Abstract. We explicitly write down the Eisenstein elements inside the space of modular symbols for Eisenstein series with integer coefficients for the congruence subgroups Γ 0 (pq) with p and q distinct odd primes, giving an answer to a question of Merel in these cases. We also compute the winding elements explicitly for these congruence subgroups. Our results are explicit versions of the Manin-Drinfeld Theorem [Thm. 9].

For any odd prime p, we construct an infinite family of imaginary quadratic fields whose class numbers are divisible by p. We give a corollary that settles Iizuka’s conjecture for the case n=1 and p>2.

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We prove a formula for the coefficients of a weight 3/2 Cohen-Eisenstein series of squarefree level N . This formula generalizes a result of Gross, and in particular, it proves a conjecture of Quattrini. Let l be an odd prime number. For any elliptic curve E defined over Q of rank zero and square-free conductor N , if l | |E(Q)|, under certain conditions on the Shafarevich-Tate group X D , we show that l divides |X D | if and only if l divides the class number h(−D) of Q( √ −D). the coefficients of the Cohen-Eisenstein serieswhere g i and w i are certain orders defined in Sec. 2. We work with certain definite quaternion algebras ramified at finitely many primes p 1 , p 2 , . . . , p k , and we compute the coefficients of H for square-free level in Theorem 5.1. As a consequence, we deduce Conjecture 2.3 (Corollary 5.3). The estimation of the number of imaginary quadratic fields whose ideal class group has an element of order l ≥ 2 and the analogous questions for quadratic twists of elliptic curves has been the center of interest in many results. For elliptic curves E of prime conductors, using the theory of p-adic L-functions and Eisenstein quotients, Mazur [10] showed that under certain conditions, the quadratic twist of E by a primitive, odd quadratic Dirichlet character χ has finite Mordell-Weil group of order not divisible by a prime l if and only if the quadratic field associated to χ has class number prime to l. In [5], Frey obtained the information about the elements of order l in the Selmer group of E D , the quadratic twist of E by −D, by assuming the elliptic curve E over Q contains a Q-rational torsion point of prime order l. In [7], James proved that 3 divides the order of the Selmer group of X 0 (11) D if and only if 3 divides the class number h(−D) under the similar assumption that the elliptic curve E contains a rational torsion point of order 3. In [17], Wong showed that there are infinitely many negative fundamental discriminants −D such that the twist X 0 (11) D of the modular curve X 0 (11) has rank 0 over Q and an element of order 5 in its Shafarevich-Tate group. Using the circle method and results of Frey and Kolyvagin, Ono [11] proved a result for the nontriviality of class groups of imaginary quadratic fields and results on the nontriviality of the Shafarevich-Tate groups of certain elliptic curves. It is also known that for almost all primes l, there exist infinitely many square-free integers D such that l |X D | [8].We prove that (Theorem 2.6) if E is an elliptic curve with square-free conductor N and l is an odd prime dividing |E(Q)|, under certain conditions on the Shafarevich-Tate group X D , the proportion of X D in the family, divisible by l, is the same as the proportion of class numbers h(−D) divisible by l in the family of negative quadratic fields Q( √ −D) with the same Kronecker conditions.To prove Theorem 5.1, we follow the strategy of Gross and we use Eichler's formula. The contents of this paper are as follows. In Sec. 2, we discuss some preliminaries. In Sec. 3, we compute th...