Reduction of CM elliptic curves and modular function congruences (original) (raw)
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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 ...
Modular curves and the eisenstein ideal
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Analytic Number Theory, 1996
Congruences for Fourier coefficients of integer weight modular forms have been the focal point of a number of investigations. In this note we shall exhibit congruences for Fourier coefficients of a slightly different type. Let f (z) = P ∞ n=0 a(n)q n be a holomorphic half integer weight modular form with integer coefficients. If is prime, then we shall be interested in congruences of the form a( N ) ≡ 0 mod where N is any quadratic residue (resp. non-residue) modulo . For every prime > 3 we exhibit a natural holomorphic weight 2 + 1 modular form whose coefficients satisfy the congruence a( N ) ≡ 0 mod for every N satisfying`− N ´= 1. This is proved by using the fact that the Fourier coefficients of these forms are essentially the special values of real Dirichlet L−series evaluated at s = 1− 2 which are expressed as generalized Bernoulli numbers whose numerators we show are multiples of . ¿From the works of Carlitz and Leopoldt, one can deduce that the Fourier coefficients of these forms are almost always a multiple of the denominator of a suitable Bernoulli number. Using these examples as a template, we establish sufficient conditions for which the Fourier coefficients of a half integer weight modular form are almost always divisible by a given positive integer M. We also present two examples of half-integer weight forms, whose coefficients are determined by the special values at the center of the critical strip for the quadratic twists of two modular L−functions, possess such congruence properties. These congruences are related to the non-triviality of the −primary parts of Shafarevich-Tate groups of certain infinite families of quadratic twists of modular elliptic curves with conductors 11 and 14.
University of Groningen Modular invariants for genus 3 hyperelliptic curves
2018
In this article we prove an analogue of a theorem of Lachaud, Ritzenthaler, and Zykin, which allows us to connect invariants of binary octics to Siegel modular forms of genus 3. We use this connection to show that certain modular functions, when restricted to the hyperelliptic locus, assume values whose denominators are products of powers of primes of bad reduction for the associated hyperelliptic curves. We illustrate our theorem with explicit computations. This work is motivated by the study of the values of these modular functions at CM points of the Siegel upper half-space, which, if their denominators are known, can be used to effectively compute models of (hyperelliptic, in our case) curves with CM.
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2008
The classical theory of elliptic modular equations is refor mulated and extended, and many new rationally parametrized modular equations are dis covered. Each arises in the context of a family of elliptic curves attached to a genus-zero congrue nce subgroupΓ0(N), as an algebraic transformation of elliptic curve periods, which are parame triz d by a Hauptmodul (function field generator). Since the periods satisfy a Picard–Fuchs equat ion, which is of hypergeometric, Heun, or more general type, the new equations can be viewed as algeb r ic transformation formulas for special functions. The ones for N = 4, 3, 2 yield parametrized modular transformations of Ramanujan’s elliptic integrals of signatures 2, 3, 4. The case of signature 6 will require an extension of the present theory, to one of modular equations for general elliptic surfaces.
2016
Let F be a totally real field and χ an abelian totally odd character of F. In 1988, Gross stated a p-adic analogue of Stark's conjecture that relates the value of the derivative of the p-adic L-function associated to χ and the p-adic logarithm of a p-unit in the extension of F cut out by χ. In this paper we prove Gross's conjecture when F is a real quadratic field and χ is a narrow ring class character. The main result also applies to general totally real fields for which Leopoldt's conjecture holds, assuming that either there are at least two primes above p in F , or that a certain condition relating the L-invariants of χ and χ −1 holds. This condition on L-invariants is always satisfied when χ is quadratic. Contents S. DASGUPTA, H. DARMON, and R. POLLACK 4. Galois representations 477 4.1. Representations attached to ordinary eigenforms 477 4.2. Construction of a cocycle 480 References 482