Abstract. We introduce a height measure on Q to count rational numbers. Through it, we prove a density result on the average value of the root numbers of families of twists of elliptic curves.Zagier and Kramarz computed in [11] the rank of the curves x 3 + y 3 = m, with m an integer < 70, 000. These data suggest that the rank is even for exactly half of the twists of x 3 + y 3 = 1. This conjecture has been proved (conditionally on the Birch and Swinnerton-Dyer conjecture) by Mai in [4]. Define, as usual, the root number W (E) of an elliptic curve E as the sign of the functional equation of the L series associated to E (see C.16 of [9]). According to the parity conjecture,rank(E) . Given a proper definition of average, we can express Mai's result as saying that the average value of the root numbers of the quadratic twists of x 3 + y 3 = 1 is 0. Let H(m/n) = max{|m| , |n|} be the height of m/n, where m and n are relatively prime integers. Definition. The average value of a function ψ : Q → R is Av ψ(t) = limT →∞ H(t)

What do you do when a change in enrollment policies leaves you with more than 600 students in a first-term university calculus class, three-quarters of those students had a failing mark in mathematics in the pre-enrollment test, you planned a series of remedial activities for the second term, and the COVID-19 pandemic shuts the university down with a two-day notice? The pandemic hit instruction with might, forcing schools and universities that were timidly experimenting with digital tools to reinvent themselves in days. The pandemic also offered incentives for creative solutions that, in normal times, would have been considered fit for submission to the committee for recursive committee submissions at best. This paper narrates a teaching experience of how we proposed and managed an at-distance remedial course in August that not only catered to more than twice the number of students expected by our best forecasts, but was a very good success once its effectiveness was compared to the outcomes predicted by the pre-enrollment test scores. We expose the design of the course and link its measured effectiveness with both its design and student engagement; in particular, we show that a different approach to the examination of cognitive load and to fostering student–teacher and student–student communication thanks to digital mediation could be effective in countermanding the math-induced drop-out phenomenon in STEM.

Abstract. We give some examples of families of elliptic curves with nonconstant j-invariant where the parity of the (analytic) rank is not equidistributed among the fibres. Mathematics Subject Classifications (2000). Primary: 11G05; secondary: 11G07, 14Gxx.Key words. elliptic curves, root numbers.Assuming the Birch and Swinnerton-Dyer conjecture, the root number of an elliptic curve E=Q is À1 to the rank of EðQÞ, the group of rational points of E. Given a 'generic' algebraic family E t of elliptic curves, one would expect to find the same numbers of curves with even and odd rank (see, for example, the graph in [16]). If E t is a family of twists of a given curve (i.e., the j-invariant is constant), then there are known counterexamples: assuming Selmer's Conjecture, Cassels and Schinzel prove in [2] that ð7 þ 7t 4 Þy 2 ¼ x 3 À x has odd rank for any t 2 Q. Given E=Q and a polynomial fðtÞ 2 Q½t , we can build the family E fðtÞ of twists of E by fðtÞ; then Rohrlich [11] proves that, if E acquires everywhere good reduction over some Abelian extension of Q, then WðE fðtÞ Þ ¼ WðEÞsgnð fðtÞÞ. Given any E=Q, the author ([8, 9]) has shown that the set fAv Q WðE fðtÞ Þg is dense in the interval ½À1; 1 , where f ðtÞ varies over all polynomials in Q½t and Av Q W denotes the average value of the root numbers for t 2 Q.It has been suggested by Silverman -see the final remarks in [14] -that this kind of phenomenon could occur only for constant families: we present here some counterexamples with t 2 Z.? This work was supported by a EU TMR fellowship 'Arithmetic Algebraic Geometry', contract ERB FMR XCT 960006.