Bounded gaps between primes with a given primitive root

Fix an integer $g \neq -1$ that is not a perfect square. In 1927, Artin conjectured that there are infinitely many primes for which $g$ is a primitive root. Forty years later, Hooley showed that Artin's conjecture follows from the Generalized Riemann Hypothesis (GRH). We inject Hooley's analysis into the Maynard--Tao work on bounded gaps between primes. This leads to the following GRH-conditional result: Fix an integer $m \geq 2$. If $q_1 < q_2 < q_3 < \dots$ is the sequence of primes possessing $g$ as a primitive root, then $\liminf_{n\to\infty} (q_{n+(m-1)}-q_n) \leq C_m$, where $C_m$ is a finite constant that depends on $m$ but not on $g$. We also show that the primes $q_n, q_{n+1}, \dots, q_{n+m-1}$ in this result may be taken to be consecutive.Comment: small corrections to the treatment of \sum_1 on pp. 11--1

A common phytoene synthase mutation underlies white petal varieties of the California poppy.

The California poppy (Eschscholzia californica) is renowned for its brilliant golden-orange flowers, though white petal variants have been described. By whole-transcriptome sequencing, we have discovered in multiple white petal varieties a single deletion leading to altered splicing and C-terminal truncation of phytoene synthase (PSY), a key enzyme in carotenoid biosynthesis. Our findings underscore the diverse roles of phytoene synthase in shaping horticultural traits, and resolve a longstanding mystery of the regaled golden poppy

An algebraic version of a theorem of Kurihara

Let E/Q be an elliptic curve and let p be an odd supersingular prime for E. In this article, we study the simplest case of Iwasawa theory for elliptic curves, namely when E(Q) is finite, III(E/Q) has no p-torsion and the Tamagawa factors for E are all prime to p. Under these hypotheses, we prove that E(Q_n) is finite and make precise statemens about the size and structure of the p-power part of III(E/Q_n). Here Q_n is the n-th step in the cyclotomic Z_p-extension of Q