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Comp."],"abstract":"<p>\n                    Let\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper E\">\n                        <mml:semantics>\n                          <mml:mi>E<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">E<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    be an elliptic curve over\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper Q\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mi mathvariant=\"double-struck\">Q<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {Q}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with Mordell\u2013Weil rank\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2\">\n                        <mml:semantics>\n                          <mml:mn>2<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    be an odd prime of good ordinary reduction. For every imaginary quadratic field\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper K\">\n                        <mml:semantics>\n                          <mml:mi>K<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">K<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    satisfying the Heegner hypothesis, there is (subject to the Shafarevich\u2013Tate conjecture) a line, i.e., a free\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper Z Subscript p\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mi>p<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {Z}_p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -submodule of rank\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"1\">\n                        <mml:semantics>\n                          <mml:mn>1<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper E left-parenthesis upper K right-parenthesis circled-times double-struck upper Z Subscript p\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>E<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>K<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>\n                              \u2297\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mi>p<\/mml:mi>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">E(K)\\otimes \\mathbb {Z}_p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    given by universal norms coming from the Mordell\u2013Weil groups of subfields of the anticyclotomic\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper Z Subscript p\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mi>p<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {Z}_p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -extension of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper K\">\n                        <mml:semantics>\n                          <mml:mi>K<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">K<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ; we call it the\n                    <italic>shadow line<\/italic>\n                    . When the twist of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper E\">\n                        <mml:semantics>\n                          <mml:mi>E<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">E<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    by\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper K\">\n                        <mml:semantics>\n                          <mml:mi>K<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">K<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    has analytic rank\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"1\">\n                        <mml:semantics>\n                          <mml:mn>1<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , the shadow line is conjectured to lie in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper E left-parenthesis double-struck upper Q right-parenthesis circled-times double-struck upper Z Subscript p\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>E<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Q<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>\n                              \u2297\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mi>p<\/mml:mi>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">E(\\mathbb {Q})\\otimes \\mathbb {Z}_p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ; we verify this computationally in all our examples. We study the distribution of shadow lines in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper E left-parenthesis double-struck upper Q right-parenthesis circled-times double-struck upper Z Subscript p\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>E<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Q<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>\n                              \u2297\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mi>p<\/mml:mi>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">E(\\mathbb {Q})\\otimes \\mathbb {Z}_p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    as\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper K\">\n                        <mml:semantics>\n                          <mml:mi>K<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">K<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    varies, framing conjectures based on the computations we have made.\n                  <\/p>","DOI":"10.1090\/mcom\/4110","type":"journal-article","created":{"date-parts":[[2025,7,31]],"date-time":"2025-07-31T13:59:11Z","timestamp":1753970351000},"page":"2539-2557","source":"Crossref","is-referenced-by-count":1,"title":["Shadow line distributions"],"prefix":"10.1090","volume":"95","author":[{"given":"Jennifer","family":"Balakrishnan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mirela","family":"\u00c7iperiani","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Barry","family":"Mazur","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Karl","family":"Rubin","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2025,7,31]]},"reference":[{"key":"1","isbn-type":"print","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1007\/978-3-319-30976-7_2","article-title":"Shadow lines in the arithmetic of elliptic curves","author":"Balakrishnan, J. 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