**Title:** Local Points on Quadratic Twists of the Classical Modular Curve

**Speaker:** Ekin Ozman

**Speaker Info:** Wisconsin

**Brief Description:**

**Special Note**:

**Abstract:**

Let X^d(N) be the modular curve described as quadratic twist of the classical modular curve, X_0(N) by a quadratic field K=Q(sqrt{d})$ and Atkin-Lehner involution w_N. Rational points on this twist are K-rational points of X_0(N) that are fixed by sigma composed with w_N where sigma is the generator of Gal(K/Q). Unlike X_0(N), it's not immediate to say that there are points (global or local) on X^d(N). Given (N,d,p) we give necessary and sufficient conditions for existence of a Qp-rational point on X^d(N), answering the following question of Ellenberg:For which d and N there exists points on X^d(N) for every completion of Q?

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