Reduction of CM elliptic curves and modular function congruences (original) (raw)
On congruences for the coefficients of modular forms and some applications
1997
We start with a brief overview of the necessary theory: Given any cusp form f=∑ n≥ 1 an (f) qn of weight k, we denote by L (f, s) the L-function of f. For Re (s)> k/2+ 1, the value of L (f, s) is given by L (f, s)=∑ n≥ 1 an (f) ns and, one can show that L (f, s) has analytic continuation to the entire complex plane. The value of L (f, s) at s= k/2 will be of particular interest to us, and we will refer to this value as the central critical value of L (f, s).
1997
In the last 15 years, many class number formulas and main conjectures have been proven. Here, we discuss such formulas on the Selmer groups of the threedimensional adjoint representation ad(f) of a twodimensional modular Galois representation f. We start with the p-adic Galois representation f0 of a modular elliptic curve E and present a formula expressing in terms of L(1, ad(f0)) the intersection number of the elliptic curve E and the complementary abelian variety inside the Jacobian of the modular curve. Then we explain how one can deduce a formula for the order of the Selmer group Sel(ad(f0)) from the proof of Wiles of the Shimura–Taniyama conjecture. After that, we generalize the formula in an Iwasawa theoretic setting of one and two variables. Here the first variable, T, is the weight variable of the universal p-ordinary Hecke algebra, and the second variable is the cyclotomic variable S. In the onevariable case, we let f denote the p-ordinary Galois representation with values ...
Arithmetic of certain hypergeometric modular forms
Acta Arithmetica, 2004
In a recent paper, Kaneko and Zagier studied a sequence of modular forms F k (z) which are solutions of a certain second order differential equation. They studied the polynomials e F k (j) = Y τ ∈H/Γ−{i,ω} (j − j(τ)) ord τ (F k) , where ω = e 2πi/3 and H/Γ is the usual fundamental domain of the action of SL 2 (Z) on the upper half of the complex plane. If p ≥ 5 is prime, they proved that e F p−1 (j) (mod p) is the nontrivial factor of the locus of supersingular j-invariants in characteristic p. Here we consider the irreducibility of these polynomials, and consider their Galois groups.
Modular invariant and good reduction of elliptic curves
Mathematische Annalen, 1992
An element d ~ K*/K *" acts on r by "twisting" the models of the curves [5]. Let Z be the set of finite places of K. An interesting problem in the arithmetic theory of elliptic curves is the description of the set r of elliptic curves over K with good reduction at all places outside a given finite set S ___ Z. The sets 811s c~ ~llK(j) are principal homogeneous spaces over the subgroup of H 1 (GK, la,) of the elements which are unramified at all places outside S. Thus, ~lls is completely determined by the set:
Mathematics of Computation, 2002
We prove that there are exactly 149 genus two curves C defined over Q such that there exists a nonconstant morphism π : X 1 (N) → C defined over Q and the jacobian of C is Q-isogenous to the abelian variety A f attached by Shimura to a newform f ∈ S 2 (Γ 1 (N)). We determine the corresponding newforms and present equations for all these curves.
Linear Relations Between Modular Form Coefficients and Non-Ordinary Primes
Bulletin of the London Mathematical Society, 2005
Here we generalize a classical observation of Siegel by determining all the linear relations among the initial Fourier coefficients of a modular form on SL 2 (Z). As a consequence, we identify spaces M k in which there are universal p-divisibility properties for certain p-power coefficients. As a corollary, let f (z) = ∞ n=1 a f (n)q n ∈ S k ∩ O L [[q]] (note: q := e 2πiz) be a normalized Hecke eigenform, and let k ≡ δ(k) (mod 12), where δ(k) ∈ {4, 6, 8, 10, 14}. Reproducing earlier results of Hatada and Hida, if p is a prime for which k ≡ δ(k) (mod p − 1), and p ⊂ O L is a prime ideal above p, then we show that a f (p) ≡ 0 (mod p).
The arithmetic of the values of modular functions and the divisors of modular forms
Compositio Mathematica, 2004
We investigate the arithmetic and combinatorial significance of the values of the polynomials j n (x) defined by the q-expansion ∞ n=0 j n (x)q n := E 4 (z) 2 E 6 (z) ∆(z) • 1 j(z) − x. They allow us to provide an explicit description of the action of the Ramanujan Thetaoperator on modular forms. There are a substantial number of consequences for this result. We obtain recursive formulas for coefficients of modular forms, formulas for the infinite product exponents of modular forms, and new p-adic class number formulas.