24 research outputs found
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
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 . 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 as 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
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
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]