On the modular curves YE(7)
نویسندگان
چکیده
Let E denote an elliptic curve over Q and Y E (7) the modular curve classifying the elliptic curves E over Q such that the representations of Gal(Q/Q) in the 7-torsion points of E and of E are symplectically isomorphic. In case E is given by a Weierstraß equation such that the c 4 invariant is a square, we exhibit here nontrivial points of Y E (7)(Q). From this we deduce an infinite family of curves E for which Y E (7)(Q) has at least four nontrivial points.
منابع مشابه
ELLIPTIC CURVES AND MODULAR FORMS Contents
1. January 21, 2010 2 1.1. Why define a curve to be f rather than V (f) ⊂ P(k)? 3 1.2. Cubic plane curves 3 2. January 26, 2010 4 2.1. A little bit about smoothness 4 2.2. Weierstrass form 5 3. January 28, 2010 6 3.1. An algebro-geometric description of the group law in terms of divisors 6 3.2. Why are the two group laws the same? 7 4. February 2, 2010 7 4.1. Overview 7 4.2. Uniqueness of Weier...
متن کاملOn a unit group generated by special values of Siegel modular functions
There has been important progress in constructing units and Sunits associated to curves of genus 2 or 3. These approaches are based mainly on the consideration of properties of Jacobian varieties associated to hyperelliptic curves of genus 2 or 3. In this paper, we construct a unit group of the ray class field k6 of Q(exp(2πi/5)) modulo 6 with full rank by special values of Siegel modular funct...
متن کاملASPECTS OF COMPLEX MULTIPLICATION Contents
1. Preview 2 Complex multiplication on elliptic curves over C 2 Traces of singular moduli 3 Class field theory 3 The Kronecker limit formula and Kronecker’s solution of Pell’s equation 4 Application to Diophantine equations (Villegas) 4 L-series and CM modular forms 5 Other topics 6 2. Complex Multiplication on Elliptic Curves over C 6 Elliptic Curves over C 6 Elliptic functions 7 Complex multi...
متن کاملPeriods and Special Values of L-functions
Introduction 1 1. Modular forms, congruences and the adjoint L-function 2 2. Quaternion algebras and the Jacquet-Langlands correspondence 6 3. Integral period relations for quaternion algebras over Q 8 4. The theta correspondence 12 5. Arithmetic of the Shimizu lift and Waldspurger’s formula 16 6. Hilbert modular forms, Shimura’s conjecture and a refined version 19 7. Unitary groups and Harris’...
متن کامل18.783 Elliptic Curves Spring 2013 Lecture #24 05/09/2013
Andrew V. Sutherland In this lecture we give a brief overview of modular forms, focusing on their relationship to elliptic curves. This connection is crucial to Wiles’ proof of Fermat’s Last Theorem [7]; the crux of his proof is that every semistable elliptic curve over Q is modular.1 In order to explain what this means, we need to delve briefly into the theory of modular forms. Our goal in doi...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Math. Comput.
دوره 69 شماره
صفحات -
تاریخ انتشار 2000