Biased statistics for traces of cyclic p-fold covers over finite fields

In this paper, we discuss in more detail some of the results on the statistics of the trace of the Frobenius endomorphism associated to cyclic p-fold covers of the projective line that were presented in [1]. We also show new findings regarding statistics associated to such curves where we fix the number of zeros in some of the factors of the equation in the affine model

Ramanujan graphs in cryptography

In this paper we study the security of a proposal for Post-Quantum Cryptography from both a number theoretic and cryptographic perspective. Charles-Goren-Lauter in 2006 [CGL06] proposed two hash functions based on the hardness of finding paths in Ramanujan graphs. One is based on Lubotzky-Phillips-Sarnak (LPS) graphs and the other one is based on Supersingular Isogeny Graphs. A 2008 paper by Petit-Lauter-Quisquater breaks the hash function based on LPS graphs. On the Supersingular Isogeny Graphs proposal, recent work has continued to build cryptographic applications on the hardness of finding isogenies between supersingular elliptic curves. A 2011 paper by De Feo-Jao-Pl\^{u}t proposed a cryptographic system based on Supersingular Isogeny Diffie-Hellman as well as a set of five hard problems. In this paper we show that the security of the SIDH proposal relies on the hardness of the SIG path-finding problem introduced in [CGL06]. In addition, similarities between the number theoretic ingredients in the LPS and Pizer constructions suggest that the hardness of the path-finding problem in the two graphs may be linked. By viewing both graphs from a number theoretic perspective, we identify the similarities and differences between the Pizer and LPS graphs.Comment: 33 page

Averages of central L-values of Hilbert modular forms with an application to subconvexity

We use the relative trace formula to obtain exact formulas for central values of certain twisted quadratic base change L-functions averaged over Hilbert modular forms of a fixed weight and level. We apply these formulas to the subconvexity problem for these L-functions. We also establish an equidistribution result for the Hecke eigenvalues weighted by these L-values.Comment: 57 pages, minor changes made to version 1. The final version of this article will be published in the Duke Mathematical Journal, Vol. 149, No. 2, published by Duke University Pres

Explicit construction of Ramanujan bigraphs

We construct explicitly an infinite family of Ramanujan graphs which are bipartite and biregular. Our construction starts with the Bruhat-Tits building of an inner form of $SU_3(\mathbb Q_p)$. To make the graphs finite, we take successive quotients by infinitely many discrete co-compact subgroups of decreasing size.Comment: 10 page

The fluctuations in the number of points of smooth plane curves over finite fields

In this note, we study the fluctuations in the number of points of smooth projective plane curves over finite fields $\mathbb{F}_q$ as $q$ is fixed and the genus varies. More precisely, we show that these fluctuations are predicted by a natural probabilistic model, in which the points of the projective plane impose independent conditions on the curve. The main tool we use is a geometric sieving process introduced by Poonen.Comment: 12 page

Statistics for traces of cyclic trigonal curves over finite fields

We study the variation of the trace of the Frobenius endomorphism associated to a cyclic trigonal curve of genus g over a field of q elements as the curve varies in an irreducible component of the moduli space. We show that for q fixed and g increasing, the limiting distribution of the trace of the Frobenius equals the sum of q+1 independent random variables taking the value 0 with probability 2/(q+2) and 1, e^{(2pi i)/3}, e^{(4pi i)/3} each with probability q/(3(q+2)). This extends the work of Kurlberg and Rudnick who considered the same limit for hyperelliptic curves. We also show that when both g and q go to infinity, the normalized trace has a standard complex Gaussian distribution and how to generalize these results to p-fold covers of the projective line.Comment: 30 pages, added statement and sketch of proof in Section 7 for generalization of results to p-fold covers of the projective line, the final version of this article will be published in International Mathematics Research Notice

Structure and function of preQ1 riboswitches

PreQ1 riboswitches help regulate the biosynthesis and transport of PreQ1 (7-aminomethyl-7- deazaguanine), a precursor of the hypermodified guanine nucleotide queuosine (Q), in a number of Firmicutes, Proteobacteria, and Fusobacteria. Queuosine is almost universally found at the wobble position of the anticodon in asparaginyl, tyrosyl, histidyl and aspartyl tRNAs, where it contributes to translational fidelity. Two classes of PreQ1 riboswitches have been identified (PreQ1-I and PreQ1-II), and structures of examples from both classes have been determined. Both classes form H-type pseudoknots upon PreQ1 binding, each of which has distinct unusual features and modes of PreQ1 recognition. These features include an unusually long loop 2 in PreQ1-I pseudoknots and an embedded hairpin in loop 3 in PreQ1-II pseudoknots. PreQ1-I riboswitches are also notable for their unusually small aptamer domain, which has been extensively investigated by NMR, X-ray crystallography, FRET, and other biophysical methods. Here we review the discovery, structural biology, ligand specificity, cation interactions, folding, and dynamics, and applications to biotechnology of PreQ1 riboswitches

hLARP7 C-terminal domain contains an xRRM that binds the 3\u27 hairpin of 7SK RNA

The 7SK small nuclear ribonucleoprotein (snRNP) sequesters and inactivates the positive transcription elongation factor b (P-TEFb), an essential eukaryotic mRNA transcription factor. The human La-related protein group 7 (hLARP7) is a constitutive component of the 7SK snRNP and localizes to the 3\u27 terminus of the 7SK long noncoding RNA. hLARP7, and in particular its C-terminal domain (CTD), is essential for 7SK RNA stability and assembly with P-TEFb. The hLARP7 N-terminal Lamodule binds and protects the 3\u27 end from degradation, but the structural and functional role of its CTD is unclear.We report the solution NMR structure of the hLARP7 CTD and show that this domain contains an xRRM, a class of atypical RRM first identified in the Tetrahymena thermophila telomerase LARP7 protein p65. The xRRM binds the 3\u27 end of 7SK RNA at the top of stem-loop 4 (SL4) and interacts with both unpaired and base-paired nucleotides. This study confirms that the xRRM is general to the LARP7 family of proteins and defines the binding site for hLARP7 on the 7SK RNA, providing insight into function