Abstract
Abstract In recent years, a number of papers have been devoted to the study of zeros of period polynomials of modular forms. In the present paper, we study cohomological analogues of the Eichler–Shimura period polynomials corresponding to higher L-derivatives. We state a general conjecture about the locations of the zeros of the full and odd parts of the polynomials, in analogy with the existing literature on period polynomials, and we also give numerical evidence that similar results hold for our higher derivative “period polynomials” in the case of cusp forms. The unimodularity of the roots seems to be a very subtle property which is special to our “period polynomials”. This is suggested by numerical experiments on families of perturbed “period polynomials” (Section 5.3) suggested by Zagier. We prove a special case of our conjecture in the case of Eisenstein series. Although not much is currently known about derivatives higher than first order ones for general modular forms, celebrated recent work of Yun and Zhang established the analogues of the Gross–Zagier formula for higher L-derivatives in the function field case. A critical role in their work was played by a notion of “super-positivity”, which, as recently shown by Goldfeld and Huang, holds in infinitely many cases for classical modular forms. As will be discussed, this is similar to properties which were required by Jin, Ma, Ono, and Soundararajan in their proof of the Riemann Hypothesis for Period Polynomials, thus suggesting a connection between the analytic nature of our conjectures here and the framework of Yun and Zhang.
Highlights
Derivatives of L-functions, and especially higher order derivatives, remain mysterious objects
P (x) dx, V Q(x) where V is a domain in Rn defined by polynomial inequalities with coefficients in Q and P, Q ∈ Q[X1, . . . , Xn]. This definition accounts for numbers that are clearly very important for number theory and other areas of mathematics, but are not necessarily algebraic, e.g
Based on the numerical data leading to Prop. 1.2, we find that, if one normalizes the odd parts of the our period polynomials to be monic, the signs of the polynomials just to the right of 1 are −
Summary
Derivatives of L-functions, and especially higher order derivatives, remain mysterious objects. Smyth and R.J.Wang [27] proved that all non-real zeros of the odd part of the period polynomial of an Eisenstein series lie on the unit circle This is a natural line of symmetry for the period polynomials, given that they are reciprocal polynomials thanks to the functional equation for completed L-values, and so as explained by S. Imamoglu [6] have proved that, apart from some “trivial” real zeros, all zeros of the odd part of the period polynomial of a cusp form lie on the unit circle. A discussion of possible follow-up work will be given
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