Add Serre invariants to E/Q pages

lmfdb/elliptic_curves/elliptic_curve.py

@@ -231,6 +231,10 @@ def ctx_elliptic_curve_summary():
 def ctx_gl2_subgroup():
     return {'gl2_subgroup_data': gl2_subgroup_data}
 
+@app.context_processor
+def ctx_ec_reduction_type():
+    return {'ec_reduction_type': lambda x: ["nonsplit multiplicative","additive","split multiplicative","good"][x+1] if x >= -1 and x <= 2 else "unknown" }
+
 @ec_page.route("/stats")
 def statistics():
     title = r'Elliptic curves over $\Q$: Statistics'

lmfdb/elliptic_curves/templates/ec-curve.html

@@ -388,7 +388,7 @@ <h4> {{ KNOWL('ec.q.modular_degree', title='Modular degree') }}
     <h2> Local data </h2>
 
 <style type="text/css">
-#local_data th, #local_data td {
+#local_data th, #local_data td, #serre_data th, #serre_data td {
 padding : 4px;
 text-align: center;
 }
@@ -398,39 +398,31 @@ <h2> Local data </h2>
   This elliptic curve is {{ '' if data.semistable else 'not' }} {{KNOWL('ec.semistable', title='semistable')}}.
   There
   {% if data.num_bad_primes==1 %}
-  is only one prime
+  is only one prime $p$
   {% else %}
-  are {{data.num_bad_primes }} primes
+  are {{data.num_bad_primes }} primes $p$
   {% endif %}
   of {{KNOWL('ec.q.reduction_type', title='bad reduction')}}:
 </p>
 
 <table id = "local_data" class="ntdata"><thead>
 <tr>
-<th>prime</th>
+<th>$p$</th>
 <th>{{KNOWL('ec.q.tamagawa_numbers', title='Tamagawa number')}}</th>
 <th>{{KNOWL('ec.q.kodaira_symbol', title='Kodaira symbol')}}</th>
 <th>{{KNOWL('ec.q.reduction_type', title='Reduction type')}}</th>
 <th>{{KNOWL('ec.local_root_number', title='Root number')}}</th>
-<th>{{KNOWL('ec.conductor_valuation', title='ord($N$)')}}</th>
-<th>{{KNOWL('ec.discriminant_valuation', title='ord($\Delta$)')}}</th>
-<th>{{KNOWL('ec.j_invariant_denominator_valuation', title='ord$(j)_{-}$')}}</th>
+<th>{{KNOWL('ec.conductor_valuation', title='$v_p(N)$')}}</th>
+<th>{{KNOWL('ec.discriminant_valuation', title='$v_p(\Delta)$')}}</th>
+<th>{{KNOWL('ec.j_invariant_denominator_valuation', title='$v_p(\mathrm{den}(j))$')}}</th>
 </tr>
 </thead><tbody>
 {% for pr in data.local_data %}
 <tr>
-<td> ${{pr.prime}}$</td>
+<td>${{pr.prime}}$</td>
 <td>${{pr.tamagawa_number}}$</td>
 <td>${{pr.kod}}$</td>
-<td>
-{% if pr.reduction_type==0 %}
- Additive
- {% elif pr.reduction_type==1 %}
-  Split multiplicative
-  {% elif pr.reduction_type==-1 %}
-   Non-split multiplicative
-{% endif %}
-</td>
+<td>{{ec_reduction_type(pr.reduction_type)}}</td>
 <td>{{pr.root_number}}</td>
 <td>{{pr.conductor_valuation}}</td>
 <td>{{pr.discriminant_valuation}}</td>
@@ -488,6 +480,26 @@ <h2> {{KNOWL('ec.galois_rep', title='Galois representations')}} </h2>
 <p>The torsion field $K:=\Q(E[{{data.adelic_level}}])$ is a degree-${{data.data.adelic_image_size}}$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/{{data.adelic_level}}\Z)$.
 {% endif %}
 
+<p> The table below list all primes $\ell$ for which the {{KNOWL('ec.q.serre_invariants', title='Serre invariants')}} associated to the mod-$\ell$ Galois representation are exceptional. </p> 
+
+<table id = "serre_data" class="ntdata"><thead>
+<tr>
+<th>$\ell$</th>
+<th>{{KNOWL('ec.reduction_type', title='Reduction type')}}</th>
+<th>{{KNOWL('ec.q.serre_invariants', title='Serre weight')}}</th>
+<th>{{KNOWL('ec.q.serre_invariants', title='Serre conductor')}}</th>
+</tr>
+</thead><tbody>
+{% for l,r,k,M in data.serre_data %}
+<tr>
+<td>${{l}}$</td>
+<td>{{ec_reduction_type(r)}}</td>
+<td>${{k}}$</td>
+<td>{{M}}</td>
+</tr>
+{% endfor %}
+</tbody>
+</table>
 
 <h2>{{ KNOWL('ec.isogeny', title='Isogenies') }} </h2>
 {% if data.data.isogeny_degrees != "unknown" %}
@@ -577,8 +589,9 @@ <h2>{{KNOWL('ec.q.torsion_growth', title='Growth of torsion')}} in number fields
 <td align=center>{{tgd.f | safe}}</td>
 <td align=center>{{tgd.t}}</td>
 <td align=center>
-{% if tgd.bc_label == 'Not in database' %}Not in database
-{% else %}<a href={{tgd.bc_url}}>{{tgd.bc_label}}</a>
+{% if tgd.bc_url %}<a href={{tgd.bc_url}}>{{tgd.bc_label}}</a>
+{% else %}
+{{tgd.bc_label}}
 {% endif %}
 </td>
 </tr>

lmfdb/elliptic_curves/web_ec.py

@@ -337,6 +337,12 @@ def make_curve(self):
         data['disc_latex'] = web_latex(D)
         data['cond_latex'] = web_latex(N)
 
+        def red(p):
+            ld = [ld['reduction_type'] for ld in local_data if ld['prime'] == p]
+            return ld[0] if len(ld) else 2
+
+        self.serre_data = [(l,red(l),k,web_latex_factored_integer(M,equals=True)) for l,k,M in self.serre_invariants]
+
         # retrieve data about MW rank, generators, heights and
         # torsion, leading term of L-function & other BSD data from
         # table ec_mwbsd:
@@ -758,7 +764,7 @@ def make_torsion_growth(self):
 
         for tgd in tgdata:
             tg1 = {}
-            tg1['bc_label'] = "Not in database"
+            tg1['bc_label'] = "not in database"
             tg1['d'] = tgd['degree']
             F = tgd['field']
             tg1['f'] = formatfield(F)