Another version of the Modularity Theorem states that every complex elliptic curve with a rational value is the homomorphic holomorphic image of a Jacobian. This connection and the already mentioned versions of Modularity help to create two more versions of the Modularity Theorem.
Modular Forms, Elliptic Curves, and Modular Curves
First definitions and examples
So when one thinks of ∞ as lying far in the imaginary direction, is defined as beholomorphic at ∞ if g extends holomorphically to the puncture point q = 0, i.e. The Laurent series sums over n∈N. Show that Gk(τ) is therefore bounded as Im(τ)→ ∞. A tip for this exercise is at the end of the book.).
Congruence subgroups
Since f[α]k is holomorphic on H and weakly modular with respect to α−1Γ α, again a congruence subgroup of SL2(Z) (exercise 1.2.5), the idea of its holomorphism at∞ makes sense. The inequality is strict because c ≡0 (mod 4) and d is odd.) Use the identity. The inequality is strict due to the properties of c and d modulo 4.) Each multiplication reduces the positive integer min{|c|,2|d|}, so the process must stop at c= 0 ord= 0.
Complex tori
The double composition of isogenes is the composition of doubles in reverse order. Since ˆϕ◦ϕ = [deg(ϕ)] and the matrix of a composition is the right-to-left product of the matrices, dual isogeny must induce the matrix.
Complex tori as elliptic curves
For (b), use part (a) to show that the non-positive terms in the Laurent series on both sides are equal. Reason similarly with Λ = Λµ3 to find the zeros of the corresponding Weierstrass function℘and to show that℘(1/2) is real.
Modular curves and moduli spaces
A complex valued function F of the improved elliptic curves for Γ is degree-k homogeneous with respect to Γ if for any non-zero complex number. Showing that if :H −→Cis is weight-kinvariant with respect toΓ, then formula (1.13) gives a well-defined function F on improved elliptic curves of the special type.
Modular Curves as Riemann Surfaces
- Topology
- Charts
- Elliptic points
- Cusps
- Modular curves and Modularity
See Exercise 2.1.3(c) for a less elementary way of proving this. The exercise establishes some facts that will make proposition 2.2.2 clearer, to be followed.). Transferring the fundamental domain D (as shown in Figure 2.3) to the Riemann sphere via stereographic projection yields a triangle with a missing vertex. The reader must sketch this before proceeding.) It is clear that the triangle must be compacted by moving the point to infinity, to be thought of as infinitely far from the imaginary axis on the plane or as the north pole on the sphere. Retain the coordinate partsπ(U) and the maps ϕ:π(U)−→V from Section 2.2 for the neighborhoods U ⊂ H. The subscript Γ is suppressed when the context makes the group clear.) This notion is dual to the period of an elliptic point, being inversely proportional to the size of an isotropy subset.
Dimension Formulas
The genus
Letε2 andε3 denote the number of elliptical points of periods 2 and 3 inX(Γ), andε∞ the number of cusps ofX(Γ). The number of elliptical points of period 2 inX0(p) is therefore the number of solutions of the congruencex2+ 1≡0 (modp). The number of elliptical points of period 3 in X0(p) is therefore the number of solutions of x2−x+ 1≡0 (modp).
Automorphic forms
The transformation wetf(γ(τ)) =j(γ, τ)kf(τ) shows that an automorphic formf with respect to a congruence subgroupΓ is generally not well defined. The exceptional cases are when k = 0 or f = 0.) However, the order of disappearance of any automorphic formf seems to make sense on the quotient. Show that the definition of νs(f) given in the text is independent of the choice of α and is well defined on the quotient X(Γ). Show that the derivative of the modular invariant lies in A2(Γ). non-zero elements for all even positive integers k.
Meromorphic differentials
Since ω is meromorphic at the vertices of X(Γ), the local differential ω|V has the form g(q)(dq)n, where g is meromorphic at 0. Thus the differential ωj onVj retracts under ψj to a suitable one for each Uj⊂ H. limitation of the global difference(τ)(dτ)n onH. Show that if the local meromorphic differentials ωj retract under ψj∗to restrictions of some ordinary global meromorphic differentials f(τ)dτ onH, then they are compatible.
Divisors and the Riemann–Roch Theorem
Then ω has a local representation ωx =fx(q)(dq)n for each point x∈X, where q is the local coordinate around x. The next result gives information about canonical divisors and a simpler version of the Riemann-Roch Theorem for divisors of sufficiently large degree. For any even positive integer, the meromorphic differential λk/2 belongs to Ω⊗k/2(X(Γ)) and its divisor has degree k(g−1).
Dimension formulas for even k
Now let an even integer satisfyfyk≥2 and let f be an arbitrary nonzero element ofAk(Γ) (exercise 3.2.3 showed that such f exists). If div(f) were integral, we could now calculate dim(Mk(Γ)) using the Riemann–Roch theorem, but it is not. Then formulas (3.8) and (3.9), which connect the vanishing orders ω and f, show that in accordance with the identification of ω and f(τ)(dτ)k/2 and the rule that the divisor of the product is the sum of the divisors (exercise 3.5. 2(a)).
Dimension formulas for odd k
If −I /∈ Γ, let g be the gender of X(Γ), ε3 the number of elliptic points with period 3, εreg∞ the number of regular cusps and εirr∞ the number of irregular cusps. Let any nonzero element of Ω1(X(Γ)), for example, be the divisor drawn on j(τ)dτ in H where j is the derivative of the modular invariant j. After identifying C(X(Γ)) with its image, the extension of the function fields has the same degree as the original surjection, namely 2, and the Galois group Γ/Γ acts as {±1} in the field extension generators. .
More on elliptic points
This is the union of two conjugacy classes under Γ0(N) (Exercise 3.7.1(c)), so the number of period 2 (or period 3) elliptic points of Γ0(N) is the number of extended conjugacy classes of order. 4 (or order 6) elements of Γ0(N). Counting the elliptic points of Γ0(N) by counting these conjugacy classes is done in the same environment as the proof of Proposition 2.3.3. Statement 3.7.1. The period 2 elliptic points of Γ0(N) are in bijective agreement with the ideals J of Z[i] such that Z[i]/J∼=Z/NZ.
More on cusps
A hint for this exercise is at the end of the book.). d) Likewise show that the period 2 elliptic points of Γ0(N) is given by formula (3.15). But φ(N/2) is odd only for N = 2, when the whole pairing process collapses anyway, and for N = 4, when only four of the five “×”s in the relevant table pair are off, which is three reps late (Exercise 3.8) .3). To count the points of Γ0(N), recall from Theorem 3.8.3 that for this group, vectors [ac] and a.
More dimension formulas
A hint for this exercise can be found at the end of the book.). A hint for this exercise can be found at the end of the book.). In the table, each “±” refers to the dimensions for the modular and cusp shapes, respectively.
Eisenstein Series
Eisenstein series for SL 2 (Z)
This argument is not really different from the non-intrinsic proof of Section 1.1 (and both require the absolute convergence shown in Exercise 1.1.2 to rearrange the sum), but the intrinsic method is used for elegance here and in the next section and because it remains manageable in more general situations. For any congruence subgroup Γ and any integer k, define the weighted Eisenstein space of Γ to be the quotient space of the modular forms through the cusp forms. Recall from Chapter 3 that ε∞ denotes the number of cusps of the compact modular curve X(Γ) and εreg∞ denotes the number of regular cusps.
Finally, the computation of the Fourier expansion that follows will produce coefficients that satisfy condition (3) of Proposition 1.2.4, indicating that Eq also satisfies condition (3) in the definition. Excluding these cases for the remainder of this paragraph, Evk does not vanish at∞ifv=±(0,1) and vanishes at∞otherwise. When N = 1, the Fourier expansion matches Gk(τ) from Chapter 1 fork even and vanishes to zero when k is odd (Exercise 4.2.6).
Dirichlet characters, Gauss sums, and eigenspaces
Every Dirichlet signχ modulo d raises to a Dirichlet signχN moduloN defined by the ruleχN(n(modN)) =χ(n(modd)) for all n∈Z that are relatively prime to N. Every Dirichlet signχmoduloN has a conductor, the smallest positive divisorN such that χ=χd◦πN is some sign χd modulod, or, equivalently, such that χ is trivial on the normal subgroup. The only sign moduloN with lead 1 is the trivial sign1N, and the trivial sign1N moduloN is primitive only for N= 1.
Gamma, zeta, and L -functions
The hint for this exercise is at the end of the book.) 4.4.3. This exercise presents Euler's argument for the functional equation ζ(s). The ideas here can be turned into a rigorous proof of meromorphic continuation and the functional equation ζ, cf. For any of the cut branches, determine a) Show that (−r)s = (−1)srs for every positive real number r and any s∈C, although the rule (zw)s=zsws does not hold in general.
Eisenstein series for the eigenspaces when k ≥ 3
For any two Dirichlet signs ψ modulou and ϕmodulov such that uv=N and (ψϕ k (here the signs are raised to level uv so their product makes sense) and ϕ is primitive, consider a linear combination of the Eisenstein series forΓ(N ),. For any positive integer N and any integer k≥3, let AN,k be the set of triples (ψ, ϕ, t) such that ψ and ϕ are primitive Dirichlet signs modulo andv with (ψϕ k, other is a positive integer , so that the tuv|N. Eisenstein series of Theorem 4.5.1 will be repeated at the end of the book in a context that includes the Modularity Theorem.
Eisenstein series of weight 2
The formula dim(E2(Γ)) =ε∞−1 in (4.3) at the beginning of this chapter shows that asv runs through a set of cusp representatives forΓ(N), the set {g2v} has one element too many to represent a base. Thus, E2(Γ(N)) is the set of linear combinations of gv2 whose coefficients sum to 0, which nullifies the occurrence of the correction, so that we can use Gv2 instead, subject to the most symmetric constraint. If ψ or ϕ is non-trivial, the coefficients sum to 0 inGψ,ϕ2 (τ), and as in section 4.5.
Bernoulli numbers and the Hurwitz zeta function
Substituting the generating function of the Bernoulli polynomials into the left-hand side of this relation shows that (Exercise 4.7.2(b)). For any positive real number ε, let γε be the complex contour that intersects the "bottom" of the branch (where arg(z) =−π) traverses from −∞to −ε, and then a counterclockwise circle of radius ε around the origin, and finally cut the "top" of the branch (where arg(z) =π) from−εto−∞. For Re(s)>1, show that asε→0+ the sum of the two linear pieces of the contour integral in the section converges to−2isin(πs)Γ(s)ζ(s, r).
Eisenstein series of weight 1
Since the constant term and the Fourier coefficients of Eψ,ϕ1 are symmetric inψandϕ, the series depends only on the two characters as an unordered pair, and it makes sense to have for each triple ({ψ, ϕ}, t) ∈AN,1 to be defined . Prove that the right side is G2(τ). b) Check whether the degree-1 function is homogeneous with respect to Γ(N). Argue that the Fourier coefficients of the Eisenstein series in Ms/2(Γ1(4)) dominate those of the cusp shapes in Ss/2(Γ1(4)) as n → ∞, and that the methods of this exercise thus yielding an asymptotic solution to solve the ssquares problem by ignoring the cusp shapes.
The Fourier transform and the Mellin transform
The properties of the Riemann zeta function are established by examining the Mellin transform of (essentially) the theta function. Set l = 1 and write ϑ(τ) rather than ϑ(τ,1) for the duration of this section, consider the Mellin transform of the function ∞. That is, the Mellin transform of the 1-dimensional theta function is the Riemann zeta function multiplied by a well-understood factor.
Nonholomorphic Eisenstein series
As in the previous section, the transformation law shows that the part of the integral is The discrete part contains cusp shapes and a little more, the residual spectrum, so named because it consists of residuals of the Eisenstein series in the half-plane to the right of the line of symmetry for the functional equation. Show that the sum (4.46) of the Eisenstein series stops symmetric better than a forum,.
Modular forms via theta functions
That is, the computations of the solutions of the cubic equation C are a system of eigenvalues emerging from a modular form. From now until the end of the section the symbol d is unrelated to the d of the cubic equation C. That is, the Fourier coefficients are the number of solutions (4.47) of equation C as predicted at the beginning of the section.
Hecke Operators
The double coset operator
With the finiteness of the orbital space established, the double cosetΓ1αΓ2 can act on modular forms. Taking α=Generate dual coset operator bef[Γ1αΓ2]k =f, a natural inclusion of the subspace Mk(Γ1) into Mk(Γ2), an injection. If we take α=I and let {γ2,j} be the set of coset representatives for Γ1\Γ2, we get the dual coset operator bef[Γ1αΓ2]k.
The d and T p operators
For each sign χ : (Z/NZ)∗ −→ C, the space Mk(N, χ) from Chapter 4 is exactly the χ-eigenspace of the diamond operators. In summary, we have four compatible notions of the Hecke operator Tp, starting from the double cosetΓ1(N)1 0. Fourth, Tp is an endomorphism of the divisor group of the moduli space S1(N), induced by.
![Figure 5.1. E τ [5] as a union of six 5-cyclic subgroups](https://thumb-ap.123doks.com/thumbv2/9libinfo/10469617.0/188.661.186.484.93.282/figure-5-1-e-τ-union-cyclic-subgroups.webp)
The n and T n operators
A little inspection shows that this regroups into the desired formula (5.14) with n=pr (exercise 5.3.4). A hint for this exercise can be found at the end of the book.).
The Petersson inner product
Inspection of the argument that the integral defining the Petersson inner product converges shows that in fact this holds as long as the product f g vanishes at every vertex. At the end of this chapter, however, we will briefly review Eisenstein series, showing among other things that the Petersson inner product of an Eisenstein series and a cusp form is always 0. That is, Eisenstein series and cusp forms are in some sense orthogonal , but this statement is an abuse of language since the Petersson inner product does not converge on all Mk(Γ), possibly diverging for two non-cusp shapes.
Adjoints of the Hecke Operators
The formulation of Definition 5.4.1 and various results in the next section to include this additional generalization is a bit cumbersome, so we do not bother since our main objects of study via the Petersson inner product are edge forms. Since every such vector is a modular form, we say eigenform instead, and the result is Theorem 5.5.4. The space Sk(Γ1(N)) has an orthogonal basis of simultaneous eigenforms for the Hecke operators {n, Tn: (n, N) = 1}. Use Proposition 5.5.2 and coset representatives to show thatTp∗=wNTpw−N1 and soTn∗=wNTnwN−1 for alln.
Oldforms and Newforms
![Figure 1.3. E[5]: the 5-torsion points of a torus](https://thumb-ap.123doks.com/thumbv2/9libinfo/10469617.0/41.661.250.419.561.715/figure-1-3-e-5-torsion-points-torus.webp)
