Most phenotypic traits are controlled by many genes, but a global picture of the genotype-phenotype map (GPM) is lacking. For example, in no species do we know generally how many genes affect a trait and how large these effects are. It is also unclear to what extent GPMs are shaped by natural selection. Here we address these fundamental questions using the reverse genetic data of 220 morphological traits in 4,718 budding yeast strains, each of which lacks a nonessential gene. We show that 1) the proportion of genes affecting a trait varies from <1% to >30%, averaging 6%, 2) most traits are impacted by many more small-effect genes than large-effect genes, and 3) the mean effect of all nonessential genes on a trait decreases precipitously as the estimated importance of the trait to fitness increases. An analysis of 3,116 yeast gene expression traits in 754 gene-deletion strains reveals a similar pattern. These findings illustrate the power of genome-wide reverse genetics in genotype-phenotype mapping, uncover an enormous range of genetic complexity of phenotypic traits, and suggest that the GPM of cellular organisms has been shaped by natural selection for mutational robustness.

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For every odd integer n ≥ 3, we prove that there exist infinitely many number fields of degree n and associated Galois group S n whose class number is odd. To do so, we study the class groups of families of number fields of degree n whose rings of integers arise as the coordinate rings of the subschemes of P 1 cut out by integral binary n-ic forms. By obtaining upper bounds on the mean number of 2-torsion elements in the class groups of fields in these families, we prove that a positive proportion (tending to 1 as n tends to ∞) of such fields have trivial 2-torsion subgroup in their class groups and narrow class groups. Conditional on a tail estimate, we also prove the corresponding lower bounds and obtain the exact values of these averages, which are consistent with the heuristics of Cohen-Lenstra-Martinet-Malle and Dummit-Voight.Additionally, for any order O f of degree n arising from an integral binary n-ic form f , we compare the sizes of Cl 2 (O f ), the 2-torsion subgroup of ideal classes in O f , and I 2 (O f ), the 2-torsion subgroup of ideals in O f . For the family of orders arising from integral binary n-ic forms and contained in fields with fixed signature (r 1 , r 2 ), we prove that the mean value of the difference |Cl 2 (O f )| − 2 1−r1−r2 |I 2 (O f )| is equal to 1, generalizing a result of Bhargava and the third-named author for cubic fields. Conditional on certain tail estimates, we also prove that the mean value of |Cl 2 (O f )| − 2 1−r1−r2 |I 2 (O f )| remains 1 for certain families obtained by imposing local splitting and maximality conditions.There is a height ordering on R H arising from the height ordering H on Sym n (Z 2 ), where H(f ) is defined as the maximum absolute value of the coefficients of f . Note that although two rings in R H may be isomorphic, their heights need not be equal. For example, if γ ∈ SL 2 (Z), and we define the action γf (x, y) := f ((x, y)γ) on the space of integral binary n-ic forms, then it is always true thatSuch orbits [f ] may be ordered by their Julia invariant, which is an invariant defined in [27] for the action of SL 2 (Z) on Sym n (Z 2 ) (see §3.3 for details). Thus, we also define the family R J to be the multiset of ringsordered by Julia invariant J, where J(R [f ] ) := J([f ]). Asymptotics on the size of R J were obtained by Bhargava-Yang [13].In this paper, we compute averages taken over certain families contained in R H or R J . Let R r 1 ,r 2 H ⊂ R H and R r 1 ,r 2 J ⊂ R J be the respective subfamilies consisting of all Gorenstein 1 integral domains whose fraction field has signature (r 1 , r 2 ), i.e., has r 1 real embeddings and r 2 pairs of conjugate complex embeddings. Also, let R r 1 ,r 2 H,max ⊂ R r 1 ,r 2 H (resp. R r 1 ,r 2 J,max ⊂ R r 1 ,r 2 J ) be the subfamily containing all maximal orders. It is worthwhile to note that a given order O in a number field with signature (r 1 , r 2 ) may occur in R r 1 ,r 2 H or R r 1 ,r 2 H,max an infinite number of times (up to isomorphism) but only occurs with finite multiplicity in R r 1 ,r 2 J or R r 1...

The budding yeast Saccharomyces cerevisiae is the best studied eukaryote in molecular and cell biology, but its utility for understanding the genetic basis of phenotypic variation in natural populations is limited by inefficient association mapping due to strong and complex population structure. To overcome this challenge, we generated genome sequences for 85 strains and performed a comprehensive population genomic survey of a total of 190 diverse strains. We identified considerable variation in population structure among chromosomes and identified 181 genes that are absent from the reference genome. Many of these nonreference genes are expressed and we functionally confirmed that two of these genes confer increased resistance to antifungals. Next, we simultaneously measured the growth rates of over 4,500 laboratory strains, each of which lacks a nonessential gene, and 81 natural strains across multiple environments using unique DNA barcode present in each strain. By combining the genome-wide reverse genetic information gained from the gene deletion strains with a genome-wide association analysis from the natural strains, we identified genomic regions associated with fitness variation in natural populations. To experimentally validate a subset of these associations, we used reciprocal hemizygosity tests, finding that while the combined forward and reverse genetic approaches can identify a single causal gene, the phenotypic consequences of natural genetic variation often follow a complicated pattern. The resources and approach provided outline an efficient and reliable route to association mapping in yeast and significantly enhance its value as a model for understanding the genetic mechanisms underlying phenotypic variation and evolution in natural populations.

Most systematic tables of data associated to ranks of elliptic curves order the curves by conductor. Recent developments, led by work of Bhargava and Shankar studying the average sizes of n-Selmer groups, have given new upper bounds on the average algebraic rank in families of elliptic curves over Q, ordered by height. We describe databases of elliptic curves over Q, ordered by height, in which we compute ranks and 2-Selmer group sizes, the distributions of which may also be compared to these theoretical results. A striking new phenomenon that we observe in our database is that the average rank eventually decreases as height increases.