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

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

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

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

Exact averages of central values of triple product L-functions

We obtain exact formulas for central values of triple product L-functions averaged over newforms of weight 2 and prime level. We apply these formulas to non-vanishing problems. This paper uses a period formula for the triple product L-function proved by Gross and Kudla

Molecular Mechanism of GTPase Activation at the Signal Recognition Particle (SRP) RNA Distal End

The signal recognition particle (SRP) RNA is a universally conserved and essential component of the SRP that mediates the co-translational targeting of proteins to the correct cellular membrane. During the targeting reaction, two functional ends in the SRP RNA mediate distinct functions. Whereas the RNA tetraloop facilitates initial assembly of two GTPases between the SRP and SRP receptor, this GTPase complex subsequently relocalizes ∼100 Å to the 5′,3′-distal end of the RNA, a conformation crucial for GTPase activation and cargo handover. Here we combined biochemical, single molecule, and NMR studies to investigate the molecular mechanism of this large scale conformational change. We show that two independent sites contribute to the interaction of the GTPase complex with the SRP RNA distal end. Loop E plays a crucial role in the precise positioning of the GTPase complex on these two sites by inducing a defined bend in the RNA helix and thus generating a preorganized recognition surface. GTPase docking can be uncoupled from its subsequent activation, which is mediated by conserved bases in the next internal loop. These results, combined with recent structural work, elucidate how the SRP RNA induces GTPase relocalization and activation at the end of the protein targeting reaction

Galois representations and Galois groups over Q

In this paper we generalize results of P. Le Duff to genus n hyperelliptic curves. More precisely, let C/Q be a hyperelliptic genus n curve, let J(C) be the associated Jacobian variety and let ¯ρℓ : GQ → GSp(J(C)[ℓ]) be the Galois representation attached to the ℓ-torsion of J(C). Assume that there exists a prime p such that J(C) has semistable reduction with toric dimension 1 at p. We provide an algorithm to compute a list of primes ℓ (if they exist) such that ¯ρℓ is surjective. In particular we realize GSp6 (Fℓ) as a Galois group over Q for all primes ℓ ∈ [11, 500000]