On the ratio of the sum of divisors and Euler’s Totient Function I

We prove that the only solutions to the equation σ(n)=2φ(n) with at most three distinct prime factors are 3, 35 and 1045. Moreover, there exist at most a finite number of solutions to σ(n)=2φ(n) with Ω(n)≤ k, and there are at most 22k+k-k squarefree solutions to φ (n)|σ(n) if ω(n)=k. Lastly the number of solutions to φ(n)|φ(n) as x→∞ is O(x exp(-½√log x))

Higher order congruences amongst hasse-weil L-values

For the (d+1)-dimensional Lie group G=Z×pZp⊕d we determine through the use of p-power congruences a necessary and sufficient set of conditions whereby a collection of abelian L-functions arises from an element in K₁Zp[G]. If E is a semistable elliptic curve over Q, these abelian L-functions already exist; therefore, one can obtain many new families of higher order p-adic congruences. The first layer congruences are then verified computationally in a variety of cases

On Iwasawa λ\lambda-invariants for abelian number fields and random matrix heuristics

Following both Ernvall-Mets\"{a}nkyl\"{a} and Ellenberg-Jain-Venkatesh, we study the density of the number of zeroes (i.e. the cyclotomic $\lambda$-invariant) for the $p$-adic zeta-function twisted by a Dirichlet character $\chi$ of any order. We are interested in two cases: (i) the character $\chi$ is fixed and the prime $p$ varies, and (ii) $\text{ord}(\chi)$ and the prime $p$ are both fixed but $\chi$ is allowed to vary. We predict distributions for these $\lambda$-invariants using $p$-adic random matrix theory and provide numerical evidence for these predictions. We also study the proportion of $\chi$-regular primes, which depends on how $p$ splits inside $\mathbb{Q}(\chi)$. Finally in an extensive Appendix, we tabulate the values of the $\lambda$-invariant for every character $\chi$ of conductor $\leq 1000$ and for odd primes $p$ of small size.Comment: 159 pages, 2 figure

On trivial p-adic zeroes for elliptic curves over Kummer extensions

We prove the exceptional zero conjecture is true for semistable elliptic curves E/Q over number fields of the form F(e²ⁿⁱ⁄ｑⁿ, ∆₁¹⁄ｑⁿ , . . . , ∆₁¹⁄ⁿd) where F is a totally real field, and the split multiplicative prime p ≠ 2 is inert in F(e²ⁿⁱ⁄ｑⁿ) ∩ R

Transition formulae for ranks of abelian varieties

Let A/k denote an abelian variety defined over a number field k with good ordinary reduction at all primes above p, and let K∞ =∪n≥1 Kn be a p-adic Lie extension of k containing the cyclotomic Zp-extension. We use K-theory to find recurrence relations for the λ-invariant at each σ-component of the Selmer group over K∞, where σ : Gk → GL(V ). This provides upper bounds on the Mordell-Weil rank for A(Kn) as n → ∞ whenever G∞ = Gal(K∞/k) has dimension at most 3

Congruences modulo ρ between ρ-wisted Hasse-Weil L-values

Suppose E₁ and E₂ are semistable elliptic curves over Q with good reduction at p, whose associated weight two newforms f₁ and f₂ have congruent Fourier coefficients modulo p. Let RS(E*, ρ) denote the algebraic padic L-value attached to each elliptic curve E, twisted by an irreducible Artin representation, ρ, factoring through the Kummer extension Q(μp∞, Δ1/p∞). If E₁ and E₂ have good ordinary reduction at p, we prove that RS(E₁, ρ) ≡ RS(E₂, ρ) mod p, under an integrality hypothesis for the modular symbols defined over the field cut out by Ker(ρ). Under this hypothesis, we establish that E₁ and E₂ have the same analytic λ-invariant at ρ. Alternatively, if E₁ and E₂ have good supersingular reduction at p, we show that |RS(E₁, ρ) − RS(E₂, ρ)|ₚ < p ᵒʳᵈᵖ⁽ᶜᵒⁿᵈ⁽ρ⁾⁾/². These congruences generalise some earlier work of Vatsal [Duke Math. J. 98 (1999), pp. 399–419], Shekhar–Sujatha [Trans. Amer. Math. Soc. 367 (2015), pp. 3579–3598], and Choi-Kim [Ramanujan J. 43 (2017), p. 163–195], to the false Tate curve setting

On λ-invariants attached to cyclic cubic number fields

We describe an algorithm for finding the coefficients of F(X) modulo powers of p, where p ≠2 is a prime number and F(X) is the power series associated to the zeta function of Kubota and Leopoldt. We next calculate the 5-adic and 7-adic λ-invariants attached to those cubic extensions K/Q with cyclic Galois group A₃ (up to field discriminant <10⁷), and also tabulate the class number of K(e2πi/p) for p=5 and p=7. If the λ-invariant is greater than zero, we then determine all the zeros for the corresponding branches of the p-adic L-function and deduce Λ-monogeneity for the class group tower over the cyclotomic Zp-extension of K

Non-commutative Iwasawa theory for elliptic curves with multiplicative reduction

Let E/ℚ be a semistable elliptic curve, and p ≠ 2 a prime of bad multiplicative reduction. For each Lie extension ℚ FT / ℚ with Galois group G∞ ≅ℤр ⋊ ℤ p ×, we construct p-adic L-functions interpolating Artin twists of the Hasse–Weil L-series of the curve E. Through the use of congruences, we next prove a formula for the analytic λ-invariant over the false Tate tower, analogous to Chern–Yang Lee's results on its algebraic counterpart. If one assumes the Pontryagin dual of the Selmer group belongs to the ℳℌ(G∞)-category, the leading terms of its associated Akashi series can then be computed, allowing us to formulate a non-commutative Iwasawa Main Conjecture in the multiplicative setting

Corrigendum: A conjecture of De Koninck regarding particular values of the sum of divisors function

The proof of Lemma 7 of [1]is made complete by giving the proof of a missing Case (4). This omission was pointed out to the authors by Min Tang, to whom we are most grateful. The same definitions and notation are employed as in [1], and one should replace the first paragraph of the proof by the following argument