λ\lambda-invariant stability in families of modular Galois representations

Consider a family of modular forms of weight 2, all of whose residual $\pmod{p}$ Galois representations are isomorphic. It is well-known that their corresponding Iwasawa $\lambda$-invariants may vary. In this paper, we study this variation from a quantitative perspective, providing lower bounds on the frequency with which these $\lambda$-invariants grow or remain stable.Comment: final version; to appear in Research in the Mathematical Science

Statistics for Iwasawa Invariants of elliptic curves, II\rm{II}

We study the average behaviour of the Iwasawa invariants for Selmer groups of elliptic curves. These results lie at the intersection of arithmetic statistics and Iwasawa theory. We obtain unconditional lower bounds for the density of rational elliptic curves with prescribed Iwasawa invariants.Comment: 23 pages, minor changes. Accepted for publication in the International Journal of number theor

Heuristics for anti-cyclotomic Zp\mathbb{Z}_p-extensions

This paper studies Iwasawa invariants in anti-cyclotomic towers. We do this by proposing two heuristics supported by computations. First we propose the Intersection Heuristics: these model `how often' the $p$-Hilbert class field of an imaginary quadratic field intersects the anti-cyclotomic tower and to what extent. Second we propose the Invariants Heuristics: these predict that the Iwasawa invariants $\lambda$ and $\mu$ usually vanish for imaginary quadratic fields where $p$ is non-split.Comment: v2: Incorporated changes as per the referee reports; accepted for publication (Experimental Math

Studying Hilbert's 10th problem via explicit elliptic curves

N.Garc\'ia-Fritz and H.Pasten showed that Hilbert's 10th problem is unsolvable in the ring of integers of number fields of the form $\mathbb{Q}(\sqrt[3]{p},\sqrt{-q})$ for positive proportions of primes $p$ and $q$. We improve their proportions and extend their results to the case of number fields of the form $\mathbb{Q}(\sqrt[3]{p},\sqrt{Dq})$, where $D$ belongs to an explicit family of positive square-free integers. We achieve this by using multiple elliptic curves, and replace their Iwasawa theory arguments by a more direct method.Comment: Comments very welcome

Derived pp-adic heights and the leading coefficient of the Bertolini--Darmon--Prasanna pp-adic LL-function

Let $E/\mathbf{Q}$ be an elliptic curve and let $p$ be an odd prime of good reduction for $E$. Let $K$ be an imaginary quadratic field satisfying the classical Heegner hypothesis and in which $p$ splits. In a previous work, Agboola--Castella formulated an analogue of the Birch--Swinnerton-Dyer conjecture for the $p$-adic $L$-function $L_{\mathfrak{p}}^{\rm BDP}$ of Bertolini--Darmon--Prasanna attached to $E/K$, assuming the prime $p$ to be ordinary for $E$. The goal of this paper is two-fold: (1) We formulate a $p$-adic BSD conjecture for $L_{\mathfrak{p}}^{\rm BDP}$ for all odd primes $p$ of good reduction. (2) For an algebraic analogue $F_{\overline{\mathfrak{p}}}^{\rm BDP}$ of $L_{\mathfrak{p}}^{\rm BDP}$, we show that the ``leading coefficient'' part of our conjecture holds, and that the ``order of vanishing'' part follows from the expected ``maximal non-degeneracy'' of an anticyclotomic $p$-adic height. In particular, when the Iwasawa--Greenberg Main Conjecture $(F_{\overline{\mathfrak{p}}}^{\rm BDP})=(L_{\mathfrak{p}}^{\rm BDP})$ is known, our results determine the leading coefficient of $L_{\mathfrak{p}}^{\rm BDP}$ at $T=0$ up to a $p$-adic unit. Moreover, by adapting the approach of Burungale--Castella--Kim in the $p$-ordinary case, we prove the main conjecture for supersingular primes $p$ under mild hypotheses.Comment: 34 page

STRUCTURE OF FINE SELMER GROUPS IN ABELIAN p-ADIC LIE EXTENSIONS

This paper studies fine Selmer groups of elliptic curves in abelian p-adic Lie extensions. A class of elliptic curves are provided where both the Selmer group and the fine Selmer group are trivial in the cyclotomic Zp-extension. The fine Selmer groups of elliptic curves with complex multiplication are shown to be pseudonull over the trivializing extension in some new cases. Finally, a relationship between the structure of the fine Selmer group for some CM elliptic curves and the Generalized Greenberg's Conjecture is clarified

CD4 Deficiency Causes Poliomyelitis and Axonal Blebbing in Murine Coronavirus-Induced Neuroinflammation.

Mouse hepatitis virus (MHV) is a murine betacoronavirus (m-CoV) that causes a wide range of diseases in mice and rats, including hepatitis, enteritis, respiratory diseases, and encephalomyelitis in the central nervous system (CNS). MHV infection in mice provides an efficient cause-effect experimental model to understand the mechanisms of direct virus-induced neural-cell damage leading to demyelination and axonal loss, which are pathological features of multiple sclerosis (MS), the most common disabling neurological disease in young adults. Infiltration of T lymphocytes, activation of microglia, and their interplay are the primary pathophysiological events leading to disruption of the myelin sheath in MS. However, there is emerging evidence supporting gray matter involvement and degeneration in MS. The investigation of T cell function in the pathogenesis of deep gray matter damage is necessary. Here, we employed RSA59 (an isogenic recombinant strain of MHV-A59)-induced experimental neuroinflammation model to compare the disease in CD4-/- mice with that in CD4+/+ mice at days 5, 10, 15, and 30 postinfection (p.i.). Viral titer estimation, nucleocapsid gene amplification, and viral antinucleocapsid staining confirmed enhanced replication of the virions in the absence of functional CD4+ T cells in the brain. Histopathological analyses showed elevated susceptibility of CD4-/- mice to axonal degeneration in the CNS, with augmented progression of acute poliomyelitis and dorsal root ganglionic inflammation rarely observed in CD4+/+ mice. Depletion of CD4+ T cells showed unique pathological bulbar vacuolation in the brain parenchyma of infected mice with persistent CD11b+ microglia/macrophages in the inflamed regions on day 30 p.i. In summary, the current study suggests that CD4+ T cells are critical for controlling acute-stage poliomyelitis (gray matter inflammation), chronic axonal degeneration, and inflammatory demyelination due to loss of protective antiviral host immunity. IMPORTANCE The current trend in CNS disease biology is to attempt to understand the neural-cell-immune interaction to investigate the underlying mechanism of neuroinflammation, rather than focusing on peripheral immune activation. Most studies in MS are targeted toward understanding the involvement of CNS white matter. However, the importance of gray matter damage has become critical in understanding the long-term progressive neurological disorder. Our study highlights the importance of CD4+ T cells in safeguarding neurons against axonal blebbing and poliomyelitis from murine betacoronavirus-induced neuroinflammation. Current knowledge of the mechanisms that lead to gray matter damage in MS is limited, because the most widely used animal model, experimental autoimmune encephalomyelitis (EAE), does not present this aspect of the disease. Our results, therefore, add to the existing limited knowledge in the field. We also show that the microglia, though important for the initiation of neuroinflammation, cannot establish a protective host immune response without the help of CD4+ T cells

Iwasawa Theory of Fine Selmer Groups

Iwasawa theory began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur considered generalizations of Iwasawa theory to Selmer groups of elliptic curves (Abelian varieties in general). At the turn of this century, Coates and Sujatha initiated the study of a subgroup of the Selmer group of an elliptic curve called the \textit{fine} Selmer group. The focus of this thesis is to understand arithmetic properties of this subgroup. In particular, we understand the structure of fine Selmer groups and their growth patterns. We investigate a strong analogy between the growth of the $p$-rank of the fine Selmer group and the growth of the $p$-rank of the class groups. This is done in the classical Iwasawa theoretic setting of (multiple) \ZZ_p-extensions; but what is more striking is that this analogy can be extended to non-$p$-adic analytic extensions as well, where standard Iwasawa theoretic tools fail. Coates and Sujatha proposed two conjectures on the structure of the fine Selmer groups. Conjecture A is viewed as a generalization of the classical Iwasawa $\mu=0$ conjecture to the context of the motive associated to an elliptic curve; whereas Conjecture B is in the spirit of generalising Greenberg's pseudonullity conjecture to elliptic curves. We provide new evidence towards these two conjectures.Ph.D