No abstract

We study rationality properties of geodesic cycle integrals of meromorphic modular forms associated to positive definite binary quadratic forms. In particular, we obtain finite rational formulas for the cycle integrals of suitable linear combinations of these meromorphic modular forms.

Referring to Ramanujan's original definition of a mock theta function, Rhoades asked for explicit formulas for radial limits of the universal mock theta functions g 2 and g 3 . Recently, Bringmann and Rolen found such formulas for specializations of g 2 . Here we treat the case of g 3 , generalizing radial limit results for the rank generating function of Folsom, Ono, and Rhoades. Furthermore, we find expressions for radial limits of fifth order mock theta functions different from those of Bajpai, Kimport, Liang, Ma, and Ricci.

We show that the minimum h min of the stable Faltings height on elliptic curves found by Deligne is followed by a gap. This means that there is a constant C > 0 such that for every elliptic curve E/K with everywhere semistable reduction over a number field K, we either have h(E/K) = h min or h(E/K) ≥ h min + C. We determine such an absolute constant explicitly. On the contrary, we show that there is no such gap for elliptic curves with unstable reduction.

No abstract