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Register a new userVariants of Lehmers speculation for newforms
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In the spirit of Lehmers unresolved speculation on the nonvanishing of Ramanujans tau-function, it is natural to ask whether a fixed integer is a value of $tau(n)$ or is a Fourier coefficient $a_f(n)$ of any given newform $f(z)$. We offer a method, which applies to newforms with integer coefficients and trivial residual mod 2 Galois representation, that answers this question for odd integers. We determine infinitely many spaces for which the primes $3leq ellleq 37$ are not absolute values of coefficients of newforms with integer coefficients. For $tau(n)$ with $n>1$, we prove that $$tau(n) ot in {pm 1, pm 3, pm 5, pm 7, pm 13, pm 17, -19, pm 23, pm 37, pm 691},$$ and assuming GRH we show for primes $ell$ that $$tau(n) ot in left { pm ell : 41leq ellleq 97 {textrm{with}} left(frac{ell}{5}right)=-1right} cup left { -11, -29, -31, -41, -59, -61, -71, -79, -89right}. $$ We also obtain sharp lower bounds for the number of prime factors of such newform coefficients. In the weight aspect, for powers of odd primes $ell$, we prove that $pm ell^m$ is not a coefficient of any such newform $f$ with weight $2k>M^{pm}(ell,m)=O_{ell}(m)$ and even level coprime to $ell,$ where $M^{pm}(ell,m)$ is effectively computable.
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We consider natural variants of Lehmers unresolved conjecture that Ramanujans tau-function never vanishes. Namely, for $n>1$ we prove that $$tau(n) ot in {pm 1, pm 3, pm 5, pm 7, pm 691}.$$ This result is an example of general theorems for newforms with trivial mod 2 residual Galois representation, which will appear in forthcoming work of the authors with Wei-Lun Tsai. Ramanujans well-known congruences for $tau(n)$ allow for the simplified proof in these special cases. We make use of the theory of Lucas sequences, the Chabauty-Coleman method for hyperelliptic curves, and facts about certain Thue equations.
We study several variants of Euler sums by using the methods of contour integration and residue theorem. These variants exhibit nice properties such as closed forms, reduction, etc., like classical Euler sums. In addition, we also define a variant of multiple zeta values of level 2, and give some identities on relations between these variants of Euler sums and the variant of multiple zeta values.
In this note, we consider the newforms of integral weight, level 4 and of trivial character, and prove that all of them are actually level 1 forms of some non-Dirichlet character. As a byproduct, we can prove that all of them are eigenfunctions of the Fricke involution with eigenvalue -1.
We discuss two different systems of number representations that both can be called base 3/2. We explain how they are connected. Unlike classical fractional extension, these two systems provide a finite representation for integers. We also discuss a connection between these systems and 3-free sequences.
SIDH is a post-quantum key exchange algorithm based on the presumed difficulty of finding isogenies between supersingular elliptic curves. However, SIDH and related cryptosystems also reveal additional information: the restriction of a secret isogeny to a subgroup of the curve (torsion point information). Petit (2017) was the first to demonstrate that torsion point information could noticeably lower the difficulty of finding secret isogenies. In particular, Petit showed that overstretched parameterizations of SIDH could be broken in polynomial time. However, this did not impact the security of any cryptosystems proposed in the literature. The contribution of this paper is twofold: First, we strengthen the techniques of Petit by exploiting additional information coming from a dual and a Frobenius isogeny. This extends the impact of torsion point attacks considerably. In particular, our techniques yield a classical attack that completely breaks the n-party group key exchange of Azarderakhsh et al. for 6 parties or more, and a quantum attack for 3 parties or more that improves on the best known asymptotic complexity. We also provide a Magma implementation of our attack for 6 parties. We give the full range of parameters for which our attacks apply. Second, we construct SIDH variants designed to be weak against our attacks; this includes backdoor choices of starting curve, as well as backdoor choices of base field prime. We stress that our results do not degrade the security of, or reveal any weakness in the NIST submission SIKE.
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