The Ramanujan-Serre differential operators and certain elliptic curves
نویسندگان
چکیده
منابع مشابه
MPHS I: Differential modular forms,Elliptic curves and Ramanujan foliation
In this article we define the algebra of differential modular forms and we prove that it is generated by Eisenstein series of weight 2, 4 and 6. We define Hecke operators on them, find some analytic relations between these Eisenstein series and obtain them in a natural way as coefficients of a family of elliptic curves. Then we describe the relation between the dynamics of a foliation in C indu...
متن کاملOn the rank of certain parametrized elliptic curves
In this paper the family of elliptic curves over Q given by the equation Ep :Y2 = (X - p)3 + X3 + (X + p)3 where p is a prime number, is studied. Itis shown that the maximal rank of the elliptic curves is at most 3 and someconditions under which we have rank(Ep(Q)) = 0 or rank(Ep(Q)) = 1 orrank(Ep(Q))≥2 are given.
متن کاملThe Ramanujan differential operator, a certain CM elliptic curve and Kummer congruences
Let τ be a point in the upper half-plane such that the elliptic curve corresponding to τ can be defined over Q, and let f be a modular form on the full modular group with rational Fourier coefficients. By applying the Ramanujan differential operator D to f , we obtain a family of modular forms Dlf . In this paper we study the behavior of Dl(f)(τ) modulo the powers of a prime p > 3. We show that...
متن کاملon the rank of certain parametrized elliptic curves
in this paper the family of elliptic curves over q given by the equation ep :y2 = (x - p)3 + x3 + (x + p)3 where p is a prime number, is studied. itis shown that the maximal rank of the elliptic curves is at most 3 and someconditions under which we have rank(ep(q)) = 0 or rank(ep(q)) = 1 orrank(ep(q))≥2 are given.
متن کاملCertain maximal curves and Cartier operators ∗
In general, this bound is sharp. In fact if q is a square, there exist several curves that attain the above upper bound (see [4], [5], [14] and [23]). We say a curve is maximal (resp. minimal) if it attains the above upper (resp. lower) bound. There are however situations in which the bound can be improved. For instance, if q is not a square there is a non-trivial improvement due to Serre (see ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 2013
ISSN: 0002-9939,1088-6826
DOI: 10.1090/s0002-9939-2013-11917-9