Due to his motivation and knowledgeable guidance, I was able to gain a lot of knowledge in number theory. I would also like to thank all the teachers in the Mathematics Department for their constant support and encouragement and imparting knowledge. Last but not the least, I would like to thank my family for believing in me and giving me the opportunity to study in IIT Hyderabad.
The Manin-Drinfeld theorem is an important result of the theory of modular forms about the existence of an isomorphism between the space of Eisenstein series E2(Γ0(N)) and a first homology group H1(X0(N ),Z) of a modular curve X0(N). Explicitly, such an isomorphism was first constructed in [1] in the case when N =p is prime. In the thesis we review the structure of an isomorphism in the case when N =p2 andp≥3 is prime. The description of the structure of spaces Mk(Γ) is the main goal of the theory of modular forms.
There are still many open problems and conjectures about modular forms, such as the Birch and Swinnerton-Dyer conjecture. They proved that the Eisenstein element corresponding to the Eisenstein series E ∈ E2(Γ0(N)) is a combination of certain modular symbols.
Congruence subgroups
Modular curves
Having defined fundamental systems of neighborhoods, we can define the topology on H∪P1(Q): a set G⊂H∪P1(Q) is open if and only if for every τ ∈G there is a neighborhood B ∈ Uτ is such that B ⊂G. As a conclusion, the space H∪P1(Q) is endowed with a topology by specifying the fundamental system of neighborhoods of each point. For this topology to be properly transferred to the modular curve, we check that every element γ ∈ SL2(Z) is a homeomorphism of H∪P1(Q).
Then γ is a bijection H∪P1(Q). Continuity for each τ ∈H follows from the relation γ(H) ⊂ H and the fact that H is equipped with the usual topology of the upper half-plane. With the quotient topology, X(Γ) is a Riemann surface with local coordinates around the vertex s ∈ P1(Q)∪ {∞}, given by the composition of the straight map δ ∈ SL2(Z), where δ(s)∞and the wrap mapρ(z ) = e2πiz/h with width he, which is equal to the index of the subgroup Γs that fixes the point s [5, §2.4.].
Modular forms
In this case, the series expansion in powers of qh is called the Fourier expansion of f[α]k. If ff is weight-k-weakly modular with respect to Γ, then f[α]k for any α ∈SL2(Z) function is weight-k weakly modular with respect to α−1Γα. If, in addition, for each α ∈SL2(Z) the zero term in the Fourier expansion of f[α]k is zero, then f is called a peak form.
For each α ∈ SL2(Z), the zero Fourier coefficient off[α]k is called the Fourier coefficient at ∞, and it is independent of the choice of f. It is assumed that all Fourier coefficients at ∞ of an Eisenstein series E belongs to some number filed K, i.e.
Homology and cohomology groups
General definitions
Let Cn(X) be a free abelian group based on the set of all singular n−simplices inX. One of the advantages of a cohomology is that it is clear how to define the product of cohomology. As a consequence of this formula, the coproduct is factorized through coboundary maps and gives the coproduct of cohomologies.
If all coefficients at ∞ of the corresponding Eisenstein set E belong to K, we obtain a homomorphism. Finally, the intersection number for two cycles is defined as. here we assume that the cycles intersect transversely at each point). The definition depends only on homology classes and thus gives the pairing of homology groups H1 [6, ch.
Poincare's duality theorem [6, Ch.0, §4] states that the pairing is perfect, that is, every linear functional on H1(Y0(N), K) is given by pairing with some element of.
Modular symbols
Projective line over the ring
General construction and a projective line over Z /N Z
The question remains whether this does not divide. If not, then =lp and.
Special values of L-functions and the function F E
We will show that it can be expanded to a meromorphic function on C with possible poles only at s= 0 or s= 2. To do this we use the Mellin transformation of the function f. From the relation −b0c0 ≡1( mod N) it follows that (c0, N) = 1. Let P(c0) be the set of all distinct simple multiples of c0. By the Chinese remainder theorem there exists an integer t such that for every p∈P(c0),. The core of this mapping consists of matrices that are equivalent to the modulus of the identity matrix p, i.e.
These mappings will allow transferring the Eisenstein element from the modular curve XΓ to the modular curve X(Γ0(p2)). For any analytic functionf, the order off◦π0 at Γ1 coincides with the order (f◦π)2 and the ratio (f◦π)f◦π02 has no zeros or poles. To express the value of πE in terms of L−functions, we will use the Atkin-Lehner involution law.
The next two cases show that there is no such simple description of the projective line over Z/NZ if N is not the square of a prime number. Specifying values of Eisenstein's basis series on the cusp will extend the representation to the modular curve X0(p2). Since βkp∞= kp1, it must be shown that the constant terms of the Fourier expansion at ∞ of E2[βkp]2 are zero.
Using a complete set of coset representatives of P1(Z/p2Z) from Lemma 2.1.1, we write the limit of an element π∗(E0) as follows. Let E ∈H1(X0(p2),vertices, K) be the Eisenstein element corresponding to the fundamental Eisenstein series E (ie E is one of the series E1, E2 or E2φ,φ¯). By Lemma 4.3.3, any holomorphic differential is indeed integrable to zero over E −cπ∗(E0), which implies that E =cπ∗(E0) (the last conclusion follows from the fact that the crossover is perfect).
Representation (1) is the general form of an element of H1(X0(p2), cusps, K), because the first relative homology is the set of curves with endpoints on set cusps. The exact expression for ep2 is obtained by integrating a holomorphic form ω over the Eisenstein element E2, where E2 corresponds to the basic form E2. We recall that from the main theorem, For each g we find the modular symbol ζ(g). 0,1) is the second row of the identity matrix I. kp,1) is the second row of the matrix.
Manin, "Parabolic points and zeta-functions of modular curves", Matematika USSR-Izvestiya, vol. Banerjee, “A note on Eisenstein's elements of the root-square plane,” Proceedings of the American Mathematical Society, vol.
