 a 1 + a 1 a + 3a = 7 4 a 1 + a 1 a + 3a = 8 a 1 + a 1 a + 3a = 9 0 a 1 + a 1 a + 3a = 10 We can construct the function: Θ Q (z) := R Q (m)q m m=0 1 + 0q + q + 4q 3 + 0q 4 + 0q 5 + 0q 6 + 4q 7 + q 8 + 0q 9 + q 10 + Again, we see that this is a moular form of weight k = 1 an level N. We shall see a rigorous argument as to why this is in Section 4 - Backgroun Material. Some further questions which arise through this worke example are: A proof which shows that using the representation numbers as the coefficients of some type of Fourier expansion is inee a moular form. Generalisation to other totally real number fiels?

FIRST YEAR PHD REPORT 10 4. Backgroun Material To make further progress with the problem we shall be neeing some theory relate to moular forms. We primarily use [Diamon an Shurman(005)] for basic efinitions, an [Hanke(013)] for etails on the relationship between quaratic forms an moular forms. 4.1. Some Basic Definitions an Notation. We begin by outlining some basic efinitions, notation an theorems to form a founation for us to then unerstan an efine a moular form. Definition 4.1. Let H = {z C I(z) > 0} enote the complex upper half plane. ( ) a b Definition 4.. Let γ SL (R) where γ =. We efine the c map SL (R) H H (γ, z) γ z = az + b cz + which is the group action of SL (R) on H. Definition 4.3. Let f be a meromorphic function on H, an let k Z. We say that f is weakly moular of weight k if: ( ) az + b f = (cz + ) k f(z) cz + ( ) a b for all z H an SL c (Z). Some notation: We have the slash action, ( ) az + b (f k γ)(z) = (cz + ) k f. cz + Definition 4.4. A moular form of weight k is a holomorphic function f : H C which is weakly moular of weight k an holomorphic at infinity. A moular form can be expresse as a convergent power series (Laurent Series) which is usually calle the q-expansion of f, f(z) := a n q n where q = e πiz. n=0

FIRST YEAR PHD REPORT 11 Definition 4.5. Let N be a positive integer. The principal congruence subgroup of level N is the group: { ( ) } Γ(N) = γ SL (Z) 1 0 γ (mo N). 0 1 A congruence subgroup of SL (Z) is a subgroup Γ SL (Z) such that Γ(N) Γ for some N 1. The least such N is calle the LEVEL of Γ. Definition 4.6. Let Γ be a congruence subgroup. The set of CUSPS of Γ is the set Cusps(Γ) := Γ \ P 1 (Q) of Γ-orbits in P 1 (Q). Some notation: P 1 (Q) = Q { } is calle the projective line over Q. Definition 4.7. Let f be a meromorphic function on H, let Γ be a congruence subgroup an let k be an integer. We say that f is weakly moular of weight k for the group Γ (of level Γ or level N) if f satisfies f k γ = f for all γ Γ. A moular form of weight k for the group Γ is a holomorphic function, f : H C, weakly moular of weight k for Γ an holomorphic at all cusps of Γ. If f vanishes at all cusps of Γ then it is calle a CUSP FORM (of weight k for Γ). 4.. Relating Quaratic Forms an Moular Forms. Recall that Q is a positive efinite quaratic form by construction. Then we know that R Q (m) <, which we shall nee to construct meaningful moular forms. Then, we can efine the Theta Expansion of Q as a series expansion: Θ Q (z) := R Q (m)e πizm = R Q (m)q m. m=0 m=0 Firstly, we shall nee to show that this series has some properties with respect to convergence, an the following Lemma helps us to establish this. Lemma 4.1. The Fourier Series f(z) := m=0 a mq m converges absolutely an uniformly on compact subsets of H to a holomorphic function f : H C if all of the coefficients, a m C satisfy a m Cm r for some constant C > 0 an some r > 0. For the proof, one can use [Hanke(013)] which in turn references [Miyake an Maea(1989)] - Lemma 4.3.3 page 117. Theorem 4.1. The Theta series, Θ Q (z) of a positive efinite integervalue quaratic form Q converges absolutely an uniformly to a holomorphic function f : H C.

FIRST YEAR PHD REPORT 1 Proof. Since Q is a positive efinite quaratic form, we know that the number of integer solutions to Q are boune. The solutions correspon to the number of lattice points in a smooth boune region of R n an so we have M i=0 R Q(i) < CM n for some constant C. Therefore, we have R Q (m) < C 1 M n 1 for some constant C 1, an by Lemma 4.1, we have Θ Q (z) converges absolutely an uniformly to a holomorphic function when z H. Clearly, by the efinition of Θ Q (z), we have Θ Q (z) = Θ Q (z + 1). In the special case where Q = x, we have Θ Q ( 1/4z) = izθ Q (z) an Θ Q (z) = Θ Q (z + 1). (Θ in this case is famously known as Jacobi s Theta Series). A proof of this can be foun on page 5 of [Bruinier et al.(008)bruinier, van er Geer, Harer, an Zagier]. ( ) a b Theorem 4.. For all SL c (Z) such that 4 c, we have ( ) ( ) az + b c cz Θ Q = ɛ 1 + ΘQ (z) cz + where Q = x, π/ < arg( z) π/, { 1 if 1 (mo 4); ɛ = i if 3 (mo 4). an ( ) ( ) c c if c > 0 or > 0; = ( ) c if both c < 0, < 0. We can now generalise for a generic quaratic form Q. Theorem 4.3. Suppose that Q is a non-egenerate positive efinite ( quaratic ) form over Z in n variables, with level N. Then for all a b SL c (Z) such that N c, we have ( ) ( ) [ ( ) ] az + b et(q) c cz n Θ Q = ɛ 1 + Θ Q (z) cz + where ( ) c (z), ɛ, are efine as in the previous theorem. Corollary 4.1. Θ Q (z) is a moular form of weight n/ where n is the egree of the number fiel we began with (an hence also the number of variables in the constructe quaratic form). Θ Q (z) has level N

an character χ( ) = FIRST YEAR PHD REPORT 13 ( ( 1) n/ et(q) ), with respect to the trivial multiplier system ɛ(γ, k) := 1 when n( is) even, an with respect to the theta c multiplier system ɛ(γ, k) := ɛ 1 when n is o. We make a final remark here to say that the level N of the moular form is the same as the level of the quaratic form Q. References [Bruinier et al.(008)bruinier, van er Geer, Harer, an Zagier] Jan Henrik Bruinier, Gerar van er Geer, Günter Harer, an Don Zagier. The 1--3 of moular forms. AMC, 10:1, 008. [Diamon an Shurman(005)] Fre Diamon an Jerry Michael Shurman. A first course in moular forms, volume 8. Springer, 005. [Hanke(013)] Jonathan Hanke. Quaratic forms an automorphic forms. In Quaratic an Higher Degree Forms, pages 109 168. Springer, 013. [Miyake an Maea(1989)] Toshitsune Miyake an Yoshitaka Maea. Moular forms. Springer, 1989. [Stewart an Tall(1987)] Ian N Stewart an Davi O Tall. Algebraic number theory, 1987.

FIRST YEAR PHD REPORT 14 5. Part [B] - escription of acaemic activities The aim of this section is to outline an escribe my acaemic activities for the 013-014 acaemic year. First we shall list all of the courses an seminars/stuy groups attene, marking out those which were examine. TCC - Local Fiels (University of Bristol), examine MA4H9 - Moular Forms, examine MA46 - Elliptic Curves, examine Algebraic Geometry for Number Theory (Stuy Group) TCC - Moular Curves Mumfor Curves (Stuy Group) Galois Cohomology (Stuy Group) attenance at Number Theory Seminar, every Monay uring term time attenance at Number Theory group meetings - hel weekly uring term time Below is a list of books an papers rea uring the first year of the PhD. Algebraic Number Theory - Ian Stewart Galois Theory - Ian Stewart The Collision of Quaratic Fiels, Binary Quaratic Forms, an Moular Forms - Karen Smith - can be foun at the following page: http://www.math.oregonstate.eu/ swisherh/karensmith.pf A First Course in Moular Forms - Diamon an Shurman (first couple of chapters) Quaratic Forms an Automorphic Forms, Arizona Winter School notes, Jonathan Hanke - can be foun at the follwing page: http://swc.math.arizona.eu/aws/009/09hankenotes.pf