P-Adic L-functions and Zeta Element

Many problems in number theory have simple statements, but their solutions require a deep understanding of algebra, algebraic geometry, complex analysis, group representations, or a combination of all four. The original simply stated problem can be obscured in the depth of the theory developed to understand it. This book is an introduction to some of these problems, and an overview of the theories used nowadays to attack them, presented so that the number theory is always at the forefront of the discussion. Lozano-Robledo gives an introductory survey of elliptic curves, modular forms, and $L$-functions. His main goal is to provide the reader with the big picture of the surprising connections among these three families of mathematical objects and their meaning for number theory. As a case in point, Lozano-Robledo explains the modularity theorem and its famous consequence, Fermat's Last Theorem. He also discusses the Birch and Swinnerton-Dyer Conjecture and other modern conjectures. The book begins with some motivating problems and includes numerous concrete examples throughout the text, often involving actual numbers, such as 3, 4, 5, $\frac{3344161}{747348}$, and $\frac{2244035177043369699245575130906674863160948472041} {8912332268928859588025535178967163570016480830}$. The theories of elliptic curves, modular forms, and $L$-functions are too vast to be covered in a single volume, and their proofs are outside the scope of the undergraduate curriculum. However, the primary objects of study, the statements of the main theorems, and their corollaries are within the grasp of advanced undergraduates. This book concentrates on motivating the definitions, explaining the statements of the theorems and conjectures, making connections, and providing lots of examples, rather than dwelling on the hard proofs. The book succeeds if, after reading the text, students feel compelled to study elliptic curves and modular forms in all their glory.

The rank of a CM elliptic curve and a recurrence formula

We generalise works of Kobayashi to give a formulation of the Iwasawa main conjecture for modular forms at supersingular primes. In particular, we give analogous definitions of the plus and minus Coleman maps for normalised new forms of arbitrary weights and relate Pollack’s p-adic L-functions to the plus and minus Selmer groups. In addition, by generalising works of Pollack and Rubin on CM elliptic curves, we prove the ‘main conjecture’ for CM modular forms.

In this paper, we prove that if every supersingular elliptic curve over Fp can be lifted to a CM elliptic curve by an imaginary order OD for some D ? pθ, then θ ≥ 1 2 . We also prove that if every supersingular elliptic curve over Fp can be lifted to a CM elliptic curve by an imaginary order OD for some D ? pθ, then θ ≥ 23 as suggested by Elkies.

The Birch and Swinnerton-Dyer (BSD) conjecture establishes a deep connection between the arithmetic structure of elliptic curves and the analytic behavior of their associated L-functions, yet its conceptual interpretation remains elusive despite extensive partial results. In this work, we propose an informational–geometric reinterpretation of the conjecture within the framework of Viscous Time Theory (VTT), in which arithmetic invariants are mapped to measurable quantities governing informational coherence. Within this framework, the canonical height is interpreted as a coherence potential, the Mordell–Weil rank as the dimensionality of stable coherence directions, and the BSD regulator as an informational volume. The analytic behavior of the L-function near the critical point s = 1 is reinterpreted as a global coherence response of the underlying informational manifold. This leads naturally to a regime-dependent conservation principle for informational coherence, under which the classical BSD identity emerges as a stable balance condition. To test this formulation, we perform independent numerical validation on benchmark elliptic curves with established BSD data. An informational L-function is fitted to empirical analytic profiles, from which informational curvature and volume are derived. The results show strong quantitative agreement between informational volumes and arithmetic regulators, with correlation coefficients exceeding 0.99, and demonstrate robust, rank-dependent stability behavior across curves of varying complexity. The framework is further explored under increasing geometric and topological complexity through higher-genus and synthetic informational models, which exhibit systematic coherence suppression consistent with known arithmetic phenomena. While this work does not claim a proof of the Birch and Swinnerton-Dyer conjecture, it offers a coherent explanatory framework that clarifies its internal structure, geometrically aligns analytic and arithmetic invariants, and opens new avenues for numerical and conceptual investigation.

Let be a CM elliptic curve defined over a number field K, with Weiestrass form y^3=x^3+bx or y^2=x^3+c. For every positive integer m, we denote by the m-torsion subgroup of and by the m-th division field, i.e. the extension of K generated by the coordinates of the points in . We classify all the fields K_7. In particular we give explicit generators for K_7/K and produce all the Galois groups textrm{Gal}(K_7/K). We also show some applications to the Local–Global Divisibility Problem and to modular curves.

Bruin and Najman [LMS J. Comput. Math. 18 (2015), pp. 578–602], Ozman and Siksek [Math. Comp. 88 (2019), pp. 2461–2484], and Box [Math. Comp. 90 (2021), pp. 321–343] described all the quadratic points on the modular curves of genus 2 ≤ g ( X 0 ( n ) ) ≤ 5 2\leq g(X_0(n)) \leq 5 . Since all the hyperelliptic curves X 0 ( n ) X_0(n) are of genus ≤ 5 \leq 5 and as a curve can have infinitely many quadratic points only if it is either of genus ≤ 1 \leq 1 , hyperelliptic or bielliptic, the question of describing the quadratic points on the bielliptic modular curves X 0 ( n ) X_0(n) naturally arises; this question has recently also been posed by Mazur. We answer Mazur’s question completely and describe the quadratic points on all the bielliptic modular curves X 0 ( n ) X_0(n) for which this has not been done already. The values of n n that we deal with are n = 60 n=60 , 62 62 , 69 69 , 79 79 , 83 83 , 89 89 , 92 92 , 94 94 , 95 95 , 101 101 , 119 119 and 131 131 ; the curves X 0 ( n ) X_0(n) are of genus up to 11 11 . We find all the exceptional points on these curves and show that they all correspond to CM elliptic curves. The two main methods we use are Box’s relative symmetric Chabauty method and an application of a moduli description of Q \mathbb {Q} -curves of degree d d with an independent isogeny of degree m m , which reduces the problem to finding the rational points on several quotients of modular curves.

The transcendental Brauer group of a product of CM elliptic curves

Extremal primes for elliptic curves with complex multiplication

We define an adelic version of a CM elliptic curve $E$ which is equipped with an action of the profinite completion of the endomorphism ring of $E$. The adelic elliptic curve so obtained is provided with a natural embedding into the adelic Heisenberg group. We embed into the adelic Heisenberg group the set of commensurability classes of arithmetic $1$-dimensional $\mathbb{K}$-lattices (here and subsequently, $\mathbb{K}$ denotes a quadratic imaginary number field) and define theta functions on it. We also embed the groupoid of commensurability modulo dilations into the union of adelic Heisenberg groups relative to a set of representatives of elliptic curves with $R$-multiplication ($R$ is the ring of algebraic integers of $\mathbb{K}$). We thus get adelic theta functions on the set of $1$-dimensional $\mathbb{K}$-lattices and on the groupoid of commensurability modulo dilations. Adelic theta functions turn out to be acted by the adelic Heisenberg group and behave nicely under complex automorphisms (Theorems 6.12 and 6.14).

Let $X_0^6(1)/W_6$ be the Atkin-Lehner quotient of the Shimura curve $X_0^6(1)$ associated to a maximal order in an indefinite quaternion algebra of discriminant $6$ over $\mathbb Q$. By realizing modular forms on $X_0^6(1)/W_6$ in two ways, one in terms of hypergeometric functions and the other in terms of Borcherds forms, and using Schofer's formula for values of Borcherds forms at CM-points, we obtain special values of certain hypergeometric functions in terms of periods of elliptic curves over $\overline\mathbb Q$ with complex multiplication.

Given a non-CM elliptic curve E over Q of discriminant ∆E , define the “torsion conductor” mE to be the smallest positive integer so that the Galois representation on the torsion of E has image π−1(Gal(Q(E[mE ])/Q)), where π denotes the natural projection GL2(Ẑ) → GL2(Z/mEZ). We show that, uniformly for semi-stable non-CM elliptic curves E over Q, one has

Let $K=\mathbb{Q}(\sqrt{-q})$, where $q$ is any prime number congruent to $7$ modulo $8$, with ring of integers $\mathcal{O}$ and Hilbert class field $H$. Suppose $p\nmid [H:K]$ is a prime number which splits in $K$, say $p\mathcal{O}=\mathfrak{p}\mathfrak{p}^*$. Let $H_\infty=HK_\infty$ where $K_\infty$ is the unique $\mathbb{Z}_p$-extension of $K$ unramified outside $\mathfrak{p}$. Write $M(H_\infty)$ for the maximal abelian $p$-extension of $H_\infty$ unramified outside the primes above $\mathfrak{p}$, and set $X(H_\infty)=\mathrm{Gal}(M(H_\infty)/H_\infty)$. In this paper, we establish the main conjecture of Iwasawa theory for the Iwasawa module $X(H_\infty)$. As a consequence, we have that if $X(H_\infty)=0$, the relevant $L$-values are $\mathfrak{p}$-adic units. In addition, the main conjecture for $X(H_\infty)$ has implications toward (a) the BSD Conjecture for a class of CM elliptic curves; (b) weak $\mathfrak{p}$-adic Leopoldt conjecture.

We prove that some fields generated by j j -invariants of CM elliptic curves (of infinite dimension over Q \mathbb {Q} ) satisfy the Property (B). The singular moduli are chosen so as to have supersingular reduction simultaneously above a fixed prime q q , which provides strong q q -adic estimates leading to an explicit lower bound for the height.

Let $E/\mathbb{Q}$ be a fixed elliptic curve. For each prime $p$ of good reduction, write $E(\mathbb{F}_p) \cong \mathbb{Z}/d_p \mathbb{Z} \oplus \mathbb{Z}/e_p \mathbb{Z}$, where $d_p \mid e_p$. Kowalski proposed investigating the average value of $d_p$ as $p$ runs over the rational primes. For CM curves, he showed that $x\log\log{x}/\log{x} \ll \sum_{p \le x} d_p \ll x\sqrt{\log{x}}$. It was shown recently by Felix and Murty that in fact $\sum_{p \le x} d_p$ exceeds any constant multiple of $x\log\log{x}/\log{x}$, once $x$ is sufficiently large. In the opposite direction, Kim has shown that the expression $x\sqrt{\log{x}}$ in the upper bound can be replaced by $x\log\log{x}$. In this paper, we obtain the correct order of magnitude for the sum: $\sum_{p \le x} d_p \asymp x$ for all large $x$.